Good morning everybody I'm Tommy O'Brien coming to you live from TFN and Thursday morning just after 9 a.m. Eastern time we get the markets kicking things off in negative territory you're looking at an S&P right now negative by 47 points we start that acceleration basically it was a drift lower from 4 p.m. Eastern time things really start dropping at about one in the morning you're at 3810 at that price point you trade down about 40 points from there we've been chopping around at about 3770 for most of the session for the last six hours or so since about 230 basically since Europe's been open chopping around since about 3770 NASDAQ 100 you're off 1.3% the S&Ps are off 1.2% the Dow right now off 1.1% as well and you're looking at a Russell negative by 1.5% jumping back to the NASDAQ 100 real quick you're trading at 11,535 now I talked about on my program on Monday I believe it was I talked about all of these areas of confluence that we have excuse me let me get that back we back this up and confluence folks isn't it an area of a Fibonacci retracement okay that is two different trends you're talking about the longer term trend which is the whole acceleration remarkable to say from a November of 2008 or even you're talking about the beginning of 2009 in the S&P the low you trade up to 16,767 now this is the NASDAQ 100 when I was talking to my dad about this made a great point he pulled out the comp the comp basically almost hit it I'll pull the comp up in a moment the point being though you're talking about a price level of 10,750 that is the 382 retracement folks of the entire move higher and it's almost crazy to say that you're talking about a move from 1,000 to almost 17,000 did you hear that right that's a 17 bag almost almost we made the 16,767 but guess what folks we go from 1,000 to almost 17,000 never say the market can't do a 382 sometimes it just looks so big on a chart but nonetheless we came within we reached a low of 11,068 the 382 is 10,751 you're talking about only 300 points in the NASDAQ 100 from that level right now you're sitting about 750 points away from that level and you put this thing on a weekly just for some context here I just said we're sitting 750 points away from where we are in the 382 this week alone you're down 750 points from the high trading at 11,537 the high was 12,262 keep it in mind folks and what is the comp is that the comp there it is okay so check this out if you pull up the comp which on the thinkorswim platform it's the entire NASDAQ composite dollar sign COMP you do the same exact thing and you put this thing on a monthly okay you got lows now the comp made a low in March of 2009 at 12,065 the comp made it as high as 16,212 I may sneeze here here we go excuse me and check out the comp it basically did it you made it within about 60 points 6-0 is all the comp the 382 is about 10,500 you make it down to a low of 10,565 how about that folks so when you think that a market can't do it okay the NASDAQ's the one that's been leading us down first it'll probably be the one that makes a low first because it's been paying the penalty so harshly for some of the multiples that it's had not assure that it's going to happen but that's the way it's happened previously you're talking about NASDAQ 100 negative 150 points at 11,541 you basically hit the 382 of the entire run he had higher from 2009 if that happens in the S&P's folks 3,200 is where this market would come back to a lot of talk of that 3,200 price point we'll see if it happens but put it on your radar well within reasonable when you got a market right now again let's put that on a weekly okay you're talking about a market that is almost 200 points off its high right now and 3,200 it's less than 600 points from where you are you were trading in this market folks a month ago at 4,200 nothing stopping this market getting from 3,200 we'll see what happens all right we jump to commodities we got crude back and off a bit we'll pull up the short-term timeframe crude makes it to 114.05 yesterday you're back to 109.02 for some context here let's back it up a little bit further when you're talking about a Fibonacci retracement let's take the lows that we had a week ago Wednesday we go to the highs of 114 and you're basically chopping around right at the 382 of that retracement right now you did make it as low as 108.23 we're up back at 109 but you see this area kind of chopping around to 382 we'll see where we go the 618 if we break this area you're talking about 106.25 in the crude contract gold contract this morning there's a little action for you that's an hourly you put it to a 15 minute there's some action for you so that's going to be on some inflation data that we got at 830 I'll jump to that in a moment but we have the gold contract right now up $5.18.22 and we jump to notes and bonds we're getting a little bit a higher price and lower yield the 10 year up 19 ticks how about jumping right to where we were a week ago folks 1808 would we just hit 1808 back it up a little bit further than that that's the area that you sold off from on June 10th maybe an area we bump into a little bit of resistance we back this up further on a daily you see that that is the area that we're talking about where we fell out of bed on June 10th that's the high that we recently had in price on the 10 year just one week ago June 23rd also a pretty critical area when you think we're pumping into an area that was support back on May 9th maybe that area becomes an area of resistance we'll see where the 10 year trades right now you're trading at 1803 and we jump over to the VIX volatility index back near 30 2954 this morning all right let's jump over to the inflation data we got this morning the Fed's preferred inflation measure rose 4.7% in May now they say around multi decade highs that would be true it is not an exact high but it's pretty close to an exact high uh both these numbers though just slightly less than the estimates we're looking for so inflation held at quote-unquote stubbornly high levels in May though the monthly increase was slightly less than expected that's from the number this morning core personal consumption expenditures rose 4.7% from a year ago that is down 0.2 percentage points less than the previous month previous month coming in at 4.9 the market was looking for 4.8 so you have last month it's 4.9 this month they're saying we expect 4.8 we come in at 4.7 on a monthly basis that same measure which excludes food and energy increased 0.3 percent the market was looking for 0.4 percent so you take out food and energy which are more volatile and they are having a huge impact right now but you take out food and energy it misses the estimate as well slightly lower headline inflation however higher rising 0.6 percent for the month that was only 0.2 percent in april that kept the year over year inflation at 6.3 percent just like it was in april march was the 6.6 number which was a record since january of 1982 consumer spending which accounts for nearly 70 percent of all economic activity in the u.s personal income rose 0.5 percent in may there's a lot of numbers out this morning man we're going to be it's going to be an interesting open personally come rise 0.5 percent in may ahead of the 0.4 estimate okay so personal income beats the estimates these are pretty decent numbers right now market really didn't react that much income after taxes and other charges or disposable income declined 0.1 percent on the month and 3.3 percent from a year ago so that is income after taxes and other charges or disposable personal income okay disposable personal income declined 0.1 percent on the month and down 3.3 percent from a year ago spending adjusted for inflation falling 0.4 percent a sharp drop from the 0.3 percent gain in april i mean there's a lot here we'll finish up the conversation okay goods inflation rose 9.6 services 4.7 personal savings rate still 5.4 percent lots of data we'll be right back folks we'll talk to our man kevin hanks we'll get his take on some of the action we'll be right back this to gold owns and operates the largest undeveloped gold project in australia the mount todd gold project this to gold just completed their feasibility study resulting in a 7 million ounce gold reserve this to gold has all major permits approved and has retained cibc capital market assistance in evaluating alternatives and in completing an accreted transaction this to gold trades on the nysc american and tsx under the ticker symbol vgc this to gold executing a strategy to create shareholder value tfnn has just launched their july 4th tiger dollar sale for one week only we've doubled all the bonuses where you can now get up to 20 30 or even a 40 bonus on your tiger dollar purchase tiger dollars are good on all tfnn newsletters webinars and trading services and they never expire for all the details and to get your tiger dollars before the sale ends 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negative 41 points that's just more than one percent the red nasdaq 100 pretty similar territory negative 135 points negative 1.1 percent right now let's jump over to our man kevin hanks every trading day folks 12 noon eastern time right here on tiger tv the td ameritrade network fast market your host kevin hanks tom white the team at td ameritrade network do an outstanding job of breaking down the day's market action folks they set up hypothetical trade setups you're always talking about defined risk for every single trade that they go through kevin hanks we got some economic data this morning good morning good morning cameo brian well i'll tell you i'll i'll start your viewers out with two things number one inflation is peak and number two we're about between three and four weeks away from confirming that we're in a recession tommy so when we get second quarter gdp in about well between three and four we three and a half weeks from now it's going to confirm that we're in a recession and now with this core pce number coming down personal consumption expenditures coming down uh i think inflation has peaked tommy so that's what this market and the tenure yield is start is trying to wrestle with right now now what's