Well, hello and welcome to understand men now. I'm Jonathan as they have Jonathan as a comment I'm so excited to be shooting this live stream for you today our topic the four warning signs He's gonna end the relationship. What do men say when they break up? Already really quickly if you're new to my youtube channel, please hit the subscribe button hit the bell So you can be notified of new content And if there's anything during this video that resonates with you, please do me a favor and hit that like button and We're going to be doing content first and then the Q&A So if you just want to you're listening to the replay just know for the next five ten fifteen minutes It's gonna be content then we're gonna go straight to Q&A. All right We're gonna talk about the warning signs that men or women actually Start to develop when they're about to end the relationship now before I get into this I want to share with you something very personal to me I went through a divorce I After turning 40 and going through a divorce which seems like a million years ago, but about a decade and a half ago I remember so I went through a divorce and I'm in Family court because we had to go to court one time and I'm first up just really quickly when you go to family court you It's as lay versus as lay is what happened And you're on the docket and we happen to be fourth on the docket But we had to get there at 8 a.m. Because we didn't know which number we were in so I'm sitting we were number four So I'm listening to three other couples ahead of us And I'm listening to what caused you know what their their issue was going on in their divorces Okay, and I'm listening to the first couple. I'm listening the second couple I'm listening a third couple. I'm thinking to myself. Do you know what? Anybody who ever gets married should go to family court first to see what causes Endings of a relationship. I repeat that people should go to family court before they ever get married so they can see the problems that happen in Relationship and it occurred to me that if you could avoid if you recognize what the problems are you can avoid it in Advance you can avoid it or at least address them much sooner Does that resonate with you if it is please hit that like button? So I know this resonates with you because I the way I approach my coaching is very much reverse Engineering taking what causes most endings in a relationship. What causes breakups? What causes people to stop loving one another and Try to head it off at the past and so I've studied I have over 20,000 hours of coaching I've got over 3,000 hours of personal development workshops both workshops trainings videos I'm currently getting my cognitive behavioral therapy certificate I have a neuro linguistic programming certificate and as I said before 20,000 hours of coaching women now. I'm not saying this to impress you I just want to impress upon you that what I'm about to share is Because I've studied this I'm not selling you on some bullshit idea of reverse psychology to get the guy back Or get the girl that to me is a lot of crap I'm here to talk about human behavior and what causes people to do what they do is this sinking in is this resonating Please let me know hit that like button so more people can get to see this So one of the books that I've read it's and by the way for those who watch my videos No, I continually recommend the same books over and over and over again because there's a reason for it They're really your life if you really want to have a successful relationship I highly recommend reading about about 10 of the books that I recommend I mean, I recommend a variety of different books But these books and one happens to be by the author One book is called eight dates by Dr. John and Julie Gottman eight dates by Dr. John and Julie Gottman Now by the way, there's a link to all my Jonathan recommend books in the description but why I'm talking about John and Julie Gottman today is He did the exact same thing he watched. I think over 3,000 couples Over a period of time to learn what were the problems in relationship and then he reverse engineers it So very much I before I even knew he existed I had already been doing that And he came up with the four horsemen of the apocalypse that he says the four horsemen of the apocalypse In other words, this is what causes most tension and aggravation in relationship and the four horsemen are contempt criticism Defensiveness and stonewalling. Let me repeat that contempt criticism defensiveness and stonewalling So this is the common traits of what happens with not traits Experiences most couples have before a relationship ends So I'm actually going to add to that what men say to give you clues so you can be prepared Okay, so let's jump into it So the number first one I'm sharing and by the way, this isn't in the order of that I just shared but the first one is when a man says I'm busy I'm busy and he begins to be continually busy in other words. There's chaos going on his life There's busyness going on his life. There's always something else This is after the initial stages of when men claimed you because they were chivalrous and they Romanced you so they could get laid because that's the driving force, you know, this whole rhetoric that men love the hunt We don't love the hunt. We just like what comes after the hunt. We get laid.  It's not like I want to be in a relationship. I want to be in the world. I want the notebook I want serendipity. That's not how men operate. What are they on the hunt for sex? So they're very attentive in the beginning but when men settle into the relationship If they're ready if they're if they're losing interest in the relationship, they'll begin to start being busy That's called stonewalling. That's called stonewalling. In other words, they're prolonging things Until they either get the courage to end the relationship or find someone new or you break up with them Okay, so when you notice a pattern of busy busy busy busy, that's a clue That's a warning sign that he may be ready to end the relationship. Okay number two Now this one I said he I wrote this down so give me a second I said he walls up emotionally and doesn't share intimate things anymore. He's more factual He talks in facts not emotions. So he says things like oh, I went to work today. Oh, I did the gym today He's more curt. He's very he's operating from a place of contempt. In other words, it's like you're aggravating him Like so this contempt is this feeling of I'm better than you or you don't matter to me That's what contempt is. It's it's basically saying you don't matter anymore Your feelings don't matter and what's happening is they don't share Intimacy with anymore. It's very curt factual curt factual curt factual Have you ever experienced this if you have please hit that like button? Let me know Because these are the common traits or common experiences that happen before a breakup So let's jump into number three. Okay, I said I do content first and then we'll jump into the Q&A So number three He starts complaining about you. He's almost seems mad. There's tension. He gets irritated with you regularly This is where he starts criticizing you complaining you so remember I said, you know, there was Stonewalling contempt and now this is criticism. Okay, and he also might turn it around things that are problematic He might turn around and get defensive. Okay, so remember I said defensive is another one of things now Defensiveness is very common when there's friction going on because as human beings we often operate from a place of ego and not love Let me repeat that we operate from a place of ego not love This is why I highly recommend I have the CD version, but I recommend reading the book Return to love by Marianne Williamson return to love by Marianne Williamson Why I love this book and I have the CD version of it Is it all differentiates between ego and love ego and love and we oftentimes operate from that fear-based ego place In fact in my book what the heck is self-love anyway? And by the way, again, there's a link in the description to all the books I recommend in my book what the heck is self-love anyway I go into differentiating how our ego Operates from a place of both contempt You know for another and also defensiveness because the ego can't handle Friction the ego can't handle problems. So when you notice a man is complaining He's getting irritated with you. He starts becoming defensive. That's a sign that he's probably getting ready to break up with you And here's the fourth one, okay And this one I've heard this from a lot of women. Okay, and I'm sorry. I shouldn't be laughing I'm laughing because I've heard it so many times not that this is funny. In fact, it's quite the opposite But when a man starts pulling away Sexually he starts pulling away sexually and he starts saying things like I'm not in the mood And he stops being affectionate stops kissing you That's a sign That he might have either shame going on within him But it's a stonewalling act.  I'm talking about just the average garden variety Midlife guy who might have some ed issues, but he's also when he's detaching from you emotionally He's going to most likely start detaching from you sexually. Is this sinking in is this resonating has this ever happened to you? Please let me know by hitting that like button So as I said before when you reverse engineer problems, you can start recognize. So what happens is by understanding this You can start predicting human behavior and when you can predict human behavior, you can make better choices This is why when I designed my private coaching program and i'm so excited several women signed up today I only work with six women a month for my private boot camp and a couple signed up for today What they wanted help with is learning how to vet for emotionally mature men Learning how to vet for emotionally mature men emotionally mature men lean into the problems within a relationship versus running away Emotionally mature people lean into the problems instead of running away now anxious people Needy people Like they fight like hell to keep the relationship going and they can fight like hell to keep a really crappy relationship going I'm talking about those who want a juicy delicious healthy relationship It requires learning which guys are emotionally available and which ones are not and if you want some support on that Check out the link to a free discovery call with me in the description Because that's my area of expertise to help reeducate you on the reverse engineering of a relationship and how to vet for emotionally maturity I hope this is sinking in All right, I think it's time to take questions. So if you have a question Really quickly write the word question then write your question after the word question So I can see it in the live chat and please hit that like button if this is resonating with you What happens is the more likes more people get invited to the live stream So I'd be really grateful if you did that. All right, I'm going to grab my cup. What does my cup say?  