Hello everyone and welcome to the Information Command Center. I am Lisa Joseph and we're here for a discussion on COVID-19 protocols and the election campaign for 2021. The date for the pending general elections that was announced on the 5th of July by Prime Minister Hon. Alan Shastney. That date given was the 26th of July 2021 with nomination day set for the 16th of July. General elections in St. Lucia is always undefended by lots of mass crowd activity. Certainly individuals coming out to show their support for their various parties of choice and it also means lots of interaction taking place whether it be at a constituency level, it is on a house-to-house basis or persons just gathering with on street corners having conversations relating to the general elections. However for us here in St. Lucia we hold in a general election at a very very precarious time for the country in the backdrop of COVID-19. The pandemic is still on and the numbers in St. Lucia are still staggering although we realized a sort of lull in the numbers or a reduction in our transmission rates. In order for there to have been the general elections in a safe environment, number of protocols were drawn up and that was agreed to by various stakeholders including all political parties. We'll be having that discussion with some of those who are the sort of gatekeepers where those COVID-19 protocols and election campaign is concerned. With me in studio is the commissioner of police Mr. Milton Daisy, the deputy chief environmental health officer Mrs. Cheryl Eugene-Centrumaine and the chief fire officer Mr. Joseph Joseph. On July 12th just two days ago the chief medical officer Dr. Sharon Belmont George indicated that over the last 14 days 114 new cases of COVID-19 had been reported bringing a total of active cases in country to 127. The daily infection rate for the past week has been at 4.3 per 100,000 per day with an average of seven cases per day and certainly the day after on the 13th of July the country had already recorded 14 new cases of COVID-19. This has drawn concern and it is concerning especially given the images that we're seeing after national motorcades by both of major political parties and so we are here to discuss how we can maintain our safe environment while exercising our franchise and our democratic right. Let me turn my attention to Mr. Milton Daisy our commissioner of police. Let me just ask so in terms of being able to ensure the protocols what has been the difficulty for the Royal Central Police Force? Okay first of all the difficulty is in relation to actively enforcing the protocols. We had protocols, campaign protocols set out the all the major parties. Actually three of them will present and agreed to the protocols. However during the activities we realized that there was little compliance to the protocols especially the motorcades where persons were to remain in their vehicles and so on. Further down I will go over the protocols for especially the protocols and but one of which I shall mention now is that persons are not supposed to get off their vehicles during that motorcade. It was to proceed. It's a motorcade. It's not a political meeting. It's a motorcade and they were supposed to proceed on the route of the motorcade. However we saw vehicles diverting other than the route that was specified in the license granted by the commission of police because there's a specified route and so on on that day. So this was one of the issues that we saw. We also saw that persons order protocols like wearing of mask. Yes some persons had it. Some had it on the chin. Some were not wearing it at all. So these things were in the protocols for the campaigns and persons agreed to do in it. The reason for that is that the campaign we realized the campaign would not take the usual form as order elections because during the order elections we did not have COVID to deal with. So we had to take the health aspect into consideration so that we could have granted permission to have some campaign. However it was reduced. It was in a controlled manner. I know the CMO had very stern warnings to the parties so that they would be informing their supporters of the protocols and reminding them what they need to do and so on and everybody agreed to do in it. But I suppose as with anything else as with the police enforcing when it comes to being able to manage mass crowds it becomes quite a difficulty. It is difficult especially with the motorcade where you have say probably vehicles expanding to over a mile where you have police officers at different locations but however persons take the opportunity when there's no officer around to do things. And not just for health issues there were security issues there also which some of them the result being a collision between vehicles and I'll not say an accident it was collision caused by persons behavior the manner of driving and we saw that so the need now is for persons to observe the rules the protocols and so on and having seen the over the weekend what had happened we took the decision to bring in the main players in the various parties the persons who organized the campaigns the persons who organized the meetings and they agreed with us that the national motorcade should be banned. And so if there was consensus among the police including the traffic department the persons who represented the party and so on so it was agreed without any issues that we would ban the national motorcades however what we agreed is that we would continue with the community motorcades constituency motorcades so so that persons would remain for example if it is in castry central which is an example it would remain within the jurisdiction of castry central so that you would have less vehicles you'd have less participation and so on however your their message would be sent out so so we'll talk about how that differs through the national motorcade although you've alluded to it but we'll get a little deeper into that before I bring in the deputy chief environmental health officer let me say hello to the chief fire officer a Joseph Joseph where does your I don't think many people would think about the fire service the fire ambulance service at the time of election but you do play a critical role outlined that for us okay well thank you good morning to all our mayors of certainly the fire service plays a critical role however and ever we have you know emergencies you know once once their emergencies the fire service certainly needs to respond and we too are concerned about some of the actions we see some of the unsafe practices we see especially with the the campaigning and what is happening at those motorcade and like Mr. Daisy was saying the some of the some of those things we see happening um like you know the the persons not wearing masks and so on um the number of persons in in in vehicles and some of them in trucks in open trucks large numbers of persons in open trucks and so on um the the possibility of having an accident you know when they have so many vehicles is always greater and we have to look at the island's capacity to handle you know mass casualty incident because if you have an accident involving one of those trucks here and of course for for us um any incident involving more than 10 persons is considered mass casualty incident and in some cases you see in excess of 20 persons on one vehicle so just imagine you have an accident involving you know um one of these vehicles it will be you know overwhelming for all all all our services you know the hospitals I mean we do not have sufficient ambulances to transport you know that many persons you know in in just one incident so I mean there are certainly challenges that are brought to the fore as it as it relates to these kind of practices now in in terms of if you were to respond to an eventuality at one of these mass crowd election events how does the presence of COVID change the dynamics of how you would normally respond all right so okay right now most of our EMS responses I can't be considered hazmat responses because um you know whereas um pre-COVID we would have made responses with just our ordinary uniforms and so on right now we have to you know we're doing in in PPE to make those responses and just imagine you have a situation where you know you have a large number of persons involved in accidents we have you know to provide PPE for all of the persons responding we have to get ambulances to service this emergency request and we have a limited number of them and just imagine it's in one area on the island you may have to you know use the the services of ambulances at other stations so you would find that wherever you know they make a response that area which the ambulance covers is well remains without ambulance coverage for the period so you see that the challenge so like I said I mean we need to take a note of these these situations and besides you know we have other emergencies to attend to because while that is happening we can have a file you know we can have maybe an aircraft crashing on on because I mean we provide not just a domestic file and on EMS responses that we we are responsible for several other you know emergency and non-immigrancy functions so any anything that puts unnecessary stress on our department you know is a one that we would discourage. Let's bring in the deputy chief environmental health officer Mrs Cheryl Eugene Saint-Romain the COVID-19 protocols for the campaign of an election campaign came into effect June 29 2021 and as I said earlier there was an agreement of consensus across the board the all parties and on what that would be and want to be able to go through some of the critical elements of that Miss Saint-Romain but let me get your sentiment though I know that the talking point definitely is all those images that we saw coming out of the motorcades on Sunday last for you just someone critically involved intimately involved in Saint Lucia's COVID-19 response and for public health what were your immediate thoughts when you saw those images? Oh my that was it oh my it's really disheartening considering that like Mr Daisy the commissioner said there was consensus and the protocols we have protocols for elections and we have protocols specific to the campaign pre during and post there and the person's responsible were in session where those protocols were presented to them they knew every component of it and they had the opportunity from that then on to go you know educate the base the individuals that would be involved and even the organizers of take for instance the the motorcade okay because there are specific guidelines as to what should and should not happen and to see that revelation on Sunday was really disheartening. That's and of course because of the implications and I think that sometimes you know sort of carnival like atmosphere at the motorcades during the motorcade at several points but I think too that perhaps because the country had experienced a sort of quiet with the numbers in our war in COVID-19 infection that somehow it was out of sight and out of mind for many of the individuals there and like you rightly said it was very very disheartening and very worrying to have seen the number of people with drinking hand and that's something else we'll be talking about with drinking hand and certainly throwing caution to the wind when it came to how the mask they had the mask but the mask will not be inutilized in the proper way and