working against the market the u you know the u s dollar trading over 105 that's working against the u s against the stock market but i think this is going to get interesting tommy because stocks are down pre-open but we'll see how long that lasts when uh you when you've got a tenure now moving back towards and soon maybe under three percent tommy you beat me to it i was going to say we're going to get a two-handle today man we're back and i think it's like 3.03 right now uh 3.024 3.02 we'll call it uh on the tenure and remarkable we're right back to where we were kevin i think it was a week ago right is that what it is uh even over above that level almost now but yeah floating with three percent some pretty strong number on the s&p what do you think and we talked about a little bit yesterday we're in the pre-market action we'll find out as you said in about 10 minutes man when we hear that opening about what happens uh we did see some movement on those numbers but i agree with you man i mean they came in point one percent shy of the estimates on the headline of the core number um and it seems like those numbers pointing to actually a slowing of the consumer as well in those numbers we got but the market almost takes an stride kevin we saw 10 points maybe in the s&p but you were down 50 points so pretty marginal move on just a lot of data coming out at 8 30 do you think it's just waiting for the opening bell or i was kind of surprised to see that little movement with kind of the economic numbers we got almost in the same mentality that you were thinking about there well we'll see what when the market i don't know where the market goes uh you know i don't see i don't know yet how the market will consume and digest the information that it's getting but certainly yields are telling you something that inflation is not as scary as it was and you know tony i've been saying for a while and i've been wrong for a short time that i thought three percent on the tenure would be more stingy a level than it was because at one point it blew right through it real quickly but i'm not really surprised that it's been this level has been more sticky of a price than it was before so uh you know this is gonna be interesting how it plays out but you know crude oil is now is is soft again this morning but it's still coming it is still the u.s dollar that is kind of weighing on this market although the u.s dollar is uh flipping back back towards on changer even as we speak it will be interesting to see if uh we do get any pausing of that strength at all um maybe we get and i don't want to call you you know other currencies but man those other currencies in a week the dollar has been strong if we get a reversal in those fortunes uh interesting to see how that plays out with commodities plays on the gold contract and plays out with the general market as well i got a chart of the 10 year up here on the thinkorswim platform kevin pretty interesting you go back to on a price level basically the lows of 2018 and yeah we got quite a tail for the month of june you make it all the way to 1407 but it looks like we're going to finish out the month at about 118 which is interesting kevin just back above where we were on the lows of 2018 so yeah we got quite a shoot higher in a volatile market where you started the year off at about 130 not too surprising maybe you overshoot things a little by a few points uh we got some companies out with their numbers today we're coming into the long weekend kevin uh as traders do you see this thing trailing off i mean we got some big economic numbers today man um maybe you make it through part of the day and people start trying to sneak out for for cape cod and and the hamptons and the likes for the holiday weekend i you know i i i don't think that you know this was the last real significant economic data point that that we're getting this week this is what everyone was waiting for so it wouldn't be surprised to see it trade a little bit this morning and then really take it off tommy that that's the way i think the market's going to be um i think you know you know this is going to play out and it's going to be interesting but just it's always i always start by mentioning where we are on the calendar you always have to be cognizant i think vixx vixx being 29 and a half we'll see how long that holds up at that level uh there you know there's gravity on the vixx this time of year so the fact that we're near 30 is really defying gravity tommy for this time of the year and it shows the volatility that's out there so um yeah i expect values to lighten up here a little bit now into the end of the week and maybe midday you know and then people will start to turn off computers and move away for the weekend but remember we're going into a holiday weekend that's for sure and the vixx pretty remarkable man i got it up here get on the thinkorswim platform and you're talking about this year and i know you know this gave him for the listeners folks i mean you haven't seen a 17 handle on the vixx back to the week of january 10th which is amazing to think about kevin we're halfway through the year and we basically have a vixx that's been holding pretty much above 20 uh since the first week or two of the year absolutely extreme volatility and i would say the vixx has been dead on if if not a little light with the volatility this market has gotten on a continual basis uh we got some companies out with some action already this morning kevin we're coming into of course the holiday weekend do you guys have a lineup for what you're talking about at 12 today yet pretty good name to look at tesla Shopify and lows today so um i'll look at the home improvement space i'll look at online e-commerce and then of course why not we could do a tesla show every day line up is making some type of a new a new cycle even when he's quiet around his birthday so yeah we'll we'll trade tesla Shopify and lows today you guys had a great conversation going on yesterday about amazon man i enjoyed that one tesla another fan favorite Shopify we use Shopify kevin at tfnn as our platform outstanding platform um for small businesses but man quite a pullback from 176 to 33 dollars below where we came into folks the year 2020 which is remarkable when you think about online sales and what's happened to that industry completely well kevin we appreciate you taking the time man we appreciate the education on fast market every day we don't talk to you on friday so you have a great couple days have a great july 4th man and we will talk to you uh in the second half of the year coming up on july 4th July Tommy have a great week we can say hello to your dad i sure will man you have a great weekend weekend as well kevin folks tune in every trading day you heard it three gates straight three great stocks they'll be talking about today we still got two trading days left even though the holiday weekend approaching and yeah tesla man chopping around in the 600s right now 673 tesla gonna open down about 12 bucks well off the highs at 1243 and you heard Shopify man whoo you talk about uh a pullback from 176 to 33 dollars stay tuned folks we'll be right back for the opening bell at time of booming inflation we are purchasing powers eroded there's no better place to protect your harder and money than ain't gold this the gold's flagship asset is the monk cod gold project in the northern territory of australia this is australia's largest undeveloped gold project we are talking a world-class gold project in a tier one mining district this is a large-scale low-cost project with significant existing infrastructure in a politically safe and friendly mining jurisdiction this the 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we're above where we were at about four in the morning you're trading 11,575 the Dow is off 1% even at 30,674 and the Russell negative by 21 as a man basal Chapman says the day folks very young the day very young indeed crude backing off on that open crude sees a 108 handle yeah you just snuck into a 107 handle briefly for crude you get the gold contract excuse me spike in the 1825 you're at 1802 right now and we got the 10 year what are we talking about for a yield we're going to see a two-handle folks coming into the July 4th holiday where 3.02% hold on to your hats we could see it we'll see with the way this market is moving and with how fast it moves occasionally two basis points in the 10 year we usually get that type of action one way or the other over the span of 30 minutes to an hour we'll see where it moves okay what else we have going on folks head on over to the front page of TF and we just talked about gold you you trade gold folks you better understand what's happening with the dollar index you better understand what's happening with currencies and in this market as well okay you heard Kevin Hinks talking about it we have the tiger forex report folks okay as in Kevin Hinks talking about when this dollar when the bonds when that yield maybe pauses you don't need to trade forex to benefit from the information Teddy is putting in here every week folks he just kicked this thing off on Monday the tiger forex report he's got a weekly report out every Monday we've been talking to Teddy we talk to him every Wednesday at 40 past the hour you can always check out those interviews as well folks if you head on over to our youtube page just search TFNN you'll find our channel you can see all the videos we post we always break down the interviews themselves for their own video you can watch those segments very informative and he is kicking this thing off it's $97 a month folks as a launching promotion okay you can become an inaugural member you lock in 25% off forever that 25% savings stays with you forever as long as you subscribe I think it brings the total down to $72.75 I gotta pull it up exactly but you get 25% off the $97 price and you still get a 30-day money back guarantee so please check it out he put out a great report this past Monday he's got another one coming up when we get back from the July 4th holiday on Tuesday and he breaks down folks he puts the 30-year t-bond in there he puts the crude oil market