Hi tina. Hi of the I can't even pronounce your name. All right Um Hi ruby All right, so it looks like we have a bashful group here. Don't you want to ask a question? Write the word question post a question. Um without question. This is going to be a short live stream And I want this to go on for a while While we're waiting for some questions. I'm going to throw in a cute Movie trivia question. Let's see if you get this right so In the movie say anything in the movie say anything What song does loy play from his boom box? What song does loy play from his boom box? Let's see who gets that right in the movie say anything. What song does loy play from his boom box? Okay um all right Let's see I can't tell doesn't look like we have a group much of a we have a bashful group today. Do you have a question? Oh, okay, so heather writes Can a man learn to be vulnerable can a man learn to be vulnerable? Um, the answer is absolutely. Yes. A man can learn that and let me just say how I know this I wasn't a very I was look at when I graduated college. I had a I had a programming Graduate college get a job meet a girl get married buy a house start a family That was my programming and I was very hyper focused on my career In fact who you see today is nothing like the person I was before in fact I was a total train wreck after my divorce.  I went I was drinking I was serial dating. I was a mess. I I couldn't be vulnerable if my life depended upon it I had a very hard time being vulnerable with myself and being vulnerable with another human being And what I realized it took a it took It takes what I call a humbling event a humbling event for a man to shift out of egoic behavior And start shifting into heart centered behavior. Let me repeat that It takes a humbling event in my case the divorce and a lot of other things But I literally used to call curl up in a ball and went to bed wishing I went in the middle of the day curl up wishing I didn't wake up I was in that much pain and I had two children And those of you know, I lost a child. So I mean the fact that I didn't want to be around I mean could you like now losing Connor how much? All right now I'm going to get emotional because I was just thinking I can't even imagine what those my boys would have felt If I didn't choose to survive. So my point is now I'm really getting choked up Is It's absolutely possible But a man most likely needs a humbling event and listen I was taken down to my knees after my divorce. I got wiped out in the market crash I was a train wreck and as I started to rebuild my life My knees were cut out again when Connor passed away and I can tell you that the only way I survived it The only way I survived it was doing all the personal development work spiritual work. This is why I'm constantly recommending books In fact, one of my all-time favorite books that I highly recommend Is reading the untethered soul by michael singer the untethered soul by michael singer This is what this is the this was what helped me prepare to write my book What the heck is self love anyway, and this is all about carter I mean, it's all about the experience not all about but he broke my heart open. So going back to vulnerability um I just want to say to you that it took it took losing my son to really break my heart open That's a picture of him in the book by the way um So I just to go back to answer your question. Yes, men can learn to be vulnerable It typically takes a humbling event. Does this resonate? I It it happened. This is true for human beings alike All right. Now we got a lot of questions coming in. So let me uh, give me a second to catch up to this um Bear with me everyone because a lot of content is going on between a couple people which makes it hard for me to find Okay. All right. Here we go um Question kim writes how soon is too soon to start dating a man that is still married But in the process of divorce how soon is too soon to start dating a man who's still married and in the process of divorce so You know, there's an expression Uh You know, well actually I forgot to express I was literally about to share an expression. I forgot it So let me just tell you when you date a man who's still divorced or going through a divorce You're not just dating him.  Okay until the until the ink is signed He's still married and I would recommend scheduling a telephone date with her Because you're going to be dating her whether you like it or not Okay, so this is why most women have learned the hard way not to date someone who isn't Completely divorced because the tapest here I want you to imagine this was what marriage looks like and now you're unraveling the tapestry of this old life And until it's completely healed and then until he's actually spent some time on his own What often happens is you become what's known as the transitional girlfriend. In fact I'll never forget this after my divorce or when we filed for divorce. I was still I was I wasn't divorced. We had just filed for divorce I met a woman online and I really liked her profile and she asked me how long are you divorced? I said we're in the process of divorce and this is what she wrote me. She said jonathan Reach out to me when it's been 18 to 24 months after your divorce and you've had One to two transition girlfriends. Let me repeat that reach out to me 18 to 24 months after your divorce And you've had one or two transition girlfriends. I'm like, I'm ready. I'm ready. I'm ready Sure enough, I run into her. I see her online three years later and I now finally divorced and I saw her profile And I told her what she wrote me and I said you're absolutely right I had two three transition girlfriends until I could actually be in the space I am today to even be ready for a relationship.  All right kim. I hope I answer your question Mike writes, please help how okay, Mike Question, please help. How do you learn to get to let go of someone who is not in love with you? How do you learn to let go of some?  Here's the thing You have a choice you can be here.  They're not right for you for whatever reason They're not right for you.  You just need to make a decision Okay, if you feel like the relationship is not healthy for you and it is misaligned for you You have a choice. You can either be a child or you can step in your adulthood This is why I highly recommend reading the book. I'm about to share with you How to be an adult in a relationship Children stick around in a child mind stick around in unhealthy things adults people that are victors consciousness not victim consciousness lean into adulthood So you don't need to learn anything.  And now I'm spitting Okay, so you have a choice be a child or be an adult Or be a parent But if you're not if this relationship isn't right for you then end it or go seek therapy or something But you said you need to learn how this isn't anything you learn. This is something you do I hope that resonates with you if it does please hit that like button Moni writes uh vulnerability is so hard for me. By the way, vulnerability is as equally as hard for women as it is for men Tina lee. Thank you. How should I um, okay? Thank you either way Ivanka writes, how should I tell a man? I haven't dated in the past five years How should I tell a man? I haven't dated in the past five years because of a disability So if I understand the question correctly There's a there's a man who has a disability and you don't want to date him Is it because of the disability or something else? But how should you tell him you simply say, you know what tim? I really I really appreciate who you are I just don't feel like we're aligned and I don't have that those feelings for you Ladies, why do you have to make this complicated? It's really simple You know, we have to stop worrying about other people's feelings now By the way chapter one in my book Speak your truth do it with kindness speak your truth do it with kindness. You can simply be kind And still speak your truth But this fear around speaking your truth Stop it Start leaning into it because if you keep saying if you keep stuffing your feelings And I'm not I'm using this as an example for a variety of other things, but if you stuff your feelings You know, what's the point of living? You know get busy living or get busy dying start expressing your feelings. Just do it in a kind way That's all you have to do is be kind And yes, you you know it someone might get you might say well Jonathan They might get hurt you can't hurt someone only they can hurt themselves That's memory. I talked about the ego. You know, that's why return to love is a great book to understand that the ego makes up stories You don't have to follow that patterning anymore. Is this sinking in is this resonating? Please hit that like button. Thank you All right Hugs from texas. Thank you. Tina lee Okay, linda writes happening after a 33 marriage. Do I confront? What do I say? Linda your question doesn't make sense Happening after a 33 year marriage. What when do I confront? What do I say? I don't understand what's happening All right, next question tanja Um bear with me. Just moved If a man says he's Confirmed bachelor, what does that really really mean? Is he or can he be in a relationship? All right, when a man says he's a confirmed bachelor, let's just think about george cloney, right the quintessential confirmed bachelor Now he eventually made a choice to decide to partner with someone But here's the thing. He was very upfront with women. I'm a confirmed bachelor. Okay So he basically says that basically i'm willing to sleep with you and spend time with you But there is no effing way. I'm going to commit to you and usually it takes someone Uniquely it takes a woman and I think his wife is a perfect example of this She probably didn't she acted like she didn't care about him now I don't mean that she didn't but she had so much self-confidence self-esteem self-worth Self-love she was sovereign in herself that that intoxicated him is what I suspect So you roll the dice when a guy says he's a confirmed bachelor now take me for example My dating profile here. Listen. I'm going to go on match.com. Okay.  