not being worn at all. So Ms Saint Romain I would like to take us through some of the pertinent sections of the protocols. Let's start if you will with the campaign period and give us an idea because the protocols are very specific all the way down to office meetings being held so just to give the listening public an opportunity to understand how much consideration went into and detailing went into the protocols if you can just go over that with us some of the key points that were outlined in the constituency office meetings if you will if you have that before you. Okay whilst the period was short and like you rightly said the green light was given on the 29th and immediately following that with just a few days before the first mass event was implemented rather but what should have happened because we have a category that says pre-election campaign and is specific to the political party and administration so what should have happened every political party should have presented to the environmental health division a copy of their response plan and what the protocols do is it serves as a guideline yeah and what you need to do is to look at your own individual scenario and then you prepare a response plan to that. Stay with us when we come back we'll continue the conversation on the COVID-19 protocols and election campaign for 2021. We trust take two doses take our life by vaccine it for you for yours for us let's take our life back with the safe and effective AstraZeneca COVID-19 vaccine the mobile vaccination drive is coming to you visit the Ministry of Health and Wellness and Bureau of Health Education Facebook pages for more information let's take our life back vaccinate for you for yours for us can't vaccinate and this message is brought to you by the Bureau of Health Education of the Ministry of Health and Wellness and this station and welcome back we're continuing our discussion with the Deputy Chief Environmental Health Officer Cheryl Eugene Saint-Romain as well as the Commissioner of Police Mr. Milton Desi and the Chief Fire Officer Joseph Joseph as we do a rundown of the COVID-19 protocols for the election campaign 2021. Before we went to break I was going to pose the question to Ms. Saint-Romain regarding the protocols and how detailed the protocol was given and and being finalized upon consensus by all parties involved to give us a sense of what is contained and how what the details of the protocol documented. Thank you Lisa again the the document the protocols establishes several components of the election period okay so you have the pre-election phase which will be the campaign and it also went goes further to categorize as well the different components so there's a door-to-door that's called the canvassing there's a set of protocols for that there's protocols for meetings rallies motorcades and constituency meetings so all these areas are articulated there and with guidelines presented with them. Before you go any further can you give us the sort of indication as to what must happen especially for the door-to-door because in as much as we see in the numbers out there at the motorcades and the little meetings constituency meetings a vast number of solutions are remaining home and we do see the canvassing taking place within the various communities so what needs to happen at that levels for the individuals who are at home and for the canvassers themselves. So just let me run through some of the guidelines there it says that first people conducting door-to-door campaigns must practice risk mitigation measures to limit person-to-person interaction and mitigate risk from transferring leaflets so what we are saying here is instead of the interaction where you with pre-covid you would actually go to the person's door step have a discussion with them and then hand them the whatever leaflet or information that you want to leave with them we are recommending that that information be left at appropriate place maybe at the door step maybe somewhere that they can collect after and we're also recommending and that is all the time that person's wear mask or appropriate first place covering or other protective equipment we are recommending that they adopt ways of delivering material well I said that that reduce unnecessary touching and that is leaving materials outside the door rather than handing them directly to the residents and even for those receiving so you canvasser places material at my doorstep my balcony under a stone whatever it is I am coming outside of my house should I now wait until the individuals have totally cleared my yard my space and then for me to go and see what that material is you can do that and also you need to practice proper hand hygiene so after you have handled that material and you have gone through it and satisfy yourself with whatever information is there you would then proceed to put it in an appropriate place because you know the virus remains on on surfaces for a period of time but also you need to practice proper hand washing before and after handling that material and that's the important thing always wash the hands once that is done okay and what else during the door to door campaigning we're asking as much as possible so you practice because instead of well I think prior to COVID persons would actually go into their homes of persons well that is when you are invited but we are recommending that you do it in an open air environment so if the person is able to come out you have that interaction outside with them okay instead of going in where the space is limited and and you you know you um and I think that would be very important as well because we do have uh the number of shut-ins uh in and around St. Lucia elderly persons the senior citizens who for one reason or the other may not be able to walk and go to the door to to to perhaps stand in the doorway and to speak with that canvasser from a distance so it would be impressed upon the canvassers now to understand that if you do have individuals who are in that critical um risk brackets that certain measures must be followed must be followed so in most of these shut-ins they have somebody a caregiver so you interact with that caregiver on behalf of the individual or if there is no way that that if there's no caregiver there you try your best to limit you're exposing the person and yourself to any of the conditions and now we come to the very tricky part which is the outdoor open air meetings or large public meetings what obtains under those so under that we have um at all times throughout you need to ensure that all applicable sections of the COVID-19 prevention control act is implemented at all times so um wearing of masks we as that persons wear masks of appropriate coverings first covering sorry by followers and all candidates on stage um we ask that you ensure and maintain physical distances between persons on stage so you ensure that you you you um adhere to the six foot distancing also before and after persons between um making their presentations we ask that um all mics like turns are sanitized after each speaker um you provide adequate measures to facilitate hand hygiene within candidates on stage and within the crowd as much as you're as possible because you know in the crowd that can pose a challenge so as much as possible and you have that constant um reminders of people to maintain um the the social distancing to wear the mask and as much as possible if they have to be interacting or maybe going to the facility toilet facility that they you know practice the hand washing and so on so that that reminder must be constant throughout the meeting but at least in terms of the numbers then because everyone will be well it's outdoor so any number can apply what applies in terms of the numbers of people allowed at these uh open outdoor open air meetings it's it's almost unrealistic to give a number even so that is why we are advocating for open air so um you have the open air meeting but the constant advisories the constant reminders that is what is important and what we had asked as well is to designate an individual so each party designate an individual almost like a covid officer or somebody yes so they can one or two depending on the crowd because they would know to go through the crowd so you can you know as reminders to ensure that persons are in adherence to to to the protocols you may need an absolute heavy team there and i have turned to a police commissioner uh daisy into now being able to enforce and police these protocols at the open air meetings what takes place in that regard for the police force do you do you need a list in advance for you to be able to give permission for for the open air meetings okay um for every public meeting in fact the law provides that the the applicant provide a list of the speakers to um when um applying for for the permission and that was not because of covid it has always been the law so that they um the police would be aware of how many persons that you have speaking on the day in question the duration of your meeting and so on and that way the police won't get any surprise that um you would get somebody who who may not be permitted to speak because of what generally the person normally communicates and so on so but at most times or i should say at all times that the speakers are allowed and permission is granted actually the office of the commissioner what we have decided especially now with the few few days left for the campaign that we will not deny permission to hold any meeting however guidelines you need to follow the guidelines and so on if how do how would you how would you be able to enforce those guidelines and ensure that they've been adhered to um because i say that given because of the nature of covid-19 what it has how it has impacted every aspect of our lives including the journal election process it will mean now that you may need to have several uh community meetings taking place all at once in order for the parties to believe that the message is getting out so how does that place the um additional stress and the resources and is the intent to be able to police all of these community meetings actually the community meetings we would be policing them and that is why uh we we believe that the rallies the national rallies were not permitted so we do not have a rally where persons from all the constituencies are gathering so you have the cross cross constituency and then um issues with if there is an outbreak so um so the more meetings you have a day once we can provide the security the better for um ministry of health i would say because um if you have um 17 meetings you have less less persons congregated however if the less um the the less meetings you have the more persons you would find attending and this is what we will be um discouraging persons from so no um national rally would be permitted in fact um there would be no application for it because that was one of the things that was agreed when we brought the um the representatives the representatives from the parties um together mr. Joseph how does that that little bit of good news how does it now how's it been received by by the fire service the no national rallies to be worried about but you do have the various constituency meetings or even bring it down to the community levels there are several of them so traditionally what would occur at these meetings for the fire ambulance service do you attend all of them and have a unit