in there he puts the dollar yen in there the dollar index the euro dollar all of these very important especially if you train commodities and like check it out and that special will only run for the month of July as it's running right now into the month of July so it's a one month launch you become a charter member to teddy service with the tiger 4x report and lock in those savings check it out and what else do we have going on folks the tiger dollar sale this one only runs through the weekend and it's Thursday okay it ends July 5th the tiger dollar sale that's Tuesday when we get back you can get up to a 40 bonus on your tiger dollars folks if you're out there listening and you're a subscriber to any of our newsletters please buy some tiger dollars apply them to your account and they're used for all forward transactions if you're thinking about signing up for anything check it out the front page that's only running a few more days so buy some tiger dollars maybe you check out the tiger 4x report when you do you can still get a 30-day money back guarantee folks you spend those tiger dollars you sign up for any service that you'll you want to try a tfnn we'll refund those tiger dollars if you're not happy after 30 days so check those out on the front page of tfnn two great promotions tiger dollar sale only running through this weekend and the tiger 4x report kicking off this week with our man teddy keg stat all right let's talk a little bit of inflation so the markets are singling uh that yeah we might have an inflation victory that's the opinion piece from john authors weakening economic growth has taken over as their biggest concern i would agree there that's a lot of what kevin was just talking about i had this article pulled up pre-show going over the markets thinking about what i was going to talk about maybe kevin read this article too i don't know if he did uh but he made some singular similar arguments not sure he was using the same data he was using some of the data out this morning now this article prior to the data we got this morning okay but it does have some important data in here i wanted to take a look at so first we talk about gdp taking a look at gdp we got the third quarter uh excuse me the third revision to gdp yesterday for the first quarter negative 1.6 percent when you put that on a chart folks okay there's your covet ridiculous pullback uh as you see we got a negative number and they don't happen often folks and we just got a 1.6 pay attention to it because look on this chart okay very seldom do they come out on negative territory and we are in one of those periods of time right now with huge exception of covid this was the weakest us growth since the spring of 2009 there's your acceleration that you saw there in the spring of 2009 so as mr author's rights in his opinion piece had the effect of seemingly eliminating concerns about inflation even though the battle against it has barely begun well said i like that it's we just got the numbers this morning right we're right near record highs now let's get into some of the data that he talks about though in here five year five year break evens okay the five year five year forward break even which aims to capture this is the part i'm reading right here check it out okay it aims to capture the average inflation for the five years starting five years hence and is the measure most closely followed by the fed has now dropped below its level for much of 2018 did you hear that below where we were at in 2018 before the pandemic fast approaching the two percent number the bond market is saying inflation is no longer anything to worry about check out that pullback folks okay the problem is an inflation forecast have shifted because growth has shifted now you get into when the fed may start easing this is an important one folks because man time flies it's already june 30th tomorrow is july before you know it we're going to be forward some of those rate hikes that are already priced into this market rightfully so okay the easing cycle if you believe the fed funds futures market this is great stuff so pay attention will be well advanced by the end of next year did you hear that the easing cycle will be well advanced by the end of next year the market likes to get ahead of hiking cycles and easing cycles if the easing cycle is going to be well advanced by the end of next year when's the market going to start pricing that in not today but at some point it will the chart shows the number of 25 basis point hikes implicitly priced in for each fed meeting until early 2024 over the last week the belief in a swift start to easing has gained hold and that sadly is because of the economy that might need some easing unfortunately for the recession hit that it is possibly about to take so we have the blue line which is what the market was thinking a week ago june 24th and we have the black line which is the number of hikes that are priced in now what you have is you have hikes coming through about april but then check it out folks over the next nine months you have some hikes but then what do you have you have almost a year of easing that pulls things back more than two percentage points right now we're talking about eight hikes into march of 2023 and meanwhile you're back to five and a half so that would be what two and a half quarter points so that's about 50 to 75 basis points of easing coming at you next year that happens things change folks okay we got a couple other things we'll talk about in this article coming back dollar index paying into that one as well commodity currency is very important of what's going on right now folks that's why I keep talking about that tiger forex report stay tuned we'll be right back are you in the market for buying or selling real estate in the bay area including the surrounding st petersburg tampa and clearwater markets tiger real estate llc is a firm that has extensive experience in the tampa bay area whether you're looking to sell your current property for maximum value or you're in the market for a second home or investment property tiger realty has the experience across all areas of real estate in the tampa bay area to help buyers and sellers make the most informed decisions across all price levels from the price you should be paying per square foot in certain up and coming areas to the type of cash flow investment properties are capable of creating tiger real estate can help you make the best decision when it comes 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designed to be utilized only by sophisticated investors such as traders and active investors distributor four side fund services llc this program is brought to you by vista gold traded on the nyse american ntsx under the symbol vgz welcome back folks we got a little bit of a sell-off going on right now talk about a drop on the open the s and p is now negative 1.7 percent just like that nasdaq 100 negative 2 percent man dow negative 535 that's 1.75 percent the russell off 2 percent as well bitcoin watch out folks bitcoin an 18 thousand handle coming into the long weekend always a dicey territory on the weekend man we saw that acceleration down to 17 thousand briefly and checking out the chart if you're talking about an area of support and resistance we're blowing through it folks this is the area that future starting trading at if you don't think that's an important area i would disagree with you that is the moment the futures began tradable on the cme december of 2017 five years ago almost remarkable that it's been five years you trade up to 70 000 we are now breaking below those levels in terms of 20 650 is the high there we're at 18 000 check out those last three months folks 47 000 was the print on bitcoin just in the month of april we're still in june and we are finishing basically at lows 18 903 how about those three candles now the original run on bitcoin goes from about 70 000 we make three huge candles as well down to a low of 32 000 okay so that is what 37 000 about yes 37 000 was the a to b leg you take it off the high of 48 000 folks that is an a to b that brings you down to about 11 000 10 000 is a nice round number that's where you kick things off in october and maybe that's where finally things begin to capitulate a bit at about 10 000 i mean you see an area that we chopped around in for the better part of 2019 and 2020 yeah we got a wide range there down to 4200 up to a high of 14 000 in 2019 but all things considered you broke below this 33 30 000 level i was talking about it when it was at 30 folks i wouldn't be touching until it gets to 10 you know maybe you get back above 30 and you start building support and you can trade it again but you're in no man's land here and now that you're below where this thing started trading in futures be careful out there folks be careful in a big way excuse me okay let's jump around to some of the other equities that are moving this morning um yeah and talking about finishing up this article excuse me they're talking about the yin they're talking about the dollar they're talking about the ecb in here uh continued dollar strength the key component to this is the yin paul's remarks can be taken to show that he wouldn't be happy for the japanese authorities to intervene and strengthen their currency and it would certainly be difficult for any major central bank to do so such a thing without coordinating with others uh there's a lot at stake here folks but they're talking about look at the economy look at the potential for potential fed easing uh and inflation yeah we're right in the middle of things but that's not what the market is looking at right now folks the market is looking at a print of 1.6 percent declining gdp for the first quarter okay let's jump down some of the stocks moving uh restoration hardware they were trading down about seven eight percent pre-market they lower their guidance let's see how they're trading right now probably not helped by a market selling off yeah you're off 10.7 percent some of these equities man whoo uh pre-covid you're at about where we're at right now you come into 2020 at about 215 you're trading right now at 212 quite a monthly