Okay right here These are in my words. Do you see that in my words? Okay First words Let me start with the following I would like to get remarried or at least live together and I'm seeking a woman who feels the same For me. This isn't about growing old with someone. I want to grow in life and love with some special woman I'm clear Okay You have a choice you can choose the men who are unclear and roll the dice Or at least if you want to prove improve your odds then start choosing men who at least lean into I want something serious versus i'm a confirmed bachelor. That's my invitation for you By the way, did anyone get the trivia question right? Um Okay um Heather oh my sweetness. I'm so sorry. Heather says she lost her son. Oh my heart breaks By Karen writes what are the top reasons so many relationships break up after two years or so I understand four things you described and i'm wondering the main things that lead to those characteristics.  Let's see These are four fingers right four legs to a chair. All right. I'm sitting on a stool right now The first leg of a chair is Chemistry chemistry that attraction we have for one another important leg of the chair The second most important leg of the chair is and this is in no order is shared values Shared values. Do you share values with the same person with this person or at least close to it number three? Blendable lifestyles blendable lifestyles. This is where a lot of relationships get stuck Especially at midlife because it's not like the brady bunch where carol had her kids and mike had his kids and they just blended together That's not the way it works in today and four leg in the in the in the stool Is emotional maturity so the reason why most relationships end because there's either a lack of there's there's not much shared values There's lifestyles aren't blendable or the people aren't emotionally mature.  There's no commitment other than just being present This is why it's ladies. I always say you are in charge of your relationship destiny And you are actually the emotional leaders of the relationship If you want to build a deep foundation deep roots of trust in a relationship or a guy feels like this is the most juicy relationship a couple things Read the books. I recommend and then schedule a discovery call with me So you don't fall into the trap by the way There's a link to to schedule a discovery call with me Because i'm here to teach you the roots to deeper commitment the roots to deeper commitment because when someone is deeply committed They're not going to end the relationship in two years. But again, you need chemistry shared values blendable lifestyles and emotional maturity and by the way the vast majority of the population has weak Emotional skills. This is men and women alike women. You're just as bad as this is men Stop thinking you're better at this just because women tend to be more Emotional doesn't mean your skills are any better than men men stuff their feelings There's no doubt, but you equally have a hard time expressing your feelings. In fact one woman just wrote I have a hard time with vulnerability. So that illustrates it. All right. Thank you Karen for your question Um Angie writes I dated a widower for three months. He lost his wife. He was very kind and sweet and okay. Thank you All right, boo boo writes don't forget write the word question then question boo boo boo boo boo baker writes What are the signs that my ex is remorseful after the breakup? I'm noticing he's posting pictures wearing my favorite color He's posting pictures of the city where we broke where the breakup took place You know my first reaction reading that boo is you're projecting you're projecting what he's doing you because unless you know for a but but Look at he cared about you most likely cared about you But here's the thing caring about someone isn't doesn't just because you care for someone doesn't mean your relationship is going to work out So depending on what the reason was for the breakup we all look at I have regret even after my divorce. I have regret after my breakup with my Relationship after my divorce I have regret for a lot of women that I dated and we don't see each other And I have a little bit of regret a little bit of remorse a little bit of you know missing them There's this is just part of human behavior But who cares if he's remorseful? Why are you caring about that?  When he meets someone else or you meet someone else, he's going to become or you're going to become Somebody I used to know does anyone remember that song by godi? Somebody I used to know I should play that song right now, but i'll get copyright busted Somebody I used to know does anyone know that song so stop focusing on the relationship that's not Here focus on getting the relationship you want does this sink in thumbs up, please All right, boom. Sorry. I yelled. I'm just being loving believe it or not Okay, tracy writes. How do I support my boyfriend who's going through a hard time? How much space should I give him? first off You can't give anyone space Okay, person can take space, but you can't give them space. Okay So if someone's going through a hard time I have a couple things how deeply rooted is the trust in your relationship. Do you guys have a significant commitment? Do you have a game plan of taking the relationship from being a relationship to partnership to something deeper? um Because when someone's going through a hard time And by the way, this is the challenge with most relationships. Most everybody is going through a hard time But you know what? I'm going to share something very intimate here and very personal and I do believe that when someone is of character Okay, emotional. Let me backtrack and I'll share what I'm going to share with you in a second emotional maturity means actions matching words consistently Taking personal responsibility for one's choices is victor consciousness or not victim consciousness Okay, learning how to do what I call fight fair that means when you're in a disagreement It's not about being right. It's about being happy. Happy. That's listening to the other person's point of view Accepting the other person's point of view is being true for them and you doing the same for them Number four is empathy and empathy isn't just I can feel your feelings empathy as I care about your feelings and lastly Transparency if it's material to the relationship then someone is transparent with you.  So I'm going to share something very personal On the day that Connor passed away I had a date that night. I had a date that night I had a planned date from someone from a dating site I get the we had planned we talked a couple days prior we planned the date I'd even sent her a text message that morning saying I look forward to seeing you. Okay At one o'clock in the afternoon. I get a phone call for my x y There's paramedics at the house And then as I was driving and I'm rushed over there as I'm driving over there. I get the phone call He passed away I spent the next seven hours at my ex-wife's house because there was Shit going on at the time of between police and paramedics and all that kind of stuff coroner and everything I had The respect to tell someone I couldn't make the date. I sent her a text message. I said we've had a family tragedy Unexpectedly and I can't make the date. I didn't tell her it was my son. I just had a family tragedy My point in sharing this when someone says they don't have enough time to do anything I could have blown her off And I could have just totally blown her off Called her two days later or she might have called me and said my son passed away And she would have said absolute grace and forgiveness But true character is stepping into the difficult things And this is what I'm so fucking tired of human beings not stepping into things that are important in a relationship So going back to someone having a hard time if someone's having a hard time They may not be ready for a committed relationship.   Does it just magically work out because you're just sitting in your feminine? No It takes having grown-up conversations. It takes having adult conversations to really get to know another human being I'm yelling because i'm tired of watching women over and over give their power away to men, you know I mentioned this in the last livestream.  Are you serious about this person?  I could have that shotgun You know and say look, what's your intentions? Stop choosing people ladies that have no real intentions just like I read to you My profile says I want to get remarried crystal clear Choose men who are intentional instead of you know what I'm a I'm just a confirmed bachelor I'm just looking for something casual. Oh my god.   It's called Conversation you just have to learn the right ask the right questions and most of you ladies Don't know how to do that if you want help check out a link to my that's what I do as a coach I mean, I'm offering all this great content for free But ultimately if you want to go to that next step and to be in a juicy delicious relationship You might need some help. That's what I can help you with All right, I went off on there Uh, thank you Tracy. All right Jean writes I cannot be the oh, sorry. It's skipped Bear with me gene. Sorry Okay, I cannot be the only person that finds meeting men difficult No less emotionally mature men besides online. How do people meet these days? Things are different in the 2000s All right. So jean great question. So I want you to know that Men and women are equally emotionally immature. I'm going to tell you something. I'm out in the dating realm I find nothing but emotionally immature women most of the time there There's either the the entitled woman who's the you know It's all about her or there's the doormat type of woman Who's just literally afraid and she sets her standards so low that she accepts anything The entitled woman sets such ridiculous standards and it's all about her and usually someone somewhere in between same true for men Now where are people meeting? 