stationed and how would you do it differently this time um well for our department it's not practical to um have persons posted at all of those but if we do get an application from one of the political parties to attend an ambulance to maybe a large meeting um then we do so but i think um right now on this phone we are living through a pandemic and i think we should you know let it let it be known that we are living through a pandemic and our our actions should um suggest that we are so i think um we could make use of of the the technology that has to be bought so i mean use um live streaming and all that um you can get all of what is being set out of the meeting without you leaving your home so i think it's it's really a case of persons taking personal responsibility for their own safety and i know not understanding the number of protocols we have maybe we'll get bitches in in in some cases and i think um if we now take personal responsibility for us if we we can perhaps um we draw from some of of the some of those things that we see when persons are not wearing masks to keep you know your your safe distance away from them and all of this so and like i said i mean the fire department will do its best in terms of assisting in in managing the situation however we have our limitation and to personal responsibility must really come into play personal responsibility uh commissioner dc we began speaking about the motorcades and we've indicated that the motorcades will no longer be held that has been agreed across the board by all parties but there was a sense that um perhaps we would believe that there were no protocols and if protocols had been established perhaps it was a little too loose wasn't communicated can you just give us an idea give us a full understanding of what the protocols for the motorcade the motorcades were okay yes so for motorcades the organizers should reduce the need for in-person coordination by providing written communication or telephone communication send out a blast to um supporters of the the protocols even before because um generally it's expected that persons would be briefing the participants before but if you do it even prior to assembling so it would reduce the need for persons to be going to every individual and say okay you must observe protocols then there's the number of persons in vehicles must comply with covid-19 protocols so for example if your bus is licensed to carry um 10 passengers during the um covid the covid period yeah you must comply with that and one of the things that we saw was the overcrowding of those buses so your bus you must comply with the number of persons in the vehicle also um individuals advise not to gather outside of their vehicles before or after the motorcade so um normally that is one of the things that we see persons to gather they speak to friends and so on so these are the things that causes the contamination these are the things that causes um the virus to spread so um if you remain in your vehicle there's a reduced chance of you um getting the virus transmitted participants and that one is very important participants should remain in their vehicles during the entire event and not congregate outside of the vehicle and persons agree to that everybody had an opportunity to um give feedback and so on or what they think should be everybody agreed to that if it's a motorcade that it is a motorcade it's not a it's not a um political meeting it is not a party it is a motorcade and these um things were agreed upon so we are going around the island that persons would be driving their vehicle and then vehicles going in convoy until the end each vehicle participating in the motorcade should be comprised of individuals from the same extended bubble family and um so each vehicle in that motorcade should as far as possible that it should be um should have members of the same extended bubble or family again to reduce the risk right um it's also stated that and you must have sanitizers in the vehicle vehicles should not drive parallel to each other at any time and should proceed in single file through the motorcade route we also saw that vehicles will drive in parallel to each other that in itself even without the covid protocols that is an offense you cannot drive parallel to another vehicle whilst with except if you were um overtaking and there's clay okay um participants should disperse immediately following the motorcade okay the motorcade has and then go about your um your business so we cannot after a motorcade it was not um it's not permitted to have to hold any other session after the motorcade the rules of the route must be followed at all times and that is the rules of driving the traffic rules you must observe it and the relevant regulations with respect to events per reads must be followed mask or appropriate face covering must be worn at all times so all these will um protocols for motorcades and um what we saw over the weekend was um persons having community motorcades and join then joining the national motorcade which um of course we were not um prepared we were the police we were not prepared to handle it because we know that permission is given for motorcade to go around the island so um our resources were deployed in that regard so um there were surprises but i said to the police where you find um you at a particular location and vehicles and a motorcade is coming in the opposite direction and i must remind persons that the law makes provision for application three days before the event and this is to assist us with our planning so that we would um if if there is already an event in one area that we could um advise us to another location or another day because you have applications before you for all participating parties in the motorcades from sunday we had um an application for a general motorcade to um the round the island motorcade however persons left constituencies together and to join with um hope of joining the the mean motorcade and that pose right that was yes a challenge so how does the constituency at the constituency level because we've done away with the national motorcade and we've decided that we can at the constituency level motorcades are being sanctioned it's okay for that so how does that really differ miss and remain you want to give us an idea as to why the constituency level from a public health standpoint we'll get back to commissioner in a moment from a public health standpoint why is that a better medium um let me first say thank you lisa but let me first say that every one of those guidelines that the commissioner articulated still should be adopted for the constituency meetings yeah every single one of them um with respect to the constituency meetings it is more confined to just the community okay providing they remain well not providing but they should remain within the boundaries of the constituency so in that case you do not have much traversing where persons coming from you thought would would would end up in in um castry central or baboon or or somewhere like that so whatever the constituency and motorcade would be just in that particular area so it's just hoping there's just a residence of that community that will be um partaking in in in the the constituency motorcade but i i need to say again that every single one of those guidelines that has been articulated must be in effect for the constituency motorcade that's what they still ask the couple uh commissioner and we were saying earlier that at the constituency level easier for enforcement yes and how many uh motorcades constituency motorcades would the respective parties be allowed is there a limit yes um there um there there is a limit for the um constituency motorcade um it was one one constituency motorcade yes so um person would apply and have one constituency motorcade however the main one week yes however what we would be um looking at is that persons do not make a community motorcade into a national motorcade so we are not expecting if you have a uh motorcade in a constituency for example i um i had to just a name in a same gyms you have a constituency motorcade in same gyms we are hoping that it's only members of the same gyms community or constituency to attend and not members of bridge down constituency so that it limits the number of vehicles it limits the number of persons there and it makes policing easier for us okay and i know you have those large constituencies you have grozalli, babanoi could throw it in there um as well as she was there with all of those in roads and out roads and side roads back roads so within the constituency itself doing a motorcade can as you say can seem like a national motorcade but and is there a time frame within which the the motorcade is supposed to be um out and and and the training throughout the constituency yes actually the um upon application it would give us the time that you intend to have your motorcade however the motorcade must stop at least one hour before curfew time so that you would give persons an opportunity to get back to their homes before curfew so the curfew is still in effect um we did not extend the curfew contrary to what persons were circulating that there was no curfew the curfew remains in effect at 11 p.m so at 11 we should not have any we should not have any activities in relation to whether it is politics or otherwise in the social event we're expecting at 11 persons would be out from the streets and into their homes that's what a curfew is now what's a motorcade without music how does that factor into the no uh permission for loud music how does that factor into into play yes actually the um we took everything into consideration and the the permission for loud speakers for the motorcade so um we are still under the um under the the law that permission would not be given cut blush for um loud music however for the motorcade it's included in the in the permit that that we issue okay so we uh may central main you want to jump in i just wanted to say something um with respect to case you ask the question how different is it um to the the national motorcade for um contact tracing purposes if there's an issue in i don't use i'm being real so i'm just using it arbitrarily babono yeah if there's an issue with babono you know that it is just within the confines of babono although contract tracing is never easy that you would be you know doing your your contact tracing but if it's a case where it's open to us so you'd have to be all over the place and that can really prove a challenge so that is another um positive well so from that public health standpoint from a public health standpoint it can be you know that that can be a bit easier for us right and but and you're hoping that people would know people in the in the community and making it so so much easier we did hear about the registering of names um is that practicable at all uh and is that a responsibility that we no longer placing on the various political parties the registering of names will for the office meetings for office meetings office meetings yes so because throughout the campaign there would be a number of office meetings for strategizing and whatever the case may be planning and so on so that the um the registering of names was for that because we know it is unrealistic to yes to have persons registered you don't know at what point somebody will join um with respect to the motor kids so you cannot realistically expect to have a list of those