decline and there's your acceleration so they come out with a revision last night of their forecast they drop on the open as well you're down to 211 right now in restoration they lower the outlook for the year anticipating community consumer demand for its products will continue to soften in the back half of the year and i believe i saw in here let's see the uh yeah because the way they summed it up here i think was deteriorating economy and a slowdown in home sales that's not going to be just them folks um if they're talking about economic indicators and a slowdown in home sales always interesting though when a company sort tries to say hey it's not us it's the economy right sometimes it's going to be the company and they're going to be blaming the economy this situation i'm not sure that's sure that's the case with the numbers that we're getting with the declining gdp uh this is what to be careful of folks some great conversations in the den yesterday talking about multiples right this market can become very expensive again if we start to get revisions on earnings those are the types of revisions right there we got restoration hardware dropping 10 over the night after they lower the full year financial guidance be careful all right what else do we have happening constellation they beat estimates by 14 cents 266 a share revenue above estimates upbeat full year forecast yeah they say they added one percent when i pulled it up this morning it was actually a little bit lower yeah and went down four percent by the time i pulled it up they were actually lower pre-market on those numbers now you got the market trading lower this one's been one of the strongest equities out there this year i was talking to kevin hicks about it yesterday i mean look at this chart all you've been doing is chopping around in a consolidation area between about 211 and 240 if you're looking to get into constellation uh maybe this market trades a little lower it takes constellation with it you see the consolidation it's been in about 210 212 somewhere in there if it gets to that level maybe that's an area you could start to nibble on constellation still though 20 bucks from where we're at right now strong numbers and they're getting punished man in a big way as this market is down about almost two percent of itself constellation down about four percent always worrying when you get something like that right as in you get beaten by four 14 cents a share revenue above estimates upbeat for your forecast they're a little bit higher pre-market and they sell off on the open be careful and walgreens boots out with their numbers they were a little bit lower as well let's see how they opened up down four percent pretty similar action on pretty strong numbers as well they come out with the numbers this morning you sell off a little bit you make a new low on the open walgreens boots 96 cents a share four cents above estimates revenue beat as well they reaffirm the full year guidance these are not misses folks okay not in this environment at all they beat on revenue they beat on earnings and they reaffirm their full year guidance with single digit low single digit adjusted earnings growth and the market's like four percent we're taking it from you this morning you should have done better it's a dicey market man because that's pretty decent numbers when you talk about everything going on and the surprises that we've seen in this market all right what else we got going on how about amazon they they did do a great segment yesterday folks on amazon uh this is talking to me yeah sure is they did a great segment yesterday on fast market at 12 on amazon not talking about this okay and i do have some amazon retirement account folks there was one analyst that came out yesterday talking about that amazon web services is going to have a three trillion dollar valuation itself at one point the kicker of that one was that they didn't say when well i could probably say the same thing folks amazon's going nowhere eventually it'll be worth three trillion dollars eventually aws will be worth three trillion dollars uh i'm not sure when though that's the kicker nonetheless i am a bull on amazon in the long term especially when you get these types of pullbacks and this is the part of reason why you know they can spend money on some of these content deals and this is why you've seen netflix pullback so sharply this has been one of the hampers to disney in terms of the money and the profitability now amazon just spent i think was one point six billion dollars let's see as they scoop up uh uefa deal for coverage in the uk yes so let's see the value may be about 20 percent higher than the previous cycle and the number is not out there but that's up from about 1.4 billion euros 1.5 billion dollars so you're talking about maybe 1.7 billion euros so 1.8 billion us dollars now they had just pulled out of the competition for the indian premier league which ended up going to disney and one other company over there i forget the reliance was it one of those but those went for like six point seven billion dollars okay amazon though they're in the business man and sports content there's going to be nothing like sports content for some of these streaming companies folks and they're going to have to pay for it because the sports know it stay tuned folks markets we're basically at session those s and p's off 1.7 percent we'll be right back sharpening your skills as an investor is like getting better at playing a musical instrument you have to practice sure but you also need excellent instruction from experts at tfnn you'll get advice and guidance from the authority and technical market analysis and it's not just dry tedious text either tfnn airs live financial content streamed live on tfnn.com and tfnn's youtube channel with tiger tv live every market day from 8 30 a.m. to 4 p.m. eastern for free each host is an experienced trader and gives their take on the market while taking calls and questions live from around the world from the moment the market opens until the closing bell sounds tiger tv has eight different shows with expert hosts to help you make the right moves with your money watch online at tfnn.com or on tfnn's youtube channel and become the 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think or swim for more information just click the think or swim banner on the front page of tfnn.com welcome back folks you got the markets basically a session lows right now folks interesting coming into a long weekend right now it's thursday as kevin hanks was talking about that was the last big data drop we get oil numbers at 10 30 crude oil inventory numbers at 10 30 out but we're basically past the last economic number and not a good indicator when you get the s&p's down 70 points not sure it's going to be a cascade to disaster territory but you're talking about an s&p now off 200 points from where you were trading at just yesterday let's jump around to some of the fang stocks man we're talking about amazon you're now under 104 that's but back to back 5% days pretty much for amazon you were just trading at 114 on tuesday let alone the 118 high over the weekend coming into the monday open when we traded up to 118 percentage wise just a huge pullback right now recent lows you're right there man at 10126 is the low just recently we were down there at about 10143 what is the exact 10143 i believe amazon trading lower we jumped some of the other stocks tesla there's an article up there on bloomberg about tesla talking about a wipeout of i think 350 billion dollars in market cap this quarter alone tesla negative 1.8 percent today s&p's right now approaching that level you get the nasdaq look at this nasdaq's off 2.4 percent s&p's off 1.8 percent we jump over to the big dog apple down 3 percent for apple shares you jump to microsoft shares down about 2 percent at 255 the low for microsoft 241 yeah this is getting a little dicey folks s&p's off 70 dow off 560 right now let's just see our curiosity again as i was talking about that comp nasdaq composite low 10565 remember the 382 of the entire move higher folks from the 2009 lows to the covid highs a 382 retracement 10500 we're 400 points away from that you're down 275 today you know i talked about it before we'll wrap up the program as we come into it folks put those numbers on your radar and make sure that you're at least prepared if the market goes to that level that you're okay because there's nothing stopping this s&p from hitting 3200 folks you got the 382 of the larger move the 618 of the covid acceleration from 2100 to 4800 and the nasdaq comps basically there so nothing stopping the other indices from getting there as well all right folks thanks for starting your day shipping interesting ones stay tuned don't forget about the tiger dollar sale running this week only folks dazzles up next have a great thursday


Aaron Judge with Meredith Moragovitz in front of the Yankee dugout. Thanks, Michael. Aaron, you just said to me there was never a doubt. What gives this team so much confidence that you're going to find a way to get it done?  There's never a doubt, never a doubt. You know, we were chirping a little back and forth early on. I think that kind of locked us in. You know, Kikuchi had our number for a while, but, you know, we just wanted to lock it in there at the end. Can you describe the emotion you feel when you make contact in that moment and know that ball is going out? I was a little nervous. I hit a couple of balls good and they can't get out. But, you know, I was just trying to put a good swing on it. The guys in front of me had great at bats. They made my job easy to try to square one up. Seems like you guys have been battling a little bit to start the season here with the Blue Jays. It looks like it will be a battle between you guys all season long. What makes that Blue Jays team so tough? They're an impressive team. They got a great pitching staff. They got a great lineup. They can hit the ball out of the ballpark.  That's what we want to do. You know, it's a good team to play against. Is there a better feeling than a walk-off at Yankee Stadium? Oh, this is pretty sweet right here.