50 percent or more of all new relationships Are happening through this device Through our our smartphones and through our laptops. That's where 50 percent of all people are meeting And and especially because of the pandemic that happened to be the number one place and a lot of people fell in love The real challenge is not meeting them. Although it is I get it It's vetting them. It's being it's learning how to ask the right questions It's about going deeper than the surface bullshit. So I want you to read this book If the buddha dated if the buddha dated, okay This teaches you how to date from a spiritual perspective from a heart-centered perspective And I promise you if you lean into this book, you're going to feel a renewed sense of Confidence in the dating realm because it's not about meeting him It's about meeting you and when you meet you really meet you guys will start flocking to you Now the trick is learning how to vet. That's what I teach. So I hope I help gene. Thank you Katharina says Jonathan, I totally agree. Thank you. I'm enough said there are no cookie way There are no cookie ways that a man or woman will act I don't get that one linda writes in a 33 year marriage and husband has begun stonewalling. What do I say or do 33 year marriage most likely I suspect in the 33 years. There is so much stuff under the rug And that's going to be a challenge. I'm a big proponent of couples doing couples therapy I'm a big proponent to start talking to one another Really talking to one another Most couples this is what john gotman and julie gotman said Is after a while most couples all they talk about is What time do we have to they have children? What time do we have to pick up kids? What time do we have to go to easter? What time do we have to pay the bills? What time do we have to do this? It's all the facts of the relationship.  That's one way of doing it Do it while he's happy and he might be agreeable Um, and my guess is you guys haven't had a lot of sex in your relationship So that's going to make it even doubly hard. But here's the thing You have a choice. You can say don't look Let's be real with each other our relationship kind of I'm assuming here by the way Okay, I'm just making the assumption here and our marriage kind of sucks We can stay in it or we can choose to improve it. Which one do you want? Just talk talk to one another. That's what's needed That's my impression. Anyway, linda. I hope that helps Okay, susan writes men never end relationships They just behave badly and reasonable until we women get so fed up and we finish with them Okay, mend. It's true. Men tend to nest in relationship. We nest we typically once we're in we don't go anywhere So that's true. But I'm here to say ladies. Do you want to have a juicy delicious relationship? Or do you want to have a mediocre relationship? If you want juicy delicious check out this book called spiritual partnership by gary zucca spiritual partnership This teaches you about what a really juicy delicious relationship looks like Start following game kathryn hendrix and start following john and julie gotman and start following harvell hendrix and Harvelle hendrix and where's the other book? Um, I guess I didn't bring it. Oh And helen hunt getting the love you want this is the kind of stuff you should be talking You should be talking about the five love languages.  Why is the number one search term wire men commitment phobic ladies? You are in charge of your relationship destiny Stop believing that feminine energy is just going to magically make the guy grow up It's going to take a lot more from you It's going to take being intentional and start having deeper conversations with one another and here's the thing You'll probably break up, but you're going to be much prepared for the next guy If you learn how to communicate the deeper shit instead of the surface stuff Jon gotman says most marriages and because it's all surface level stable ambiguity that ester parel says Erotic desire is lacking in so many relationships.  It was all based on expectation and you wonder why Men complain about women. This is a two-lane street.  You're just as bad at this This is why when women go through my boot camp. I get phone calls weeks later. Jonathan. I met a great guy Jonathan, I met a great guy Jonathan. I'm getting engaged Jonathan i'm getting engaged Jonathan my my marriage is in a few weeks. Where do I send the invitation?  Is this something you recommend? I can't do it, but I feel I waste time and energy dating for two months for it to go To go pear shaped. Okay Um, let me just be clear about something I just want you to think about that So you could follow the the advice of what's called duty dating or circular dating That means this is taught to women to date multiple men at the same time and wait for the cream to rise the top I just want you to think about this. Let's just assume that men are doing the same thing.     This advice is terrible advice and quite frankly, I'm going to tell you something when I meet a woman I tell them very quickly if we like each other I'd like to date and explore a relationship and I only date one person at a time. How do you operate? That's how I start off. I only date one person at a time now Let me differentiate between meeting a woman and dating a woman The first date is not if you're meeting a stranger. The first date is not a date. It's a meeting. Okay, that's it It's not a date. It's a meeting. You have to do the sniff test Like dogs, they sniff each other. You have to sniff each other That's where the pheromones and testosterone and estrogen and dopamine all get going right you sniff each other The first date is the second time you see each other is the first date But if you get to three dates with someone My philosophy is you date one person at a time if you've gone if you've seen each other once That's the meeting you've gone on a date You've gone on one more date and you're gonna in the third date is planned You don't need to be dating other people But jonathan I might get attached to the wrong guy If you get attached to the wrong guy that shits on you Because would you want him dating multiple people?  We can't make a decision So let's stop dating multiple people and date one person at a time see where it goes if it doesn't work out And by the way, if you learn to vet them like I teach You you won't waste your two or three months with someone you can cut to the quick and get on to the next one Look at that. Would you like to date a hundred men? And hope that one of them works out or would you like to narrow it down to three out of those hundred? And maybe choose your soulmate from that. That's what I teach in my private coaching So my opinion I date only one person at a time That's my invitation, but you're welcome to do follow the advice of others Multiple dating I think leads to a lot of pain and heartache later on down the road Not multiple meeting multiple dating Thank you. I appreciate your yelling. Okay, boo. I appreciate that Okay Sandy writes I bought two copies of eight dates the next guy that I get serious with has to read it before sex way to go By the way, don't forget my book.  Happy Easter. Gita. Thank you. All right. Oh, let's see. We got some questions here um All right Lisa writes my boyfriend and I are long distance. He is moving down to florida here to be with me He has a bad case of covet told me it's not my problem And to let him be that was three days ago. Why shut me out? Well, someone who has covet My god, that has to be very scary. I mean You know, I um, I remember I spent um This was six months ago. I was with my my family and my knees turned out to have covet and I thought, oh my god What if I get it? What if I get sick? What if I die?   So if someone's actually sick, they're probably scared They're probably not in a capacity to lean into the relationship at the moment But I think there's something else going on that you're not telling me. Let me read this again Well, first off, you have long distance and he moved. So he had a plan to move um And so he's asking well, what he's saying is look I have a bad case of covet Um, it's not your problem. In other words, he wants to deal with it He's not asking you to nurture him and he's asking for space That was I'm not sure three days is that out of line for someone who wants space I'm going to guess something else is going on, but you don't know what it is yet um So I would just check in with him as a loving partner depending on how long you two have been together Just say hey, sweetheart. I'm here for you when you need me. Okay, and um, but if that goes on too long There's probably something else going on and you're going to find out about it most likely sooner rather than later Um, and it'll be a bummer because I it sounds like it could be something that it could be Maybe he just doesn't want to move. I mean for someone to move That's a big commitment and maybe he's having second thoughts and he's using the covet as a smoke screen That's just my speculation Uh, Paula writes expectation begets resentment exactly Jonathan you don't yell Jonathan. You are great us women sometimes we need to be told well, that's how I feel Thanks, finna fin Uh leaf writes exactly Jonathan where chemistry ends confidence and honest communication Continues, especially as we get older exactly Angie writes Do you think a man who lost his wife three months ago is ready to fall in love with another woman? Woman, she was a she was in deep depression for one year before she passed away so I do but it depends on how in love with her if he was deeply in love with her He may never heal from that. My question is how is he healed from it? That's a very traumatic and emotional event. I don't think three months is enough time to heal doesn't mean he's by the way Co dependent men attach themselves to the first woman that happens frequently whittlers Immediately attach themselves to the first woman that happens frequently But it doesn't mean that the and by the way they get married very quickly But did you know that 65 to 70 percent of second marriages fail? That's because they haven't healed So I would caution you to date someone who hasn't healed and again I think it takes a minimum of two years on your own to truly heal But most humans are so co dependent that they'll attach themselves to the next person You may want to read this book Co dependent no more to understand what's going what could possibly going on whether it's you or him I highly recommend that book okay Wow, I I've by the way, um my book. Where's my book?  