persons but for the in-house meetings we are recommending that um a list of the names of those persons and the vital information like the address and so on be um taken into consideration now there is within the protocols you also have provisioned there for meals and the handling of meals and so forth give us an indication of what that specifies because you know as if anything else at the tradition you will have the roadside vendor um you may have some um within the executive of the parties perhaps having catered meals uh for um specific members and so forth so what obtains with the meals well let me go to very put my public health um part now so we are expecting that all persons who um is going to be preparing or serving food must be in possession of valid health certificate and the public health license with respect to um the serving of meals during those events we are asking or recommending that those meals be self-service not self-service but takeaways okay so at no time where you're going to have a good face style of a person's you know crowd themselves and so individuals maybe the meals will be laid somewhere and individuals in a strategized manner collect their meals and leave all distributed to to the respective persons so that is a recommendation that if meals are going to be served that they're going to be taken with package individually. Now I know vaccination plays a very key role well it is what has been presented now as the way out of COVID-19 being able to vaccinate your way out of COVID-19. For St. Lucia um we have seen that as of the 12th of July our vaccination numbers um 31,340 people receiving just the first dose of the AstraZeneca COVID-19 vaccine 22,653 have received their second dose so that is telling within itself while the numbers um yes we're happy that but it is still lower than what the expected um numbers should be for us to get anywhere near herd immunity so I'll ask you again from your public health standpoint Ms. Centromaine to talk to us about the importance of vaccination and impress upon St. Lucia why the need for us to go out and get vaccinated at this point in time the COVID vaccine is the only thing that can protect you from not necessarily protect you because um if you do get it there chances that um if you were vaccinated there chances that you may get but if you do get um COVID was being vaccinated um you are less likely to become sick or you are less likely to die from it okay um the the symptoms signs the symptoms will be much milder okay um so I'm just imploring on everybody especially during a time like that that persons utilizing services that is available and get vaccinated do it for you do it for us do it for your family member do it for your friend do it for your loved one this is the one we in order for us to be able to enjoy what we used to enjoy before to be able to do things without thinking because that is clearly what what's happening um we need to ensure that we get vaccinated okay so make that decision and the services are available all over um the where the locations the sites and locations where the team are they are always publicized so please um persons who are not take that opportunity to get vaccinated and we're 11 days away from the uh poll in day so I will give that opportunity to the chief um the commissioner of police with your closing remarks uh miss melton daisy your appeal to the solution public to adhere to the protocols that we have outlined yes uh even before I appeal to them to adhere there are certain offenses I think the public needs to know especially on election day yes so that our persons may be committed and one of them the first one on the um in the elections act is intoxicating liquor not to be sold supplied or given on polling day it shall not sell shall not supply shall not give any any individual intoxicating liquor because um for one to cast their vote they need to be sober if any person if it appears to the um presiding officer that on the day that someone appears to be intoxicated you can order this he or she could order this person out of the building so um because you cannot cast a vote if you are intoxicated that is one also um we usually have issues with employers not giving um individuals the opportunity to vote that is an offense persons um need and the law provides that it should be at least two hours nothing less than two hours it could be more because someone may be working in castries but it's registered to vote in souffle and it may take two hours because you spend time on the line and so on so um it's the law provides that it should not be anything less than two hours um loudspeakers on the day um burners signs and so on not permitted on the day of voting um influencing other persons or other persons to vote or not to vote in any direction it is also an offense um then there are other offenses in relation to ballot ballot papers etc and um offenses with respect to the officials who would be conducting that election for example the um the presiding officers the poll clerks and so on there are offenses to them and i am aware that all of them would be trained and would be given the um information um as it pertains to the elections act so um we are just hoping i'm just hoping and with the police especially we're responsible for law and order that persons will be sensible on that day and the days before respect each and everyone respect persons rights everybody has a right to follow whichever party they want um respect properties and also there are persons we are we have been getting reports that persons are destroying all the um properties of um individuals if they are not for example posters you are not for this um individual you are not supporting the individual so you destroy this is an offense damaged property which you could be arrested for we do not want to reach that far we just urging persons to do the responsible thing we know that persons are heated and so on but everybody just um be responsible and do things that is not in breach of the law also in terms of the protocols the health protocols because this is one of the main issues that we see that persons contracting the virus and there being an outbreak that we cannot handle do not want that so persons please enjoy yourself and be responsible thank you so much for that and just to add to what the commissioner just indicated there regarding the law and the election and election day we will be having a series of interviews and pertaining to that specifically beginning next week so you'll be able to have all of that information at hand um so that especially for the first time voters we have quite a large number of individuals voting for the first time in this election so right here in the information command center you'll be having all of that information the election law and you so you can um stay tuned for that i want to give a closing moment to the chief fire officer mr Joseph Joseph thank you very much um we all know that many persons are very passionate about elections i mean once this even comes you know that's their time and you know we are we are all willing to support you know the adventure but um we are asking about um if everything is done in a safe manner we need to remember that life continues beyond the election and um certainly for those who have not been through it i mean you don't know what it is but to be involved in an accident or even to get COVID for example i'm based on on some of the things we've heard from persons who have gotten it unsurvived it's not a pleasant experience so um we are asking every one to desist from the practices that will endanger your life and we spoke about some of them already and some of us sometimes take many things are granted so for example um and we spoke about the the shocks carrying several people um these vehicles have to you know sometimes troubles some treacherous treacherous paths and we don't know the last time these vehicles were serviced and all of that if one of them goes down a precipice we have some major issues so we are asking persons and to desist from the practices that you know put your life at risk and remember that COVID has no friends all right so irrespective of your party whether you are a labor party supporter WP or any other party if if you are exposed and um contract COVID then the risk is you know just as bad to you as it is to everybody so um we want to ask persons to you know remain safe and um all the best for me the best me the best party here thank you so much for that of the chief fire officer Joseph Joseph as well as commissioner police melton daisy and the deputy chief environmental health officer Mr. Cheryl Eugene sent remain for being here with us and discussing the COVID-19 protocols and election campaigning for 2021 and first remember that it takes personal responsibility to keep you safe out there during this campaigning period and also if you contract COVID-19 or you are harmed in any other way you will not be able to exercise that franchise so just remember that when you are out there wear your mask wear it appropriately and ensure that you do your social distancing as well as hand sanitizing I'm Lisa Joseph see you back here in the information command center


The confidence course is a series of obstacles that build a crew's confidence and allow them to negotiate obstacles that they've never seen before. All the recruits, their safety is handled by the drill instructors that are in their group. Each group has a drill instructor that's monitoring them closely while they do the obstacles. Some of the obstacles that recruits have the most trouble with are the higher obstacles. The A-frame or the stairway to heaven are the ones that you struggle with the most.


Good morning. As Mateo said, my name is Ramaswamy and I'm presently at IIT Delhi. In this course of lectures on the collective dynamics or collective behavior of some kind in complex systems, I want to largely discuss the dynamics of couple systems, and for reasons that are almost obvious. One looks at oscillatory phenomena. There is not just transient behavior that might go away after a while or not just things that never change. So we want something that changes in some fashion. Most of the time it will be periodic motion, that is motion that repeats exactly after some specified amount of time. Or it could be quasi-periodic, which is a little variation of periodic behavior. Or even chaotic behavior, because that is somehow the most common behavior in the world, right? Now the kinds of things we will talk about, many of you may already be familiar with it because synchrony especially with Steve Strogatz's book and podcasts and infinite number of TikTok videos and so on and so forth. This is a very common thing, but I'll also talk about its variations. I want to talk about things that have names like death, amplitude death, aging, stasis, etc. And also more quite interestingly symmetry breaking, which is a dynamical symmetry breaking which brings all this into the realm of statistical physics, there are phase transitions, and so on and so forth, okay? So this roughly is the plan. The topics that I will try to cover is what is synchronization? Starting with the oldest observation, at least the oldest scientific observation by Christian Huygens. I'll talk about that experiment, its theory, and how we understand it. I'll also try to then segue into a discussion on phase transitions and synchrony, what is the similarity between the two? And