hey what's up you guys welcome back to my channel if you're new here hi hello i'm lydia and if you are new here make sure you hit the subscribe button join the growing family as you can see i'm not in my flat i'm not my grandparents where the fuck am i i am on holiday with my grandparents or with my mom as well well i say holiday we're only staying till sunday night and it's currently friday we're staying for easter so what's this video about i thought i'd do a bit of an update because i haven't done a proper update in a long time so where do we start let's start with university i have my graduation date which i'm excited about because i have to graduate i'm getting the map book pro next week which i'm excited about and i'm a pro in about five years so i'll be very happy to get back to having a pro mental health now here's what we need to talk i'm currently really struggling with my eating disorder and it's honestly one of the things i struggle with the most because when i'm around my family i have to eat all right next week when i go back to my flat i don't have to eat and i've been really restricting my intake i've been nearly drinking diet coke but it's just become a real struggle living situation wise i don't know what's going on i want to move out of supported living i don't like being in supported living i don't like constantly being checked up on i don't like having my medication stored in a medication room i honestly just want it out of there so what i'm doing is i'm gonna be saving 300 pound of my pier every month and i'm saving the remainder of my student finance after i buy my laptop the laptop's costing me nearly three nearly a thousand six hundred pounds it's a lot of money but it's a worth it's getting three chip laptop it's got an sd port and sd card in port so i don't have to use my connector it's got hdmi cable so i can do stuff if i forget a tv but my long-term goal was to move closer to my family and i did actually ask my mom the other day if i could move back in with her because they're moving to my mom's gonna be running a pub and they have four bedrooms and my mom's making me a bedroom before i want to go stay with her i asked if i could move back in and honestly it was kind of upsetting my mom said it's not ideal because of my past which she's comparing me now to when i was 16 18 i haven't lived with her since i was 18 compared to that it's quite upsetting because i really have changed in that time so now i'm looking for houses in telford and they're about 750 a month but yeah things are a bit up in the air for me at the minute because i'm not happy in support living and there's nothing i can do about it i want to move after i've graduated i want to graduate first i am a bit hurt by my mom that i can't live with her because i thought she'd really like she'd understand that i've changed and not the same as i used to be i don't self have anymore i don't lock myself in bathrooms after taking overdoses i don't steal knives to cut myself with i don't steal money after them anymore i have my own money coming in because all the negative things that i used to do i don't do anymore yes i still have an eating disorder but i can manage that i just don't know what to say anymore so it has been a rough ride really but honestly i feel i feel okay with myself i'm not i know it's a video i was suicidal but i'm not actively suicidal so please don't worry about me i'm fine i'm not gonna attempt my life i just wish that there was something i could do to get back in time and not do all the things i used to do and it's having friends around me because all my friends just when i have a lana who's in australia i have my friend aron who's in telford speaking of a lana i'm ordering her a christmas a bit not christmas a birthday gift for the end of april and i really hope she likes what i send her it's something she mentioned on call to me that she wanted to get so i hope for them the right thing i also bought a new squishmallow today it's a new it's a big thought but i haven't seen it before so i had to get it my money is well spent that's the only thing i was treating myself to this month but yeah i think that's kind of all i've got for an update but i hope you guys understand that when i say every subscribed matters it really does for me every every view i get means the world to me i started uploading twice a week mondays and fridays at seven p.m and honestly it's going quite well my videos were really underperforming when i was uploading every day now they're doing a bit better they're going over 100 so that's good i'm getting comments on all my videos which is great for engagement if you want to support my channel because you like what i do and want bonus content head over to my patreon which is linked up there and when i upload patreon i upload i'm uploading eating disorder vlogs at the minute because i can't post them on youtube because they get taken down so facing my eating disorder is on my channel but anyway thank you for watching and if you are a new subscriber join the growing family thanks for watching and peace


Stevenson win tonight 8-3 over Lawrence of Christian Taylor head coach Shamanic Dawes coach another great performance how pleased are you with your team? No good I mean it's scoring some goals that's you know something that we lacked early in the year and you know some guys that haven't been given too much of an opportunity got an opportunity tonight and and played well and kind of that next man up mentality so it was a good I mean I don't know I just said to them in the room I don't know if we've swept two weekends in a row before so that's a big that's a big thing and we need to kind of keep going and build off that we got a big weekend ahead of us. A couple penalties early on but you scored shorthanded how important were those goals? Those goals are big early because that was how we talked about being disciplined and we did we took a couple penalties and but to get out get out ahead of them is good and we had a big five on three kill early which which obviously helps. So this offense I mean it's been great what's been going so well for them how they've been able to score all these goals. I think they stuck with it kind of through the first couple weeks of having so many opportunities and we outshot a few teams like so heavily and we're just weren't pucks weren't going in and we were in the right spots and I think it's just coming from from that I mean just being in the right place and and and being on team. Goates thank you very much congrats on another win. Thank you. For head coach Tom McDaws I'm Christian Taylor on Gomez Thanks Sports.com


Every night my dad was in charge of putting me to bed. He would help me brush my teeth and then would tuck me into my nice warm bed. Then he would tell me a story. Every night it was the same story, the story of Bosco and Billy. I had lots of books in my room as my mother would read to me throughout the day. But dad would never read any of those books. He would always tell me the same story. I always assumed that either the story had come from a book, but my father had told it so many times that he didn't need to read from the book anymore, or that it was a story he was told as a child. I always enjoyed the story of Bosco and Billy. I always found that it would help me go to sleep. Usually, I'd be almost asleep by the end of the story. I slowly learned all of the words to the story and would sometimes recite it along with my dad. For some reason, probably because I heard it every night for years, the story stuck with me. And even to this day, I still remember all the words to it. The story, in its entirety, is as follows. Billy the bear was an ordinary bear with an ordinary life. But he had a small friend named Bosco who got him into strife. Most of the time Bosco would behave, but sometimes he acted out. Billy tried to keep him in line, but sometimes he would break out. Like one time, Bosco was hungry and decided he would steal food. Billy tried to stop him and explain to Bosco that stealing is rude. But Bosco didn't listen. He stole the food anyway and took it to his secret spot and hid it away. Bosco took the food to the large oak tree that was in the woods. There he stuffed the food inside and that's where he hid his goods. Billy never found the stolen food. It's still hidden there to this day. Bosco wishes that it is never found. He hopes and he prays. Billy is kind and Bosco is a little bit silly. And that is the story of Bosco and Billy. Now the story seems innocent enough and I always thought it was a cute little story about two bears, but after recent events I know now it is a lot more sinister than it first appears. My dad passed away a few years ago and it took a toll on all of us, my mom especially. She began to be a little bit reclusive after my father passed and so I didn't see her as often because she shut herself off from others. I spent a long time trying to create a relationship with my mother again and slowly over the course of a couple years I did manage to reconnect with her. Things weren't as they once were, but she still did invite me over once a week for a cup of coffee. During these hour long visits we would sit in the kitchen and talk about times gone by, about my childhood, about my time at school, and about all of the good times we'd had together. The one subject that didn't come up, however, was my father. I think the pain was still too raw for my mom to talk about. On one of the weekly visits though, I thought that I would try and mention my father as I felt that it might do mom some good to talk about him.  You remember that story dad used to tell me before bed? The story of Bosco and Billy? I hesitantly asked, not knowing how mom would react to the mention of her late husband. Mom looked up from her cup of coffee that she was drinking. She looked confused, maybe because it was the first time he'd been mentioned for a number of years, or maybe because she didn't know what I was talking about. No, no, I don't remember that story. She replied, you had a lot of books. I can't remember all of them. I was a bit surprised by this. Dad told it to me every night, so I would have thought that she'd remember it. I then went on to explain that it wasn't a book. It was a story that he'd recite without reading it off a page because he knew it so well. I thought your father would read you a different book each night. She said, sounding confused. That's what he told me anyway. I thought that this was especially strange because he never read me one of my books, and I would have thought that my own mother would have known this. What was the story about? She asked me. It was about two bears, one named Billy and one named Bosco. I was about to continue to explain the story when my mom suddenly interrupted me. Did you just say the bear was named Bosco? She asked me, sounding quite serious. Yeah, that's right. His name was Bosco. I replied. That's strange, mom said. She had a look on her face that looked like she was trying to remember something. Your father used to tell me about a friend that he used to have named Bosco. He used to say that Bosco would get him in a lot of trouble when he was younger. I never met this Bosco, which your dad said was a good thing. He also said that he only met up with Bosco only once or twice after we met. And then your father told me that he was no longer going to see Bosco. He said that he finally got rid of him. I was always curious as to who he was. I've never seen any photos of him or any evidence that he existed now that I think about it.  Mom stopped talking for a second, and I didn't know how to respond. She then looked at me and said, Do you remember this story? Can you tell me it? I recited the entire story to her. She listened to the whole tale without changing her expression. She took it all in and then she asked, An oak tree. Are you sure that Bosco hid the food in an old oak tree in the woods? Yeah, it was definitely an old oak tree. Why is that? I asked her, sensing that something wasn't quite right. It's just that on mine and your father's first date, he took me to an old oak tree that was inside the Shelton Woods, only a 10 minute drive from here. She responded. He told me then that it was lucky for me that Bosco hadn't decided to interrupt their date. She then stopped talking and stared at the wall behind where I was sitting. I think it's time for you to go now. I'm sorry. She told me a hint of sadness in her voice. I thought that maybe it was for the best I did leave. I got up and gave her a small hug, which she didn't even seem to notice.  I got into my car and began to drive home. On the way home, however, the thought of the old oak tree and Bosco began to fill my head. I couldn't escape these thoughts. I didn't know what it all meant, if it meant anything at all. I don't know what compelled me, but I had to visit the Shelton Woods and find this oak tree. I turned my car around and headed in the direction of the woods. Fifteen minutes later, I found myself standing at the edge of the woods. I took a step forward and entered. I didn't know exactly where this oak tree was, but I was determined to find it. I explored the woods for around ten minutes before I saw it. Standing in front of me was a huge oak tree, its branches extending high up into the sky. I walked over to the base of the tree, a sense of fear starting to overcome me. Something didn't feel right. It felt like something terrible had happened here. I looked down at the ground, at the base of the tree, and I saw that the dirt there looked slightly different than the rest of the dirt in the woods. I don't know why I did this, but I bent down and I began to dig. The police say that if I hadn't started digging, then they probably never would have found them. There were four bodies in total. Full young women around the age of 24, all of them had been dead for over 30 years. It was undeniable, they were all murdered and buried under this tree. The DNA testing was also clear. My father was responsible for the killings. Somehow, he managed to keep all of this hidden for many years. He never told anyone. The only clues that he gave were through my bedtime story, which I think was his way of confessing. The only thing I'm not sure of now is who is Bosco. Maybe my father and Bosco are the same person, or maybe he really did have a friend with that name. I don't think we'll ever know. I don't think I want to know.