I'd gotten uh from one of the followers here She uh has a book club and this month has been my book. So I get to join them for their last Call this month on my book. I am so honored for those who purchased my book from the bottom of my heart Thank you so much for the so's that have referred my book. I'm so grateful I can't begin to tell you how much it means to me not because I make a boatload of dough. I make a few bucks every book. It's not the money It's the difference that's happening It's the invitation to do personal development self-help and spiritual work And I'm a big proponent of everyone to lean into their sovereignty to lean into their self. Oh my god. I smell Lean into their self-esteem self-confidence self-worth and self-love I hope this is resonating with you. Please hit that like button Listen, uh, I've got to get ready for the book club. So I think this is a good time to end By the way, the song in the trivia question. I didn't see it The song was in this movie say anything. What song does Lloyd play from his boombox? In your eyes by peter gabriel in your eyes by peter gabriel Uh, I love that song. Love love love that song. Do you have a favorite song? Please post it below. I'd like to hear it All right. I'm just going to repeat the four warning signs He's going to end the relationship what a man says first off I want to repeat the four horsemen of the apocalypse is contempt Stonewalling and defensiveness and men say things like this. I'm too busy at stonewalling He starts complaining in the relationship and starts turning it around. That's defensiveness or criticism Contempt is he starts pulling back on his emotions and starts let me just read what he says He starts talking facts instead of Intimacy and lastly he stops initiating sex or avoid sex all together by saying I'm not in the mood. He stops kissing. He stops being affectionate. Those are four warning signs to be cognizant of Oh deaf leper thanks, leave to be cognizant of Because it might be a sign that your relationship is on the brink Um, I hope you found value in this video if you have please share this with friends Tell all your friends give this to guys tell have guys watch this and tell me what they think I'd like to hear their thoughts. I know men starting to watch me Ah for those who know me.  I always do first off Give myself a big gigantic shot the bear hug of self love I'm going to reach into the camera and give you a hug of love if that's okay I'm going to ask you to turn to someone or a pet or a teddy bear or a pillow and give it or them a hug of love Because hugs are a great source of love and we can all use more love in our lives. Thanks a bunch. Bye. Bye now


We will continue with discussion on structural stability analysis, I would like to introduce in this lecture some elementary notions about fixed points and bifurcations in non-linear dynamical systems. See what we will quickly recall what we have been doing, we considered beam columns and we considered three situations which were departure from ideal situation like beam carrying axial load plus transverse loads and this axial load being applied eccentrically or the axial load is applied on a beam which is not initially perfect, there is initial imperfection Y naught of X, we showed that all these three problems are mathematically equivalent and for certain values of P the response of all these three systems become large the structure loses its stability, then we ask this question all these three cases represented departure from an ideal situation why such things happen if to answer that we need to consider the study of ideal situation itself then we showed that if we consider a perfectly straight beam which is loaded truly axially and we investigated the condition under which an equilibrium position in the neighborhood of Y equal to 0 is possible, so assuming that such a solution is possible we searched for the value of P which can lead to that, so under the assumption that a neighborhood an equilibrium position in the neighborhood is possible we were able to write the equilibrium equation for determinate systems we could write in the terms of bending moments whereas for statically indeterminate systems we need to use the fourth order equation, so I pointed out couple of times that this term PY that is P into this distance Y this force is computed this bending moment is computed with respect to the deformed configuration of the structure, whereas while finding reactions and other terms associated with the equations of motion we consider the undeformed geometry, so that difference we have to notice and we need to see later on how does that manifest in a more generic formulation of such type of problems. Then we plotted the mid-span transverse deflection versus the applied loads and we showed that the load deflection path has infinitely many branches and this is the, as the load is increased at this point at newer equilibrium states become possible and this is the overall profile of the equilibrium load deflection path. Now what we will do now is we will start considering certain issues related to dynamical systems and then we will relate all these two problems of stability analysis of structures as we go on, so for purpose of discussion we will start with a free vibration analysis of a single degree freedom nonlinear dynamical system, which is governed by this equation X double dot plus 2 eta omega X dot plus omega square X plus epsilon into G of X, X dot this is a nonlinear term which is equal to 0, we assume that this nonlinear term is a function of instantaneous values of X and X dot, it could as well be that this X could be function of tau where tau varies from 0 to T which is a current time, in which case the system will have hereditary nonlinearity, but right now we need not consider that, we are not going to consider that, so what we will do is we will introduce a new set of dependent variables I will call X of T as Y1 of T and X dot of T as Y2 of T, now clearly Y1 dot of T is X dot of T, but X dot of T is Y2 of T, therefore Y1 dot is Y2, Y2 dot of T is X double dot of T, X double dot of T we derive from the governing equation, so for X double dot we write minus 2 eta omega X dot minus omega square X minus epsilon G X, X dot, so that is what we are writing and in terms of Y1, Y2 this becomes the equation for Y1 and Y2 thus can be written in a compact manner like this, this is a pair of first order equations which are mutually coupled, so Y1 has terms containing Y2 and Y2 has terms containing Y1, now in general we consider equations of the form, suppose X and Y are the two dependent variables and I consider a pair of first order equations X dot is some F of X, Y and Y dot is some H of X, Y, we define the system state as a vector as X which consist of X and Y, now we define what are known as fixed points for this system, the fixed points are the system states at which the system state becomes time invariant, by that I mean those values of X and Y for which X dot and Y dot are 0, so that would mean at the fixed points we have F of X, Y equal to 0, H of X, Y equal to 0, so fixed points are those roots of this pair of equations, what do they mean? So we will consider some simple examples, a linear undamped single degree freedom system, Y1 dot is Y2, Y2 dot is minus omega square Y1, so Y2 equal to 0 and Y1 equal to 0 are the fixed points, you consider a damped system X double dot plus 2 eta omega X dot plus omega square X equal to 0, I have Y1 dot equal to Y2, Y2 dot is equal to minus 2 eta omega Y2 minus omega square Y1, again fixed points are Y2 equal to 0 and Y1 equal to 0, so origin is a fixed point, now we will consider another system X double dot minus alpha X plus beta X cube equal to 0, so Y1 dot is Y2 and Y2 dot is alpha Y1 minus beta Y1 cube, so here Y1 can be pulled out I write it as alpha minus beta Y1 square, so the fixed points are Y2 equal to 0 and this term equal to 0 means there are two possibilities Y1 equal to 0 or Y1 is equal to plus or minus square root of alpha by beta, so this system has three fixed points right, 0, 0, 0 comma plus this and 0 comma minus this, now these are linear differential equations, this is a non-linear differential equation, so one thing that we can observe immediately is for systems governed by linear systems there is only one fixed point, whereas for a non-linear system there can be more than one fixed point, now what are these fixed points of a dynamical system, what do they mean, what is their significance and how many fixed points can a system have, what happens if motion in the neighborhood of a fixed point is perturbed, okay, you have X dot equal to 0, Y dot equal to 0, now I slightly perturbed the motion, will the motion also die out or it will grow or it will remain unbounded I mean neither decays to 0 nor becomes unbounded what happens, next what is the rule of system parameters in deciding upon the number of fixed points, for example what happens if values of system parameters are varied, do the number of fixed points remain unaltered that you can quickly see here for alpha equal to beta equal to 0 the system has becomes linear and there will be only one fixed point, but beta not equal to 0 there will be three fixed point, so by changing beta I am changing the number of fixed points that is quite possible, do the nature of the motion in the neighborhood of fixed points change because of changes in values of system parameters, okay, so these questions we need to understand and of course relate them to the larger issue of stability of systems, so let's consider a linear undamped single degree freedom system, so as we saw the governing equation is X double dot plus omega square X equal to 0 and we will assume that the system starts from some initial condition X naught and X naught dot, now again I will write Y1 is Y2, Y2 dot is minus omega square Y1, fixed point is 0, 0, now we can solve this equation, so X of T we know it is given by R cos omega T minus theta and X dot of T is given by this, where R and theta depend on these two initial conditions, if initial conditions are changed R and theta would change, now what I can do now is I can plot X of T and X dot of T as a pair of time histories, that is what we have been doing, but instead I can also plot X dot on one axis and X of T and another axis and eliminate time from these plots, so if you do that what happens if you can quickly see here, if you write X square plus X dot square by omega square you can quickly see that this is R square, so that means if I plot X dot and X or X dot by omega and X, this will be a closed curve and it will be a circle if this scale is on Y axis I plot X dot by omega instead of just X dot, so that means at T equal to 0 we start the motion somewhere here and as time advances this point moves in the phase, this plane is known as phase plane and this point moves in the phase plane and traces a closed curve, for another set of initial conditions it traces another closed curve, so there can exist by slight change of initial conditions in another closed curve in the vicinity of this if I change the initial conditions slightly, okay, this has some significance we will see in due course what it is. Now let's consider damped system, so what is happening here there is no dissipation