this is particularly important in a model which can be solved exactly, it's called the Kuramoto system, the Kuramoto model. Because chaotic synchronization is in a sense one of the more interesting aspects of synchrony, how do systems that never repeat themselves and are unstable everywhere and so on and so forth. How do those systems synchronize? Now that is also an important aspect of the subject. And then there are variations called generalized synchronization, etc, etc. But I will try to talk about it from a geometric point of view, using the ideas of manifolds and so on. Then as I mentioned already, symmetry breaking is going to be important. So I will discuss what are called chimeras. Then I'll talk about networks because nowadays, all complex systems are networks, etc, etc, etc, etc. And finally, one of the more interesting recent-ish developments. The subject's going back to 1665, so from 1665 to today, there's been a lot of developments, particularly after the discovery of chaos in this area. So I'll talk about interesting thing also introduced by Steve Strogatz, called Swarmilators, namely things that swarm and sync, okay? So this is something of current interest. Now, so one of the reasons one studies synchronization is that this is one of the most common forms of collective behavior that's seen in nature. I mean, there are simple examples that surround us everywhere, particularly from biological systems. All of us are examples of something that is very heavily in sync with the sun because we depend on the sun for so much of our energy, that it's always useful to have processes that keep some kind of a 24 hour cycle, because that's where the earth's rotational period is that. So such rhythms are called circadian rhythms, and these are there in virtually all organisms from bacteria onwards. There's a lot of synchronization that's going on in our brains. A lot of neurons spike simultaneously in order to retain some forms of memory. Which ones are spiking together? So there will be, there is a correlation between the kind of memories you form and the kind of synchrony that sounds or various external stimuli induce in your brain. If possible, I will talk about some current examples that I've studied in neuroscience. Other social phenomena which are known to be synchronous is, for example, when groups of people walk together, then they fall into patterns. Particularly if they know each other well, like groups of friends of varying heights, so varying lengths of steps, you might have noticed that when people go all together, they start more or less walking together and so on. So this is a social phenomenon which is an example of synchrony. That is, that is in different units. And one of the things that's necessary is some interaction, right? Okay, before I put on the video, let me just show you another thing that happens. If you go to a stadium, you find Mexican waves forming, right? And one person starts and then suddenly it goes on. Or if you go to particularly good performance, you find synchronous clapping, people get into a mood and then that, that, that, that, you know, it goes on, right? This crowd is big enough and we could try for an example of that later on, okay? You can hear whatever sound this is. This is the Millennium Bridge in London. And when it opened on 10th of June, 2000, you can see the crowds that are, there are many videos on YouTube. So if this one is not clear enough, notice how suddenly you find groups of people who are walking together in sync. And this was bad enough that the bridge became known as the wobbly. Because it was shaking from side to side. That's particularly obvious now, right? So, this can actually be a major problem. Armies when they walk across bridges are told to not walk in step.  Because otherwise you'll just get your natural rhythm of your walking, tuned with the bridge and bridges have been known to collapse because of that. Anyway, so they had to go and fix this. This was a big disappointment for the city of London, and a huge opportunity for people to study synchrony and chaos and so on. A number of papers that have come out on this side, quite a few. Okay, but in nature, there are these fireflies in Malaysia, particularly. The fireflies are not moving themselves, or one or two of them might be. But otherwise, they are largely static. And they flash in unison because they are trying to attract mates. Only male fireflies flash. But the synchrony that emerges, you know, just go again, right? So you can see that, this is not the best example I've seen. That I've seen, you know, entire trees just light up after a while. Because all of them are flashing together in unison. Now, they are communicating with each other mostly in response to another male's flashing. This one gets into sync and tries to be brighter or something. I don't know, more attractive in some way. So, I mean, the precise reasons, etc., are only hypothesized because these are not easy experiments to do, but you can observe them. Also, the observation of this has inspired a lot of theoretical work, trying to come up with models that could start synchronization. And this eventually led to the Kuramoto model, which I will talk about in some detail. But, and here I have to actually thank our director Mateo. There are other forms of behavior that could be seen as some kind of synchronization. So, here's an example of fish that are schooling. You know, when they are not in the schooling formation, they are just moving around randomly. But something happens, maybe a shark somewhere, and then they all start forming these shoals that are all traveling together. And one can see that there is, there is order, there is short range order. So, it's random motion that is just, and because the fish are sort of long. You can think of nematic ordering if you like, if you like the language of liquid crystals or something, all right? But this definitely, you know, one can see more than just the fact that they form a group that moves together. They're all aligned in, you know, at least locally they're all aligned, right? In the same direction, because otherwise they'd be crashing into one another. So, the emergence of this kind of spatial and temporal pattern, if you like, can be seen within the framework of synchronization. And when I discuss generalized synchronization, I will talk about it. You remember your comment, I took it very seriously. All right, so here's another example, okay? And by now, anybody who studies complex systems has seen this and, you know, Nobel Prizes and so on and so forth. A lot of very beautiful work on active matter and so on. And here again, I want to point out that there is the emergence of at least short range order. They're not all flying together in some straight lines or even some generally. I mean, these are groups that form and reform and so on and so forth, okay? But the reason why I believe that it's useful to think of this as some kind of a new synchronization is that these patterns can be engineered. Swarms of drones, for example. As we have seen in recent applications, shall we say, drones can be made to swarm and that dynamics is not very different from this, right? Also, when you look at these kinds of murmurations they are called, right? When you look at them in some detail and with the eye of an artist, you find representation like you see on the right hand side over there, right? So again, what the artist is able to see, and I will show you photographs later on of the actual birds in flight, is that these, again there is short range order. There is definitely some kind of emergent property over here. Because the bird doesn't, I mean it infers a pattern to be maintained through years of evolution and all that, of course, but it's there, right? Now, as I said, the first mention of synchronization was by Christian Huygens in 1665 or thereabouts and Huygens as most of you, I presume, know did a lot of work in early days of mechanics. He invented the pendulum clock and so on and so forth, right? Lots of stuff, ray optics, right? Now what he observed was, I mean he observed this phenomenon and made some very explicit notes which he communicated to his father and so on and so forth, but what he basically saw was that if you hanged two pendulum clocks on a common, so here's the two pendulums. And here they were hung on a common beam. And what he observed was that no matter where you started the pendulums from swinging, after a while, they were almost, after a while he saw that they were swinging in a manner such that this pendulum reached its extreme, that is this position, whatever this symbol is, at a time, when it reached this, this particular pendulum was on this side. So both these pendulums were going to their extremes at exactly the same time. However, there are two ways in which they can do it. Both of them can reach their left extreme together or their right extreme together. But this was happening in a way such that both of them reached their outer extremes first and then their inner extreme so that they were oscillating essentially out of phase with one another. Now, what he observed by and large was this situation, and it's anti-phase synchronization or out of phase or whatever. And this would be the case of in-phase synchrony, namely that it would reach, the extremes would be reached at the same time. So just to make the point even more obvious, here are the two phases of the two oscillators and they are just displaced a little for clarity. So the two of them are exactly in phase, over here they are out of phase. And what Huygens saw was mostly this. He saw the other one also, but not for very long. And when you do the experiment again, three centuries later in the early 2000s, there was an interesting paper. I'm going to give you all these references so that you can read them on your own as well. What you find is that you started off any which way, and these are not long enough to see the thing, but you can see that both of them are on their outer limits at the same time. So this is the common thing that one sees. Again, from YouTube, and there are even more spectacular videos, I just let this play and then we'll talk about it. No, they're not touching each other, but notice that this baby here is moving. He's also in sync. All right, now what I wanted you to observe over here was, I'll just go back and play it if necessary. You saw how they were just set off randomly, right? And then they came into a stage where two of them, they were actually anti-phase. That wasn't stable, right? And then it went out of sync for a little while, and then it came back and where everything is now in phase, all right? So the pendulum hanging from the top was largely outer phase. The system, these are metronomes. So the metronome system, it's largely in phase, right? And you can, like I said, and this is only with four, but you can see thousands of them if one likes, all right? Now what we are seeing over here is demonstrations of this general idea of synchrony. And no, this is a periodic, right? Because this is, metronomes are used for musicians to keep time. So they need definite beat. So these are all periodic, right? You cannot actually make a chaotic metronome. I mean, I suppose you could, you can make anything chaotic. But metronome by itself, if I just left it on the table, is designed to be periodic. There's a spring that you wind up, this is actually not very expensive, right? And so you wind up the spring that releases energy and then it just keeps the time.  