One of the questions that arose in our studies was what happens if you have a degenerate system namely one in which the linearized matrix is has a zero Eigen value. Well there are several kinds of degenerate systems but let us look at a specific example the simplest of these. So let us look at one for which x dot is perhaps A x plus B y and y dot equal to some constant times the same thing A x plus B y. So of course this immediately implies that L is A B K A K B. So delta equal to determinant L is identically equal to zero then the question is what kind of critical points you get here and the answer is very simple you have a degenerate system these two equations are not linearly independent of each other is just a multiple of that and therefore in the x y plane if you plot the locus of the points on which the right hand side is a zero you get A x plus B y equal to zero or x or y equal to minus B a over B x which perhaps is some straight line of this kind and at every point on this line critical line if you like the system is in equilibrium no time change at all then of course you could ask what is the flow like what happens if you have an initial condition which perhaps starts here this would depend on what the signs of these constants are but in general the flow would be either along this line inwards everywhere into this line or perhaps outwards all the arrows going outwards. So this is a very simple example of a degenerate system it is not of particular interest to us right now more serious would be what happens if the system is not linearizable intrinsically the example that we look at is even in a one degree of freedom system for example just a single variable if you had something like x dot equal to x squared and you ask what does this do what kind of critical points do you have in this situation well we have a phase line there is a critical point at zero and we cannot tell whether it is stable or unstable in the conventional sense because for positive x the flow is outwards but for negative x also x squared is positive and therefore the flow is inwards this direction therefore this is as far as the left hand side is concerned it is an attractor but as far as the right hand side is concerned it is a repeller it is a higher order critical point now the question is can we say something else about it and how does this higher order critical point arrive well it occurs because x squared is not a generic polynomial on the right hand side if I ask you to write down a polynomial on the right hand side or something which is expandable in powers of x about the origin you would start by writing a naught plus a 1 x plus a 2 x squared and so on and so forth and unless you have a special accident so that a naught and a 1 about 0 you never start with the x squared term on the other hand if I put a constant here just a single constant you can always absorb that constant by shifting x to that point so the constant is irrelevant but certainly the linear term is missing so the sensible way to do this would be to ask instead of looking at this system directly unfold this second order 0 if you like this double 0 by writing this as equal to some epsilon x plus x squared and examine what happens for various kinds of epsilon small positive small negative or 0 epsilon equal to 0 corresponds to the case you have but epsilon positive or negative would be a generic system and you could linearize it in the origin of these points well this instance for example there is a critical point at the origin and there is also one at minus epsilon so there is one at minus epsilon out here in the instance in which epsilon is positive then of course minus epsilon is located here and you could ask what kind of flow do you have once again near the origin near the origin here in the vicinity of the origin I could write x dot is approximately equal to epsilon x which means if epsilon is positive the flow is outwards along the direction and outwards along this direction here by continuity therefore it remains in this direction all the way up to this critical point and now you ask what kind of critical point is this for that you have to linearize about this point so the sensible thing to do is to put in this neighborhood is to put u equal to x plus epsilon so that the origin in you is the point x equal to minus epsilon and then ask what happens to this system well that system now becomes you dot equal to u multiplied by x because this is x plus epsilon times x but x is u minus epsilon which is equal to minus epsilon u plus u squared once again if you linearize in the neighborhood of this point then the linearized equation is linearize its u dot is approximately equal to minus epsilon u and since epsilon is taken to be positive it is flowing in into this attracting fixed point here and it is flowing in this way the other direction so you have an attractor at minus epsilon and a repeller at the origin and the coalescence of these two when epsilon is 0 produces for you this figure this higher order critical point and this picture is valid for epsilon greater than 0 this picture therefore corresponds to epsilon equal to 0 and one could ask what happens if I have chosen epsilon to be negative to start with that is easily taken care of because if epsilon is negative the fixed points are at the critical points are at 0 once again and at the point minus epsilon which is now on the positive side since epsilon is negative and it is easy to see that linearization about the origin is always x dot is epsilon x and since epsilon is now negative the flow is inwards into the origin on both sides in its neighborhood and it is a trivial matter to check once again that the flow is outwards from this point at the point u equal to 0 or x equal to minus epsilon because epsilon is negative this number becomes positive here minus epsilon and it is a repeller so what happens effectively is that this point becomes a repeller and this becomes an attractor so now we get a much better picture it says that the system x dot is equal to x square which is intrinsically non-linear really arises by an accident it arises by the coalescence of two critical points one at the origin and one at minus epsilon as the second critical point minus epsilon crosses 0 as it crosses the origin as it crosses the value 0 you get what was initially to start with starting with positive values of epsilon and attractor at minus epsilon and a repeller at the origin becomes a higher order critical point when two of them coincide and then as you move over to negative values of epsilon you have an repeller at minus epsilon and an attractor at the origin so there's been an exchange of stability and this is called an exchange of stability bifurcation it's an honest bifurcation one of the simplest one can think of and we draw what's called a bifurcation diagram in which I would plot as a function of the parameter epsilon at plot the equilibrium or stationary value or steady value of x in this case the value at the critical point but there are two critical points one of them is always at the origin and therefore you have a line line on this axis and the other one is at the point is at the value minus epsilon and that is a straight line which is tilted at 45 degrees and of this kind and this if you like is x equilibrium on the other hand I know that for negative values of epsilon this critical point is unstable and therefore I denote that in standard notation by a dotted line as a dotted line here for negative values of epsilon it's unstable and therefore I have a dotted line of this kind and a solid line here to indicate that this is a stable critical point so this is unstable and this fixed point is stable when you cross over to positive values of epsilon what was stable becomes unstable in this fashion and what was unstable continues on as stable so this is stable and this is unstable and we have an example of a bifurcation which in this case is called an exchange of stability bifurcation we'd like to now ask a slightly more general question what kind of bifurcation can we expect in such systems this was a simple one dimensional system but it can be generalized to higher dimensions and the question is what kind of elementary bifurcation or distinct kinds of bifurcation do we have this problem too has been analyzed in great detail and bifurcations have been classified at least the elementary ones have been classified in simple dynamical systems I should mention that bifurcations which involve a single parameter which you tune in this case just epsilon they call bifurcations of co-dimension one and this is distinct this dimensionality is distinct from the actual dimensionality of the phase space that we are dealing with this is in the parameter space and there is just a single parameter here so it's a bifurcation of co-dimension one in this instance we'll see later that we have bifurcations of higher co-dimensions and classifying them is a non trivial task classifying bifurcations in general in higher dimensional dynamical systems the fairly non trivial task now I'd like to put this in put this in the framework of a slightly more general setting and that's as follows so I'd like to look at it as two dimensional phase space but just to get a physical feel for what is meant by tuning this epsilon let's cast it in the language of a mechanical example so what I intend to do is to consider a particle of unit mass moving along the x direction and look at these dynamical equations as equations of motion for this particle in some potential v of x so the equations of motion are x dot equal to p if I consider unit mass this is momentum here and p dot equal to minus dv over dx which is the force on the particle and I'm going to tune this by writing different kinds of functional forms for this v of x and asking if I can examine bifurcations in this framework here now what would you say is the simplest of these forms that you could write down here typically I'd write down various kinds of polynomials for this force here and the simplest of these that one could write down is perhaps to say that this is