in the system and both displacement and velocity are sinusoidals and they never decay to zero and in the phase plane it traces a closed curve, so in the phase plane when I plot different trajectories of the system with different initial conditions, different system parameters and things like that what we get here is known as a phase portrait of the system. Now if we consider now a damped single degree freedom system, again in this case we know the exact solution so 0, 0 is the fixed point and if I now again find X of T and X dot of D, X of T is given by E raise to minus 8 omega T into A cos omega DT plus B sin omega DT I am assuming under damped system and X dot of T will be given by this, where by using initial conditions I can find A and B, so now if the motion is initiated somewhere here in the phase plane, X dot versus X of T plane it spirals and decays to zero, so how does a time history look like? This is typical plot of X of T, so X dot of T will be something similar and when plotted in the phase plane it traces a spiral which eventually lands at the origin, because at T tends to infinity X of T goes to zero, X dot of T goes to zero and therefore this trajectory goes to zero. If you start from some another initial condition similar another spiral will emanate and it will again as T tends to infinity goes to zero. We can consider systems which are externally driven, for example if I take a damped single degree freedom system with an external excitation F of T I can write it as Y1 dot is Y2, Y2 dot is now minus 2 eta omega Y2 minus omega square Y1 plus F of T, so if I now consider system states as Y1, Y2 I can write the governing equation in the so called state space form where Y dot is given by, I will write this in the matrix form, call this matrix as A and this is A into Y plus some capital F, so the governing equations here are of the form this. On the other hand if the system has nonlinear terms this type of matrix representation will not be possible we will simply get a generic equation of the form A is a function of Y, T, whereas here A is a matrix whereas here it is a 2 cross 1 vector of functions whereas it is 2 cross 2 matrix here, so that's a difference. Now we say that system is autonomous if A is independent of T, so that would happen clearly when there are no external excitations, so autonomous systems have the autonomy to choose their own frequency and amplitude of oscillations, that is why called it is autonomous. On the other hand if the system is driven externally, for example a linear system is driven externally by harmonic force at say a frequency capital lambda, even if the system natural frequency is omega in the steady state the system will be obliged to oscillate at lambda, so it doesn't have the autonomy to decide its own frequency, so this in such systems in the representation that we have used A will be a function of time, so external forcing for example influences the frequency and amplitude of oscillation. Now we will again return to the question of equilibrium points or the fixed point they are used synonymously, so I consider now an autonomous system Y dot of T is equal to AY, with Y of 0 is Y naught, now we define fixed points of this system as points at which the system state is at rest, that is Y dot equal to 0, so these points are obtained as roots of the equation A of Y equal to C, so these are set of equations, so as many equations as the size of Y and you will be able to solve for the roots of these equations. Again we have seen all this for linear system we got 0 0 and un-damped system, for damped system again 0 0 and here as we saw there are 3 points, so I as already pointed out non-linear systems have more than 1 fixed points. We now consider the question what happens if we perturb the motion or the system states equilibrium system states by small motion, if you apply a small force to the system when the system is at one of its equilibrium states what happens, so let's consider X dot equal to F of XY, Y dot is G of XY, I am considering a 2 cross 1 system of equation this can be generalized for more larger size problems, the equilibrium points are given by the conditions F of X star, Y star equal to 0, G of X star, Y star equal to 0, so this X star, Y star is a set of equilibrium points. Now let us examine the nature of motion around each of the equilibrium points by perturbing the equilibrium states by small amounts, so what I will do, I will take X of T as X star plus eta of T, and X dot of T that is this is Y of T as Y star plus some XI of T, now let us substitute this into this equation, X dot is X star dot plus eta dot, but X star dot is 0, because X star is a fixed point, so Y star is also 0, so I get X dot of T as eta dot of T which is F of this, and Y dot is XI dot is equal to this. Now what I will do is I will perform a Taylor's expansion around the fixed point and retain only the first order terms because I am assuming that the perturbations that I am providing are small, and mind you my question, the type of question that I will answer is not to determine the time histories of eta of T and XI of T in all detail, the only question that I will ask is that T becomes large, will eta of T and XI of T grow or decay or remain neither growing nor decaying, so these are the questions that we want to answer, in which manner they decay to 0 or in which manner they grow is not of primary concern, so we are asking therefore a qualitative question about the system behavior, so we are not so much particular about performing a complete time history analysis and answering this question, so if we now use Taylor's expansion F of X star, Y star plus the two gradients with respect to X and Y for the functions F and G, now we know that for fixed point F and G are 0, F of X star, Y star is 0, G of X star, Y star is 0, so I am now left with this pair of equation eta dot is this gradient into eta of T and this gradient into XI of T, and similarly another equation for XI of T, now these two equations I can put in the matrix form, I write it as eta dot, XI dot into this matrix Jacobian or gradient matrix evaluated at the fixed points and this is a vector of eta and XI of T, so these are linear homogeneous set of ordinary differential equation, now for sake of simplicity in writing I will drop writing this subscript that we are evaluating at X equal to X star and Y equal to Y star, let's assume that we are doing we need not have to write explicitly, now that would mean this is really a constant, now we seek the solution in the form, this is a linear system homogeneous therefore an exponential is always a solution, so we seek an exponential form of solution where the parameter S is unknown, so we want to know if this solution is permissible, if so for what value of S it is permissible, so I substitute into this, so I get S alpha beta exponential ST into this matrix alpha beta exponential of ST, now exponential of ST cannot be 0, so I get this equation which is nothing but an eigenvalue problem associated with this matrix, so the eigenvalues are S and alpha and beta are the eigenvectors, so clearly alpha beta equal to 0 is a solution to this problem for all values of S, but we are interested in those values of S for which alpha and beta are nonzero, so we perform the eigenvalue analysis in this case we get two eigenvalues S1 and S2 they will be complex and therefore they will be complex conjugates, so I can write the roots as S equal to A plus IB, now the solution therefore can be written as some alpha into exponential AT plus IBT, and this I can write it as alpha into E raise to exponential AT plus into cos BT plus I sine BT, similarly I will write a solution for XI of T, now this modulus of this exponential function E raise to IBT is 1, so this also modulus is 1, therefore as time becomes large if you are interested in observing whether eta of T, X of T grow or not we should look at how this multiplier behaves, E raise to AT will grow to infinity if A is positive, so the condition if A is greater than 0 then this limit of eta and XI as T tending to infinity becomes infinity and then we say that the fixed point X star, Y star is unstable, mind you this matrix is evaluated at this X star, Y star, so if A is less than 0 on the other hand as T tends to infinity the fixed point is the perturbation die we say that fixed point is stable. Now depending on how the nature of these eigen solutions that's also a solution we have different nomenclatures for the fixed point, we say that fixed point is a node if both eigen values are real and are of the same sign, they can be positive or negative but both should be real that means B will be 0, the fixed points can be stable or unstable depending on the sign of these two roots, both roots are negative then the node is stable, if both one root is positive another root is negative we get another situation. Now if both roots are real with one root real and other root imaginary, no other root negative the fixed point is unstable, now on the other hand the roots are complex conjugates and the fixed point could be stable or unstable depending on the sign of the real part, so if on the other hand if roots are pure imaginary that means real part is 0 then the linearized stability analysis is inadequate to answer the question on whether the fixed point is stable or unstable, okay, we have to carry out further analysis. Now let's see what we learn from, I mean if we apply these ideas to simple oscillators what is that we can learn, so let's take the damped single degree freedom system X double dot plus 2 eta omega X dot plus omega square X equal to 0 and I introduce Y1 dot is Y2, Y2 dot is this and this is my F and this is G, the fixed point is 0 0 and dou F by dou Y1 is 0, dou F by dou Y2 is 1, dou G by dou Y1 is minus omega square, dou G by dou Y2 is minus 2 eta omega, so the eigenvalues of this matrix are given by this characteristic equation and if I write this I will get lambda as minus eta omega plus minus square root this. Now depending on the nature of this eta there are different possibilities now, if eta equal to 0 then lambda will be plus minus I omega, then we say that origin is a center because this is a pure imaginary number, on the other hand if eta lies between 0 and 1 lambda will be given by minus eta omega plus minus I omega 1 minus eta square and since eta is, we were assuming that eta is positive, damping is positive it lies between 0 and 1, therefore origin is a stable focus, if eta is greater than 1 this will be, both roots will be negative and we say that origin is a stable node, if eta is less than 0 I leave it as