And this is going to be the point. You know, if two identical systems, everything identical, identical, identical, perhaps one is not so surprised because the motion has, at some point at least, tell me your names so that I don't have to say him in the blue or him in the gray, all right? That you start out as close as possible, all right? But because there are small variations, even in a hanging pendulum, there's lots of technical things in making it as a good timekeeper. Huygens experiment actually made it impossible. He made these pendulum clocks that were good timekeepers to the level of something like 15 seconds in a day, all right? That was how accurate they were. But because they could synchronize with some comparable movement, they could not be used to determine longitude, all right? So that actually is a major problem and there's a beautiful book called Longitude, which you may like to read at some point in time because even though it's a historical book, but it has a lot to do with our history, how we invented basically how it was possible to go from my continent to yours, presuming that we are from different continents, of course, all right? Okay, yeah, yeah, yeah, go ahead. Is it important that they are on the same platform which starts oscillating as well or yeah, okay? That it starts oscillating is not important. That it is not infinitely heavy so that it damps out any, you know, what is necessary is for one system to recognize through the medium that the other one is there, all right? Yeah, it works. When it's very heavy, I will discuss some examples, okay? Yeah, see, the things that are important in all this is how different are the natural frequencies of these isolated systems? If they are very close to one another, then one kind of synchrony is possible. If they are not, they will, I'm just going to discuss that in a minute. There's going to be another kind of, there are other possibilities, let's say, all right? But that's important that they be close. The other thing that's important is the amount of damping in the both intrinsic damping through the medium maybe and the strength of the coupling, which is really dependent on the mass of the base or the suspension and the mass of the pendulum and so on.  The theories of all this are phenomenological, right? So it's not as if you're looking at material properties and so on and so forth, okay? But it's not important that it be on a swing. I use this video mostly to show, because the motion is very visible, right? You can see that, but it shouldn't give you the impression that that is the necessary condition. Yeah, so now two uncoupled pendulums with slightly different frequencies are, I mean, they're going to be uncorrelated, right? So if one pendulum has frequency omega 1 and the other frequency is omega 2. If they are almost similar pendulums, then let's say if that is the condition, then coupling them somehow makes these two pendulums pull each other to a common frequency. Namely, this one will go to some new frequency omega 1, this will go to some new frequency omega 2, and those will be identical, right? So that's what we are seeing, if one was to take a measurement. The two, we saw the uncorrelated pendulums when they were starting. Each one of them had frequencies which were almost the same. But then when they were in sync, you could even hear that they were at exactly the same frequency. So this process of going from to this is called generally entrainment, okay? So the two oscillators get entrained with one another, all right? This entrainment is possible. Should I move to this side? Is it easier? No, no, no, this is not meant to be a strain on your, at least not on your body, right? So okay, see the equation of motion for a pendulum is something like let's say x1 double dot plus omega 1 squared x1 is equal to 0, or let's say i. So I mean just for small oscillations, I'm just considering it to be a simple harmonic oscillator over here, all right? It turns out that of course these two pendulum, the other one, okay, both of them are out here. Omega 1 maybe is something which is like this, where p times omega 1 is roughly equal to q times omega 2 when they are identical or close to identical p and q are both 1. Now it turns out that the process of entrainment is actually powerful enough that you can actually drag this approximate relationship of p omega 1 is equal to q omega 2 or more to the point if I've got my eyes on you, yes, integers. So this kind of entrainment is possible even when p and q are, when this relationship is there, but it works best when p by q is rational, okay? So p by q is rational and they have no common divisors, so that's down to its minimum form and this kind of entrainment works very well if p is 1 and q is 2 or vice versa, 1 and 3, 2 and 3, et cetera, you get the idea, right, these, you know, 311 by 297 is not a good idea, all right, because we'll just see why that is the case, okay? So now the solution for this is basically something like xi of t is some ai cosine omega i t plus bi, right? So this is straightforward, right? And when they are uncoupled and not bothering to couple them at all, what I would like to show you is that this dynamics essentially can be written in the form of a map, right? And we'll just see how that happens. The x1 dynamics, if I write x1 versus x1 dot is basically lying on a circle. You should stop and ask, in case this is not familiar, almost everything I say is simple enough that it can be explained in a sentence or two, okay? Okay, so this is the motion for x1 and x2, all right? So the first oscillator is going around at some frequency omega 1, this one is going around at some frequency omega 2, all right? And the combined motion therefore lies on a torus, all right? So this is mathematically S1, this is S1 and this gives us a two-dimensional torus on which the motion occurs, right? Now, the combined motion is therefore going to take place and so on. And the, is this clear to everybody that the motion of the two, no? Probably run. Yeah? Maybe for the interest of those who are in the back, if you can make, bigger pictures, okay? And in the interest of Rama, maybe next time you can sit closer. There are two options, of course, either closer or far, far away, no? Okay, I will also speak louder and hopefully little clearer and so on and so forth. Okay, now, to whom is this picked, who finds this picture not immediately easy to understand, okay? So this picture, okay, great. So we understand that a simple oscillator is just something going around in a circle, all right? Because this is just folding X and X dot onto one phase space plot, right? Now, since the two of these systems are uncoupled with one another, consider the first circle. At every point of the first circle, there is another circle. And at every point, there's another circle, circle. So finally, what you have is a doughnut, all right? Or a bicycle tube or a car tube or whatever, flora, okay? All right, so now that we have this idea, the combined motion is going to go from here, it's going around one circle. So it's going around one circle. But meanwhile, on the other freedom, which is the X2 freedom, that motion is going around on this side. So finally, the combined orbit is going around and around and around and around, yeah? Now, to try to make a map out of this, Poincaré basically said, let's just cut through this torus. And let us ask, when does the orbit come and meet this cut, okay? And now, this is going to require a little bit of faith. This is actually a two-dimensional torus drawn on two dimensions in a world which is three-dimension. Actually, this is in two dimensions and this is in two dimensions. So this is actually in four dimensions. All right? So when you cut through with this plane, you actually only get a cross-section of this torus, all right?  And we'll try to see how to see the dynamics on this cross-section, all right? And this was invented by Poincaré and it's called the Poincaré section, yeah? Okay, so to make things easy, I'm going to just remove unnecessary constants, all right? And just consider the two phases of the two systems. So when you take the two phases of the system, what you have is, let's me say, X1 is equal to cosine theta 1 and X2 is equal to cosine theta 2. I've just shifted things around so that unnecessary constants are not needed. And I want to look at the motion of the phase theta 1 versus theta 2. So, theta 1 is omega 1t and theta 2 is just equal to omega 2t, which is theta 1 divided by omega 1, yeah, all right? So now let me try to understand how theta 2 varies with theta 1. I want to get rid of 2 pi, so okay? I'm just getting rid of the 2 pi, okay? And that divides out, I just get this equation. And now theta 1 and theta 2 go, they just go, so 0 to 1 is the same as 0 to 2 pi, it's in those units. Is that okay? Just because it's unimportant. So this is theta 1 and this is theta 2. Let me just draw this line, yeah? So, but all this algebra is modulo 1. Theta 1, if it's, let's say it's 1.5, that's 1.5 times 2 pi. So I've got to remove that extra 2 pi and this is the same as this number. So essentially this is just periodic boundary conditions, right? So instead of this curve over here, what I have is, I can just fold this back over here, then I can fold this over here. And I can then fold that over there, and so on and so forth, yeah? No, okay, all right. So let me start by asking, when does it first hit this line, okay? Let me call this point psi 1. This point over here, when it hits the other integer, psi 2, psi 3 and so on and so forth, all right? So the coordinates of psi 1 are the x coordinate, sorry, the theta 1 coordinate is just going to be, let me call this alpha. Right? It's alpha, psi 1, the y coordinate is 1, the x coordinate is 1 over alpha, yeah? This point I'm going to call psi 0 and this is just 0, 0. This is the origin, right? Psi 2 is just 2 over alpha, psi 3 is just 3 over alpha, 3 in general psi n is just n times omega, where omega is equal to 1 by alpha. Sorry for a little, I want to now make the point that this is 0 to 2 pi, 2 pi to 4 pi. So this line is exactly equal to this line. Same way this line is exactly equal to this line. So if I were to just now look at this dynamics on the curve, the first point is here. This is from 0 to 1 on both sides, right? Okay, so 1 is the same as 0, so the first point is here. The second point is here. The third point is here. The fourth point is here, and so on and so forth. In other words, psi n plus 1 is just equal to psi n plus omega mod. You're not happy? No, no, no. Yeah, no, I'm just, see the whole point is that all our dynamics and phase is only interesting from 0 to 2 pi. So if I look at it in units of 2 pi, the dynamics is only interested in 0 to 1, okay? So what this number does over here is just tells you how far you go on the line from 0 to 1 in one step, again on another step. But you've got to go past 1, so that's why the mod 1. And instead of this, I can just fold it up and look at the dynamics on this is psi 0, this is psi 1, this is psi 2, and psi 3, etc. So the entire dynamics on the Poincare section is just a set. This is each time it comes back after 2 pi or 1. And so the dynamics on psi is just a rotation on the circle. How much does it move each time? One, omega, and omega is the ratio of the two frequencies. Now it's not difficult to convince