a constant plus a linear term in x but it's quite clear that if you have an a plus bx here the only critical point is at p equal to zero and x equal to minus a over b and that's it there's just a single critical point and there's no possibility of any bifurcation no coalescence of singularities the next non-trivial case would correspond to putting in here some parameter and let's various symbols could be used let's use a and let's put a bx squared what kind of potential does this involve what is the shape of v of x in this case if I plot here x versus v of x remember this is the force it's minus dv over dx so this would imply that v of x itself equal to minus ax plus bx cube over 3 minus if I integrate this and change the sign I end up with this what kind of shape is that it's a cubic curve clearly we'd like to plot it and we should like to know whether a is positive or negative or what let's fix the sign of b let's take b to be positive for instance you could simply redo the whole thing for b negative there will be no essential change in what I'm about to say so let's suppose that b is always greater than 0 and a could be negative or 0 or positive let's look at all possible cases now what happens if a is negative what's the shape of this curve so I plot this for a negative and the shape of this graph is approximately linear at the origin with a positive slope and therefore the potential looks like this here but eventually this term takes over that's a large negative term so it's evident that this is going to go down and fall down on the other hand when x is large negative this term dominates and becomes positive in sign and therefore the potential has a shape of this kind so one knows immediately that the critical points of the system occur at p equal to 0 and this quantity equal to 0 and what does that look like what would this do well it's clear that there is an equilibrium point here at this point and that's a saddle point because it's a maximum of the potential and this point here is a minimum of the potential and therefore this point is a center since there's no friction in this problem you would have small oscillations about that point about the minimum of the potential whereas the maximum of a potential in the absence of any dissipation is always a saddle point this is all that a Hamiltonian system could have we have here a Hamiltonian system in which the Hamiltonian is p squared over 2 for unit mass plus v of x and these are Hamilton's equations that I have written down and this is the picture that we have so we clearly have critical points at p equal to 0 and the values here correspond to x equal to minus a over b square root of this with a plus or minus and this is the picture for a less than 0 remember the plus corresponds to the minus this point here corresponds to a center and the plus corresponds to a saddle you can write down the 2 by 2 linearized matrix and check out at each point that this would correspond this center would correspond to pure imaginary pair of eigenvalues and this would correspond to one positive and one negative eigenvalue if you linearize about these values of x we know how to do that now what happens if a is exactly equal to 0 once again if I plot at a equal to 0 I plot the potential v of x versus x what would this correspond to this term is gone and you just have a minus b x cubed and that at the origin is extremely flat it has an inflection point and it falls off in this fashion again for b positive so this point here has arisen because this maximum and this minimum of the potential have come together at the origin and it's become an inflection point where the slope is 0 in the first second derivative the curvature is also 0 at this point and now finally what happens when a is bigger than 0 so we look at the picture a greater than 0 that's the third case and if I plot x versus v of x then when a is positive you have a negative slope here and therefore this curve looks like this approximately linear and then of course as x becomes larger it falls off like a cube and it increases here like a cube there's no possibility of any equilibrium point at all in this potential there's no maximum or minimum at all there are no critical points in this dynamical system because if a and b both have the same sign there's no way this quantity can vanish so that immediately tells us that you don't have any critical points in that system you have a degenerate critical point in this system and you have here two critical points which are separated out therefore if I now plot the equilibrium values let's plot x equilibrium here versus a parameter and the critical the parameter in which you have this variation is in fact a to plot this I need one more direction I need P as well coming out of the plane of the board but since P is always 0 at the critical point I ignore this P it's always 0 so let's just plot it in a two dimensional diagram here x equilibrium versus a and what's the picture one has for a positive nothing for a negative you end up with a saddle point at x equal to plus square root of minus a over b and a center which is stable at minus the same value and this is changing or increasing in magnitude like the square root of minus a therefore it's a parabola which goes up in this fashion and falls down in this fashion however we know that this root here which corresponds to plus square root of minus b over a is unstable and this is stable so this branch which is minus square root of minus a over b is stable this branch here which corresponds to plus square root of minus a over b is unstable we could have had this parabola looking the other way had I put a plus ax here then the role of minus a and a plus a would just get interchanged but what's happening here clearly is that if you imagine changing a in parameter space from positive values to negative values no critical points at all in this region and all of a sudden at this point a pair of critical points gets created and what's happening is that if you start with this potential and start flattening it out then at the value a equal to 0 you have a cubic you have a second order 0 here you have an inflection point here execute and then below that the inflection point unfolds into a maximum and a minimum and you have this shape here and of course you go on changing a making it more and more negative these points will move out further exactly like a square root of minus a and this is what happens here this bifurcation where out of nowhere a stable and an unstable critical point emerge and move off is called a saddle node bifurcation in this case at the value a equal to 0 you see immediately that this is different from the bifurcation we looked at the exchange of stability bifurcation altogether different so a saddle node bifurcation is one where as you go across a critical value of the bifurcation parameter a pair of critical points is created typically one of which is stable and the other is unstable so much for this simple form we could make this a little more complicated let's do that in the next step is to take this potential and ask what happens if it's a x plus b x square this would correspond of course to v of x equal to incidentally the signs of these quantities a and b have taken them to be arbitrary here it doesn't matter which way these bifurcation diagrams look the physics is essentially the same thing in all cases now what happens to the potential it's minus a x squared over 2 minus b x cube over 3 if I integrate this once again we can predict what's going to happen we start again with a less than 0 a equal to 0 and finally a greater than 0 and plot in all cases the potential v of x as a function of x out here for a negative minus a x squared over 2 is an upward looking parabola because it dominates for sufficiently small x therefore this curve is going to look like this but then once x becomes sufficiently large this negative term is going to dominate and bring this potential down on the negative side this always remains negative and for x negative this number is also is positive so what happens there he seemed to have made a mistake here so let's keep let's keep be positive here this term is going to dominate so it's positive yeah there's no problem this term becomes positive yeah so that's fine this goes up here because eventually for large negative x this term dominates over this minus x cubed is negative and minus x cubed is positive so it goes up in this fashion this is fine this is fine when a is 0 exactly 0 then it's just a cubic curve exactly as I drew earlier minus x cubed which has an inflection point here goes down in this fashion and for a positive you have a downward parabola here and of course for large negative positive x this is going to become a large negative quantity go down there but then it has to eventually turn back and go off in this fashion now it's easy to see what's going to happen at a equal to 0 you have a higher order critical point here for a negative you have a maximum of the potential which corresponds to a saddle point you have a minimum which corresponds to a center at the origin on the other hand for a positive the origin corresponds to a saddle point and the minimum which occurs here corresponds to a center so what has happened what kind of bifurcation is this and where is this point this point mind you is not at x equal to 0 but it's equal to at minus a over b this point also that minus a over b it's as if the diagram has moved but what's happened is that the critical point at p equal to 0 and x equal to 0 which was initially a center has now become a saddle and the critical point at p equal to 0 and x equal to minus a over b which was a saddle point has collided with the center at the origin and has now become a center they have therefore exchanged roles and what's this bifurcation this is an exchange of stability bifurcation so it's evident immediately in the bifurcation diagram if I plot the parameter is a if I plot this versus x equilibrium as long as a is negative this center at the origin is stable so we have this picture this is stable and once a becomes