exercise, origin is unstable, if negatively damped system, origin will be unstable and you can verify whether what is the range of eta for which this would be a saddle and a node or a focus, okay. Now let's consider now the nonlinear system, a simple nonlinear system, I have X double dot plus X dot minus X plus X cube is equal to 0, now again I introduce Y1 dot as Y2, Y2 dot as minus Y2 plus Y1 minus Y1 cube, where Y1 is X and Y2 is X dot, so I get this equation, so my F is Y2 and G is minus Y2 plus Y1 into 1 minus Y1 square, the fixed points you can see Y2 equal to 0, if Y2 equal to 0 then Y1 into 1 minus Y1 square is 0, therefore Y1 equal to 0 is a fixed point, so 0, 0 becomes a fixed point and 1 minus Y1 square equal to 0 is also a fixed point, so the fixed points are origin and plus minus 1, 0. Now we can evaluate dou by dou Y1 and other functions and this will be now function of Y, therefore we have to assign the right values for the equilibrium points and then evaluate the eigenvalues, so at the origin we have the eigenvalues to be given by this equation and we get lambda to be this and we can conclude that origin is unstable and is a saddle, we now consider the other pair of equilibrium points, this is plus minus 1 and 0 and we see that there is a term Y1 square here, so it doesn't really matter whether we take plus or minus, so the Jacobian will be, this determinant will be the same, so again the characteristic equation is this and we get the lambda to be a pair of conjugate, complex conjugate numbers with negative real part, therefore we say origin is a stable focus. Now we can see here if I now plot this potential energy associated with this system, so that will be omega square X square by 2, so if I plot U of XS versus X you get this type of a valley, so the motion of the system can be perceived as a ball rolling in this potential, okay, so what we are concluding here is origin is stable, there is damping, therefore this oscillation will continue and again it will come to rest, that means this is equilibrium point I give a slight perturbation, the system oscillates and it decays to 0, whereas here origin is here and if you place an object here any slight perturbation will move the ball away from this point, so the origin is unstable, whereas if you are here small perturbations you know motion will be stable, okay, now we will consider a simple pendulum, the governing equation is theta double dot plus G by L sine theta equal to 0, so again theta 1 dot is theta 2, theta 2 dot is minus G by L sine theta 1, so this F is this and H is this, the fixed points are theta 2 equal to 0 and sine theta 1 equal to 0, sine theta 1 equal to 0 means I have N pi, comma 0 where N varies from 0 plus minus 1 plus minus 2 etc., so this system has an infinite set of fixed points. Now the eigenvalues you can evaluate all the fixed points I mean can be written as N pi and by assigning different values to N we can approach different fixed points, so we will keep it as N pi and if you perform the analysis we get this to be the characteristic equation and for E 1 N I get the fixed points are centers, okay, then for 0 and E 1 numbers, so for example origin is a center, for N equal to 1, 3, 5 etc. the fixed points are saddles, okay, so we can plot on this axis the fixed points suppose this is a origin, suppose this is origin, now the pi will be here, minus pi will be here, so if I now plot the trajectory passing through that it will be like this, so these are the pendulum if you see what we are saying is origin is a fixed point and if the pendulum occupies a vertical position it's an unstable equilibrium that this is this point, since the pendulum can rotate over several you know complete motion is also permitted there are several fixed points, okay, so these points that you are seeing are origin and after it completes one revolution there is a 2 pi, another revolution 4 pi so on and so forth they are the fixed points, whenever the pendulum returns to this vertical position we come across an unstable fixed point and they are here, so these are centers, these are saddles and this is known as separate tricks. Now we will consider, we will apply these ideas to a mechanical system, a simple mechanical system which is made up of a rigid bar AB and there is a sleeve here and it is supported on a spring to this wall or some fixed condition here, so the position of this spring doesn't change but the sleeve can slide on this, so as a system oscillate this slide, this sleeve will be sliding on this, so now the question that we are asking is will the rigid rod remain in the upright position under quasi-statically load P, okay, so for some value of P can a neighboring equilibrium position exist, it's almost like asking the questions on, the kind of questions we asked on beam columns we are asking here, but mind you AB is a rigid bar and the flexibility is accounted by this presence of this discrete spring, so what we will do is we will formulate this problem as a problem in dynamics and see what we learn from this, that means we will basically consider the oscillation of this bar around this point. Now before we do that we should understand that if this bar starts deflecting to the right, moment this sleeve crosses this edge, our interest in this system is this, right, because that bar would have escaped from the sleeves and we are not interested in that, so we limit our interest to this range, minus cos inverse A by L to cos inverse A by L, so this range of angle, so we'll consider one of the snapshot of the deformed geometry of the bar, suppose at some point in during its oscillation it occupies this position, the load P is acting here, then the sleeve would have, you know, it would be originally at this distance, now it would have moved up, okay, so now I want to construct the expressions for kinetic energy and strain energy and write the Lagrangian and derive the equation of motion, so what we will do is we'll consider a point X, Y, and along this bar I choose a coordinate XI, so I will call X as XI sine theta, so theta is this angle, the axis of the deformed rod to the vertical, XI sine theta and Y is XI cos theta, so X is here and Y is there, so now DT is the incremental strain energy stored in a chunk D psi here, at a located at a distance XI, so this is half M D psi, M is a mass per unit length, X dot square plus Y dot square, for X and Y I will substitute this, so I will get this equation and the kinetic energy stored in the D psi element is given by this, and the total energy will be I have to integrate from 0 to L, and I will get it at ML cube theta dot square. Now the strain energy is given by the energy stored in the spring and work done by this load, so the change in length of the spring if you include and compute this, this will be half KA square tan square theta, and this is contribution to the load is minus PL 1 minus cos theta, so I can now construct the Lagrangian T minus V, and if I now put it in the Lagrange's equation and perform the necessary differentiations I get this equation. Now you see here inertia term is linear, but thus stiffness terms is highly nonlinear, there is cos cube theta here, sine theta here, so it's a highly nonlinear system, so for sake of discussion we will also include that there is dissipation in the system which is notionally represented through a viscous damper, so the governing equation for free vibration of this system is given by this. Now I will again introduce the state space form theta 1 dot is equal to theta 2, theta 2 dot I will derive from this minus 3 C ML cube I will divide by ML cube by 3 and I get this equation. So what I wish to do is we will identify the fixed points of this system and study their stability, how many fixed points are there? We will determine that first and around each of the fixed point we will apply a small perturbation and see whether perturbation grows in time or not, so this can be written as theta 1 dot is theta 2 I call F theta 1 theta 2 and theta 2 dot is this and this is some G of theta 1 theta 2, the fixed points are roots of F equal to 0 and G equal to 0, if you do that theta 2 is 0, once theta 2 is 0 this term drops off and I will be getting KA square by cos cube theta 1 minus PL sine theta 1 equal to 0, so sine theta 1 equal to 0 is a possibility and theta 1 is root of this equation which is secant inverse of this to the power of 1 by 3, we are not taking N pi here because as I said already we are not interested in the bar rotating about this, our limitation our interest is only in range of theta lying in this, so that is why we are taking only the one of the roots of this equation, theta 2 equal to 0 and theta 1 is this. So now we will evaluate the matrix whose eigenvalues we need to study, so dou F by dou theta 1 is 0, dou F by dou theta 2 is this and dou G by dou theta 1 and dou G by dou theta 2 are given by this, now consider the fixed point 0, 0 first, so I get the gradients in this case to be dou F by dou theta 1 is 0, 1 and dou G by dou theta 1 will be this quantity and dou G by dou theta 2 is this, so the eigenvalues of interest are to be evaluated by finding roots of this equation, if you do this I get the eigenvalues as minus C by 2 plus minus half square root of this etc. Now suppose for sake of discussion damping is 0, next if we see that if KA square minus PL is less than 0, okay, then real lambda will be greater than 0, therefore fixed point is unstable, on the other hand if KA square minus PL is less than 0, real lambda will be less than 0 and fixed point is stable, so therefore condition for stability of 0, 0 is P must be less than this, so this is the condition on value of P for origin to be stable, now let us consider the other fixed points which are given by this, now when deriving this we use this condition to find the roots KA square by cos cube theta 1 minus PL equal to 0, and consequently dou G by dou theta 1 will be given by this quantity and you have to see here all these quantities are real, there is cos to the power of 4 sine to the power of 2A square etc. KA is positive, this is alpha square, minus alpha square however alpha square is a positive number and all other gradients are as before, so the eigenvalues are given by this where alpha square of course is this messy term, and I get now this equation for lambda and from this we can see that the fixed points are always stable, okay, so now we can plot the load deflection diagram, so on this axis I plot theta and load is plotted here and first we look at origin 0, 0, until P crosses this, this will be the only fixed point that system has, and beyond this