yourself that the only way in which it will close is that if somewhere in the future, it has to pass through the point over here or a point over here or whatever, right? So the only way in which it will close is if the slope is rational. And why rational? Because this is now broken up into integer units, right? So if it crosses itself, if it comes back to a starting point after however many squares in this direction and however many squares in that direction, then the orbit will close on itself. Yeah, big omega has to be rational, which also means that alpha is rational, which also means that omega 1 by omega 2 is rational and so on and so forth. Yeah, I'm only interested in when it crosses a particular, see the idea of the Poincare section is basically to draw a line on this curve, on this representation and say, tell me each time the orbit crosses that line. And for convenience, I'm taking this line, right? Because this is psi 1 and the next time it's moving the same distance. Because this length, since this length is 1 and the slope over here, so this has to be exactly the same amount. I mean, this is just simple geometry. But get this because this is one of the simplest examples of what's called an ergodic system. All through Statmic we study ergodic systems, right? But this is the simplest example, well, simplest in somebody's notation. But this is a simple example of an ergodic system. If omega is not equal to p by q. Because if it is equal to p by q, then let's say it's 1 by 1 by 4, right? So as the orbit goes around this torus, if omega 1 by omega 2 is just 1 by 4, you'll get four points, nothing else. And this is very important sometimes because when you want to divide a pizza between n friends, you need to move by precisely some rational number where 1 by n, let's say, okay, yeah? Now if alpha or omega is not rational, then these points will not, after 1 by 4, it means almost 1 by 4, all right? 1 by 4 plus epsilon, which will make it irrational, okay? Then we'll go around, and by the fourth time, let's say it will actually come here, over here, over here, over here, over here, over here, etc., etc., it's too far in the lectures. Let's not talk about that right now. It's just, this is a simple thing called the rigid rotation of a circle. And so the dynamics of two uncoupled systems, all right, is going to be represented by this. Composing omega 1 by omega 2, yeah? No, if you're irrational, you will go around because you'll keep missing. You will never come back, yeah? If you're rational, you will only cover it by a finite number of lines, end of story. I mean, actually, you can see that if omega is not rational, if it is irrational, right? Then the points will eventually fill up the entire circle, okay? That is what is meant by ergodic, okay? Have you all seen this book on classical mechanics by Arnold? Seen it in a library somewhere, on your laptops? Okay, so you'll see some of it in a homework, okay? So now, we are through as far as two uncoupled systems are concerned. If I want to take this case over here, where omega 1 is approximately equal to omega 2. And let me assume that this ratio is equal to 0.9998765, where is some irrational number, all right? If I don't need to go that far, then let's say I start over here. The next time around, I will only, oops, sorry, let me just draw my circle again. On iteration 2, I'm only over there. The third iteration, I'm only over here because I'm not 1, right? Omega is not 1. So it just will keep on going, going, going, going, going, etc. And then it will not hit 0, again, it will go this way and so on and so forth, yeah? So this is what is meant by ergodic, it's just going to go around. Now, let me try to see what is the effect of coupling these two oscillators, all right? I'm going to do the mechanics of it later, but just the idea of coupling them would be to say that some function's coupling them, it doesn't matter, all right? Now, so people have been studying different forms of coupling, we'll come to that. But the simplest thing that I want to do is to actually make this map. I want to just say that what effectively it will do is to change this map. Just one small thing, let me just go back, one thing I want to just give notation. Omega is called the winding number, okay? So the winding number is the number of winds that it does, all right? So I can even, so if I take psi n plus 1 is equal to f of psi n, all right? Then I want to define it now as omega is equal to limit of n going to infinity f to the n of psi minus n. How many times I have done it? When I've got this simple map, you can see that omega, this winding number, sorry, this winding number is equal to omega for the simple map, yeah? I mean, the limit makes no sense over here, but it's going to make sense for us later on. So now I want to consider this f of n where f of n is psi n plus omega and for reasons best known to Arnold, he considered this map of a circle, okay? Mod 1 is there. Mod 1 is only important when you want to represent it, but otherwise every point is equal to itself shifted by 2 pi, any numbers of 2 pi on both sides, right? So this is a very famous map, okay? And it's called the circle map or sometimes it's just to be specific. It's called the sine circle map, all right? And when k is equal to 0 is just the rigid rotation. What happens if k is small, okay? You can see that you see psi already is a variable that mixes up these two coordinates theta 1 and theta 2, right? So what I'm writing over here is a phenomenological, algebraically convenient form of some coupling between theta 1 and theta 2, okay? And it's a nonlinear coupling. You can consider all sorts of other behaviors to be important, okay? Now, the main thing over here is that as you keep changing k, you can ask how does this winding number change, yeah? See, we were interested in the case of omega 1 approximately equal to omega 2 when uncoupled. When you couple them, do they come together or not? Yeah? Oh, it's not W index and winding number, hashtag winding number. So this is what I've just been talking about, okay? This is the, so that is it, okay? So this is what was discovered and studied in great detail in the 1980s particularly. I should point out that just one second, let me just give a disclaimer. Almost everything that I will give in these lectures is the work of many, many, many other people, all right? And I've taken pictures from the Internet with attribution for the most part. When I haven't attributed it, it's not because I don't want to, it may have just forgotten where I got the picture from, all right? This is as much for the YouTube audience if there is any, it has this. Anyway, so the point about this is that what is noticed by Jensen, Bach and Bohr and many, many other people, Kadunov, Prakashya, Jensen, I've already said Jensen, just a large number of people, all right? The following. What is shown over here is the winding number calculated from that, all right? As a function of the strength of the coupling, yeah? And what we have over here, okay? There is some interesting properties just coming from the fact that it's a sine coupling and there is a symmetry between around the line half. I'm sorry, I never got back to you on the question. F is just whatever is on the left hand side or on the right hand side over here. In our case for the map, for any map, okay? For any map which is defined like this, I will define this to be the winding number. I start at some point, I just iterate for n times, divide by the number of iterations. So I find out on average how much it has moved each time, yeah? No, no, no. In all cases, w is equal to, the first graph is just this picture at this line. k is equal to one. Just one second, just one second. Winding number. No, that's not the winding number. Yeah, I know. Winding number is equal to, this is the definition. For this particular, for the case of k equals 0, that is on this line, on this line, that was, right? Next, then I increase the value of capital omega in my map. Then I calculate the winding number, all right? So if I keep calculating the winding number, it turns out that around every rational number, oops, okay? Around every rational number, 0 by 1, 1 by 1, 1 by 2, 1 by 3, 1 by, you see all the rational numbers over there, yeah? There is a small region which just, all the, it just attracts all the nearby frequencies. And so for example, if your initial omega, right, was let's say 0.11, right? So let's say it was 0.1. Initial omega is 0.1. And your k, the coupling strength is 1, right? Then after a little while, you will just attract both, both of them to have this particular, the frequency over here which is 0 by 1. Little thought will tell you it's the same as 1 by 1. So if the coupling is strong enough, even if the ratio of the two is not very close to each other, eventually they'll come close to one another. That is the entrainment. If your, if let's say your frequency, initial frequency is a half. So they are 1 by 2. And obviously your winding number is 1 by 2. But if it is half plus a small fraction or all the way actually up till almost 0.6 over here, let's say about 0.55, all right? If your coupling is strong enough, it goes exactly to 1 by, 1 by 2, okay? Yeah. Winding number equals to 0 is the same as winding number equals to 1. For the following reason, that on, in this modulo 1 algebra that I'm talking about, the 0.1 and the 0.0 are the same. Okay? So 0, so 0 by 1 is the same as 1 by 1. So that, it means identical. Yeah. You just keep on coming back to that point. Yeah. Thanks. Okay, now this is kind of important because also it's going to figure in a homework.  That's going to be the homework, but I'll write it out properly. And if I find that too many people are dreaming of the same function, that will not be a good thing. Okay, but jokes apart, it's not good. Are you clear about what is happening over here? As you keep increasing the coupling, nearby frequencies get attracted to one another if they are sufficiently close to rational numbers. Now, you all know that on the line k equals 0, the number of rationals is of measure 0. Yeah? Almost every number to start with on the real line, if you pick a number at random, it is an irrational number. Yeah? Okay, most people and most numbers are quite irrational. Please, okay, so you're listening. All right. Now, by the time you get to k equals 1, all right, and this is an important line over here, but I'm going to give you the reference so you can read it. This is a beautiful three-page paper in physical review letters. You know, it's worth at least looking at once. Okay, so on this line, what's happened is that this ratio 1 is to 1 is now actually taken up a large fraction. The precise length of that can be calculated. It is 1 over 2 pi. The width of that particular zone over here is 1 by 2 pi. This is somewhat smaller, that's smaller and so on and so forth. You can see every rational number now has a little width around it, where it has attracted all the close by frequencies. On this line, the irrational numbers have measure 0. So, at k equals 1, no matter which omega you start with, you're very likely to end up with an entrained frequency. And for lower coupling over here, you can calculate something like that. It's easy to compute and the definition is trivial. So, you can calculate something and you can see what is the width of these. Yeah, so yeah. Okay, color white is irrational tori. And color light blue