value of P the origin becomes unstable, so this cross indicates unstable points and dot represents stable points, but at this point two branches emerge, and along each one of these branch the solution is stable, okay, that is given by this, that is what this analysis of the second branch of fixed point shows. So P star which is the value of the load at which origin becomes unstable is given by KA square by L, so this is the complete load deflection analysis of this system by including large deformation and dynamics, okay. Now what we will do subsequently is we will assume small deformation and we will do a static analysis and see what happens, and then ask the question what are the type of problems where we need to include dynamic effects, what are the type of problem where we need to include large deformation effects, okay. So now if I were to do a linear dynamic analysis, that means after formulating the equation of motion I will linearize this term, sin theta 1 is written as theta 1, cos cube theta 1 is written as 1, and this equation becomes linear, and I get this equation. So again I can find out fixed points for this linear system, and perform the stability analysis, we find that, we will now only find origin, the other fixed points do not figure in this analysis, so we get the origin to be stable if this condition is satisfied, afterwards the origin will be unstable, so this is stable, and later on this is unstable. Now we can ask the question does the linearized stability analysis give qualitatively the correct picture of the phase portrait, do we, for example if a linearized analysis shows that a point is a center irrespective of value of certain system parameters, can we believe that? The answer to this question is yes provided the fixed point for linearized system is not a borderline case as in a center, if the linearized system predicts a saddle node or a focus then the fixed point is really a saddle node or a focus, but if it predicts it as a center we are not sure, so I will present an example to substantiate this argument, and this is taken from this book by Strogatz, the system here is given by X dot is minus Y plus AX into X square plus Y square, and Y dot is X plus AY X square plus Y square, now we can easily see that origin is a fixed point by equating this to 0, and Eigenvalue problem analysis shows that the origin is a center, because there are pure imaginary roots. Now let us analyze, now what we will do is we will solve this equation, it so happens that in this case it is possible to construct this solution, so what we will do we will make the substitution X equal to R cos theta and Y equal to R sine theta, and X square plus Y square is R square, so if I differentiate this it becomes 2XX dot plus 2YY dot is equal to 2RR dot, so I get XX dot plus YY dot is RR dot, so for RR dot I will write X into X dot, for X dot I will write the governing term from the governing equation, and for Y dot similarly I will write the second term here I get this, now after simplifying this I get the equation for R as R dot is equal to ARQ, so this can be solved, similarly the equation for theta dot can also be obtained and we can show that theta dot is equal to 1, so the governing equation in polar coordinates corresponding to the given equations in Cartesian coordinates is given by R dot equal to AR cube and theta dot equal to 1, now we can analyze this problem and show that if A is less than 0 as limit T tends to infinity R of T goes to 0, that means origin is a stable focus, now if A equal to 0 origin is a center, if A is greater than 0 R of T tends to infinity and origin is a non-stable focus, now whereas the linearized system analysis showed that origin is a center for all values of A, so exact solution does not support this observation, so if you get a center as an answer to your question on stability then you need to, you cannot be decisive, so what is advisable is to carry out a non-linear analysis, now we use the term bifurcations, what does this word mean, now non-linear systems are or dynamical systems are characterized by fixed points and their stability characteristics and also the nature of this fixed point, for example nodes saddle or focus etc. and they being stable or not, so and this dynamical system is characterized by set of system parameters, so as a system parameters in the dynamical system are changed, one observes that the number of fixed points can change, the nature of fixed points can change and the stability of the fixed points can change, the subject of bifurcation theory deals with these changes due to changes in the values of system parameters, bifurcation therefore is a change in the number of fixed points, their nature and their stability as a parameter of the system is changed. Now the occurrence of bifurcation is accompanied by qualitative changes in the nature of system behavior, if suppose a fixed point becomes from being stable it becomes unstable the system will behave differently, the system parameter values at which system undergoes bifurcations are called the bifurcation points, so Euler buckling phenomena can be interpreted as a phenomenon of bifurcation, so at P less than P critical the system will have certain types of fixed points and P greater than P critical it will have different types of fixed points. So we can consider to illustrate that a system with two parameters alpha and beta X dot plus alpha X plus beta X cube equal to 0, now if alpha is greater than 0, beta greater than 0 we can show that origin is a center, if alpha is less than 0 and beta is less than 0 this is center and saddle, if alpha less than 0 beta greater than 0 there will be two centers and one saddle, so you can analyze this problem and see if these statements are right. Now there is another kind of system behavior that we should be aware of what are known as limit cycles, now let's consider an equation of the form X double dot minus epsilon X dot multiplied by 1 minus X square plus X equal to 0, so here the nonlinearity term is associated with dissipation characteristics, we have been considering nonlinearity associated with stiffness they can be associated with dissipation also, this type of nonlinearity occurs for example in fluid structure interaction problems. Now if we now consider this system if X square is less than 1 then this 1 minus X square will be positive and this system will have negative damping, for small amplitudes of oscillation the system behaves as if it has negative damping, on the other hand if X is larger than 1, X square is larger than 1 then this term will be negative and negative into a negative term becomes positive and the system behaves as if it is positively damped, so if we now look at that in the phase plane, suppose if I plot a motion that is originating near the origin that is small oscillation will tend to grow, whereas a motion which is originated away from the origin will decay, so in between that there will be a periodic orbit to which the response will move towards in the steady state, okay, so that these periodic orbits are known as limit cycles, so they are isolated periodic free vibration solutions of nonlinear systems that means I am looking at free vibration solutions and irrespective of the initial conditions the same steady state periodic solution is obtained, and these in the neighborhood of this closed curve there cannot be another closed curve, a system can have several limit cycles, for example if you consider a dynamical system as given here you can show that this system has infinite number of limit cycles, so for example if you have the origin is unstable then small motions will start growing so there will be a stable limit cycle, any motion which is initiated within this will be repelled to this, whereas if there is another stable limit cycle, suppose this is another stable limit cycle at another distance, now if a trajectory gets initiated here to which orbit will it move towards, if both are stable there will be a contradiction, so there will be an unstable limit cycle sandwiched between two stable limit cycles, that means any motion within this region will go here, any motion in this region will go to that, so in systems with multiple limit cycles the unstable and stable limit cycles alternate, now I said that limit cycles are isolated close trajectories, if we now consider for example X double dot undamped single degree freedom vibrations, if I plot the phase plane plot for this, this will be one orbit for a given set of initial conditions say X naught and X naught dot, if I now change X naught by some small initial small change and similarly make a similar change on this I can realize in the neighborhood of this solution another close trajectory, this is not a limit cycle, any closed orbit in the phase plane is not a limit cycle, in the neighborhood of limit cycle there cannot be another close trajectory, so whereas in linear systems it is possible, but systems exhibiting limit cycles it is not possible, because if for example this oscillator that I discussed in the Van der Poel's oscillator, it has one limit cycle here, so if a motion is initiated here close to this it has to be absorbed by this, if it is here it is absorbed by this, so in the neighborhood there cannot exist another closed orbit, that is a major feature of a limit cycle oscillation. Now in the next part of our lecture what we will do now is we will consider energy methods for stability analysis, this analysis will be based on two axioms, here I will simply state the axioms we will consider what it means in the subsequent next lecture, so according to the first axiom a stationary value of the total potential energy with respect to the generalized coordinates is necessary and sufficient condition for the equilibrium state of the system, this is first axiom tells you condition for getting an equilibrium state, the next axiom tells a complete relative minimum of the total potential energy with respect to the generalized coordinates is necessary and sufficient for the stability of an equilibrium state of the system, so we will use these two axioms and study dynamical system or I mean here we are not considering dynamical system, we are considering basically statical systems as we will see and we will apply these problems to some of the simpler system that we have studied so far and then we will see that this can form the basis for developing stability analysis of elastic systems to start with we can use Rayleigh-Ritz type of approximations and then it will graduate to the finite element method, so a discussion on this will be taken up in the next lecture.