is, sorry, dark purple, whatever that number, that, aubergine. Okay, so the aubergine shading is for rational orbits, periodic orbits. Yeah? 1 is to 1 or is an orbit that just goes around like that. 1 is to 2 would be an orbit that goes around twice in one direction and once in one direction, like that. Yeah? If you took an omega of 0.9 or 0.8, no, not 0.8. So, if you took it at 0.9 and k equals 0.9, you'd find that the actual winding number is 1. So, this is an important diagram actually because this is a slightly more quantitative version of what Arnold did. And I just want to tell you that what happens above k equals 1 is that this map for the case of the sine circle map, there are non-differentiable points at certain parts of it. Red gives rise to overlapping, so these are called Arnold tongues because it was first described by Arnold when, and this is just for normalization of everybody, is in student work done in 1959. He was a student of Kolmogorov. So, this is what he's writing in a, it's a little difficult paper but anyway. He looked at this particular map. You can see that it's really the same map. I mean, just change some variables a little. Okay? And here he proves that the width of the tongue decreases as the power of the denominator of the, you see the point A. Sorry. Okay? So, A is the same as R omega, right? You know, without the 2 pi taken out. So, A is 2, M by N, where N is the denominator and the width of this, of each tongue, it goes down. It decreases as epsilon to the power N, epsilon being the coupling. And that's why these first ones over here, 0 by 1 and 1 by 1, they go down as exact straight lines. These are quadratic and cubic and so on and so forth. And these increasingly high powers means that the zones are narrow, narrow, narrow. So, actually this line at the top when K equals to 1 is just covered by rational windows. And there are gaps between them and those gaps contain irrationals. So, even at K equals 1, there are irrational orbits. Above that, over here, these start overlapping. Once they overlap, the dynamics gets really complex.  We don't, we almost always go for 1 is to 1 resonances or they are called resonances. But, you know, whenever you've got systems that are almost identical, then you think that the natural frequencies are the same or almost the same and then they get entrain. What this tells you is that if they are close to some small rational numbers but not identical, 1 by 2 or 1 by 3 or 2 by 3 or something like that, then also you can have a very good entrainment. And that can be done experimentally and we are living in at least one experimental system where this phenomenon of entrainment is very important. Does it mean that there are jumps in one of the directions in the trajectory? Well, it goes chaotic. When the tongues begin to overlap, okay, at k equals 1, this is called the devil's staircase. It's called the devil's staircase because at every rational number, there is a step. And if you step on each one of those rationals, it's going to take you and you're never going to get to the top. So you can see some of the big steps, but believe me, everywhere you just have step, step, step, step, step, fractal. Okay, so the system that we are living in of entrainment to which we are eternally grateful to external forces well outside our control, all of you know that you only see one side of the moon, right? Varum. I mean, this is an international class, 9. Okay, no, but I mean, certainly this is the kind of question that would have been of interest to people, right? Then why do we only see one side of the moon? I mean, if it doesn't keep you awake at night, no. Okay, so the reason is that the moon's, the reason why we only see one side of the moon is that with respect to the earth, I mean, we go around once a day. The moon goes around us once in something like 27 days, 27 points something, all right? But the moon also rotates on its own axis once in exactly the same time. Okay, so as it is going around, sorry, I can't do it, but okay, maybe we'll make this the earth and, right? So as it's going around, it just keeps rotating in exactly the same. As the moon takes 27.3 days, it just goes around in 27.3 days. So the lunar day, so to speak, all right, is the same as its period around the earth. And that's why we mostly see exactly the same part of the moon. Why we don't see mostly is because of Kepler, right? The moon's orbit, if it was perfectly circular, then we'd see exactly the same part, but it's not, right? And it's, then the tilt, there is a slight tilt of its axis with respect to our own axis. So sometimes we see a little more of the top, sometimes we see a little more of the bottom. But we don't see more than 60% of the moon anyhow. And the reason is that when the moon was formed, when the earth gave birth to the moon, so to speak, I mean, there was no fixing of angular momenta and so on and so forth, right? Entrainment for tidal forces to be somehow minimized on the moon as it evolved around over, I mean, we've been around for what, four billion years or so? I mean, not all of us, me closer to the moon, but we've been around for a long time. And the moon has had enough time to lose its energy to slow down until it just got into frequency locking, right? And this kind of tidal locking is there throughout our solar system. 20 out of 24 or 26 satellites that are known in the solar system are tidally locked, right?  Okay, now I'm going to, whatever we have, you know, I have 15 minutes for this, but I really need to do a little more. So I will introduce the subject today as to what we'd like to understand. Is everyone cool and comfortable as far as circle maps, entrainment, et cetera? Ben? I'm sorry, it was very dated Italian. Okay, so now that we have an idea why these things do that, the sort of pulling of frequencies and so on, what we'd like to understand is how do we actually implement some of this in a system, you know, we started by discussing synchrony in the Huygens' clock, right? So what is the mechanism of this, right? I'm just going to say roughly what we are going to do tomorrow, because there's no point coming back to it later on. But this is a mechanical system, and at least the first approximation, it is actually quite simple. Let's say, right, so the mechanical system I want to describe is two pendulums of identical mass, identical length, suspended from a common beam. And this common beam is, you know, for convenience just got this particular property. None of these details is important, but as we discussed earlier, if I let the mass of the beam, which is capital M over here, go to infinity, all right? Then essentially the systems get uncoupled. So the Lagrangian for this system is, I mean, one just writes it down as one does with most mechanical objects. You've got the total mass of the system, and the base is allowed to move, all right? And you can think of having a spring along the base just to restoring one so that it doesn't slide off to infinity, all right? So here is the mass of the entire system and the velocity of the base. And here is the corresponding potential energy, if you like. So you've got that. And then you've got, here is the Lagrangian part which is just corresponding to the two pendula. You've got the, you know, kinetic energy for the pendulum, the angle theta 1, sorry, phi 1 and phi 2 is shown over there. I will probably call them theta 1 and theta 2 tomorrow, just whimsical. And this term couples the two.  So that much is your Lagrangian that you start with, all right? And, okay, so once you have this Lagrangian, you can write down the equations of motion for the two systems, that is the systems being the individual pendulum and the entire base.  Sorry, if I look at the interaction term, it does not come from a potential. That's not coming from a potential. That is just the velocity of, it's a talk kind of a thing. I don't know what one would call it. But it's the velocity of your system that is moving and the velocity of the pendulum. Okay. I have another figure which I'll show tomorrow which makes it. So but you derive it from mechanical? Yeah, it's derived from first principles, so to speak. Yeah, all right. Now, ignore for a moment this term and this term. This equation of motion is got just from looking at this Lagrangian and obtaining the corresponding equations of motion for each one, Phi 1 and Phi 2, so k takes the values 1 and 2. For the entire system over here, again, ignore that particular term. I've just got that, all right. Now, what this, these two, these are actually three equations of motion. Two over here and one over here. This would be a conservative system because this Hamiltonian is conservative, all right. But we know that there is some amount of damping. There's phenomological damping that just comes from the fact that the pendulum is in air. So that is written as B times Phi k dot, all right. So this is just the, the damping on just some number that will be put in later on. Okay. I'm not getting into what kind of form it is because this is just a simple pendulum. If you actually try to model Huygens' pendulums because they were timekeeping devices and had to stay active in a sea voyage, the actual form of the damping is different and also there's a need for external forcing so that these pendulums just keep going. Otherwise, with just simple damping, it would just stop moving after a while, all right. So there are extra terms that are added. This is one extra term that is thrown in and this is the other extra term that is thrown in for the pendulums. And for the base, there is this extra phenomological damping friction, call it what you will, that is added in, all right. This will be the starting point for our discussion tomorrow. We will try to analyze this and see why the solution Phi 1 plus Phi 2 is equal to 0, all right. So you see now I've got two equations here, one for Phi 1 and one for Phi 2 and an equation for X, right. So I'm not going to be interested in the whole solution. That's something that I'll give you the references and you can look at. I just want to discuss the following situation. In-phase synchronization is given by the difference between the two variables.  Let's call variable 1 is the difference between Phi 1 and Phi 2 and the other variable has the sum of the two variables. Then we'll see that one of them goes to 0 and the one that goes to 0, if the difference between them goes to 0, then clearly the motion is Phi 1 equals Phi 2. Phi 1 minus Phi 2 is equal to 0. Contrary wise, if Phi 1 plus Phi 2 goes to 0, that says that they are exactly out of phase with one another. So aim of the game tomorrow will be just to see how we show that one solution dominates over the other. And this is complementary to the discussion of the sine circle map because the sine circle map is just all sorts of things just thrown into one. And this will be specific to two pendulums and possibly two metronomes. And with that, let's meet tomorrow. Sorry, Ram. So you should have momentum conservation of the whole system here, no? Which should be? Just one second. So can you see it clearly from this? See, what you would really like is M Phi 1 plus M Phi 2 dot plus M X dot is equal to is a conserved quantity. That's not easy to see because you've got both the input and the damping. No, but say I mean you just look at the Lagrangian. If you look at the Lagrangian, then...


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