positive that point becomes unstable and therefore you have a dotted line here on the a axis but the critical point at minus a over b this thing here which is just a straight line with slope minus 1 over b and we have taken b to be positive something of this kind is unstable there and is stable at this point that's the location of this critical point and for a negative it was clearly unstable and for a positive it's a center and is therefore stable and we have an exchange of stability bifurcation at a equal to 0 this bifurcation has another name it's also called exchange of stability or transcritical the saddle note bifurcation incidentally is also called a tangent bifurcation this is a matter of terminology but there are these alternative names we've looked at two distinct bifurcations and one could go on and ask are there any other bifurcations of co-dimension one because the critical parameter here is a the one that you're tuning well the next step would be the following one could play this game could continue the next step would be to say what if the shape of the force of the potential corresponding potential what if this was a x squared and this was b x cubed what then this would of course imply that the potential v of x let's make this a little stable so it looks physical let me make this minus make this also a minus sign you'll see in a minute why because I want to draw convenient pictures this case v of x then becomes equal to a a x squared over 2 plus v x 4 over 4 so the three cases that we've looked at were constant and quadratic function x squared then we had the next situation was an x and an x squared the next case I'm looking at is an x and an x cubed here the potential corresponds to minus the integral of this function the primitive of this function which is a x squared over 2 plus b x 4 over 4 I differentiate it and take a minus sign I get precisely this what kind of picture do I have now and what would you expect once again the simplest way to do this is to plot the potential as a function of x you plot v of x in all three cases so v of x versus x and let's do the same thing v of x versus x for respectively a less than 0 a equal to 0 and a greater than 0 I'm actually going the other way but it doesn't matter it's easiest to plot this potential because it's always positive we've taken b to be positive always in which case for sufficiently large x this is going to dominate and that's just an x 4 curve moving up steeply and near the origin this is a parabola turned upwards concave upwards for a positive and then after that it increases more sharply so it's parabolic at this point and increases very sharp this fashion so one immediately knows what the critical points are there at the origin there's only one critical point and that's at p equal to 0 as well as x equal to 0 and it must be a center because this is a minimum of a potential at a equal to 0 this term is absent and you have a very flat potential at that point not only is the derivative 0 the second derivative is also 0 as is the third derivative it's a minimum but it's not a simple minimum because you can see it goes like this point b x cubed equal to 0 at this point and what happens here when a is negative for small x this term dominates and since a it's negative is negative it's an inverted parabola so it's a curve like this but eventually the x 4 term will dominate and take over you have a symmetric graph of this kind unlike these cases where you just had a minimum at the origin now you have a maximum at the origin but you have two minima at these points and what are these points these points are given by the vanishing of this quantity other than x equal to 0 and those roots are at x squared equal to minus a it a over b therefore x equal to plus square root of minus a over b and this is at minus square root of minus a over b those are the locations of these points and it's evident that these are centers and this point is a saddle point in between we can therefore draw a bifurcation diagram now without further ado and this as a function of a I plot x equilibrium and what does it look like for a positive there's just a single critical point at p equal to 0 x equal to 0 and here since p is always 0 we haven't drawn it so x equilibrium is 0 and it's this and it's stable so this point this critical point is stable when a becomes negative that critical point becomes unstable so we have to replace this with a dotted line on the other side and this is unstable however two new critical points emerge and move away from the origin and they are at plus or minus square root of minus a over b so as a function of a they go like square root of minus a which would correspond to a parabolic shape there and a parabolic shape here this is stable this is unstable this is also stable sorry this is also stable with an unstable critical point in the middle and that's obvious because you can't have two minima of the potential without a maximum in between so automatically you realize that stable and unstable critical points would tend to alternate now what does this figure remind you of it's a different kind of bifurcation altogether from either the saddle note or the exchange of stability bifurcations here we have a stable critical point coming along and bifurcating continuing as an unstable one and a pair of stable ones is born what is this figure remind you of it's a combination of both in some sense but this figure because it resembles a pitch fork is called a pitch fork bifurcation a saddle note bifurcation or a tangent bifurcation had no critical point at all and then the creation of a stable unstable pair an exchange of stability bifurcation was when a stable and unstable one collided and exchange stability's and a pitch fork bifurcation is when a stable bifurcation bifurcates into a pair of stable ones and an unstable one in the middle well one could go on and ask suppose I go on increasing the powers here what would happen while it's true that you could in principle get what looks like new kinds of bifurcations the fact is these are the only three generic elementary bifurcations of co-dimension one in such systems in continuous time systems so the three elementary bifurcations of co-dimension they are a the saddle note or tangent the transcritical or exchange of stability bifurcation and see the pitch fork we've illustrated these bifurcations in the framework of a simple mechanical system drawing pictures using these potentials but as we saw right in the beginning when I gave the example of the transcritical bifurcation they could occur in dissipative systems as well so there's nothing which says these are exclusively restricted to Hamiltonian systems or potential problems or anything like that it's the phenomenon that's important and this is what basically happens these are elementary because you could have more complicated coalescences of bifurcations so in exactly the same way as saying for instance the elementary functions which I would have of a variable x expandable in power series would be one x x squared x cubed and so on and from these I can construct polynomials I can construct more complicated combinations in exactly the same way in some sense these are the basic things that happen no the pitch work is not a combination of the others no it's not so very different shape all together you can see from the potential example that I gave is very different thing all together if you go back to the potential example you had a cubic potential in that cubic curve could have a minimum and a maximum it need not a general cubic curve need not have a cubic and a specific cubic curve need not have a minimum and a maximum but it could depending on what the parameters are and that's exactly what led in the to the saddle node bifurcation similarly if you took a cubic curve then the position of the minimum and maximum could get exchanged this is what led to the exchange of stability bifurcation a fourth order curve of this kind could have three extrema but it could also have one just one right here and as it transits from the situation where it has one extremum to where it has three which is the largest number it can have you end up with a pitch fork bifurcation so clearly what has happened is depending on what kind of polynomial you have what kind of unfolding of these singularities you have you have different kinds of bifurcations and of course one could ask what would happen if I didn't have an x cube here and an x here but I had an x cube here and x to the five here for example etc those would not be the most elementary ones they'd be more complicated versions of what we already have so these are the only three elementary bifurcations for continuous time systems as I said earlier the number of bifurcations possible the classes of bifurcations possible in general dynamical systems is not known in general especially if the core dimension increases beyond two or three or four then it's hard to classify these bifurcations although a lot of work has been done in this along these lines but for the elementary cases we are looking at these are the basic ones the bifurcations we'll come across a few more we're going to study in this course a few more bifurcations which are also elementary bifurcations but which are not of these types and they occur in slightly different dynamical systems as we will see there's one more which is very very important goes along with these and that's called a hope bifurcation and to lead to that I need to introduce yet another concept namely that of a limit cycle so let me do that now yes the question is what's the correct way of unfolding a given degenerate form and there is an elaborate mathematical machinery to do this all these bifurcations are cast in what's called the normal form namely the minimal form or the minimal expression which leads to that phenomenon the bifurcation these are already the ones I've written down here are already the normal forms apart from some constant multiples or scaled months which could be scaled out some constant factors these are already the minimal forms you could make them a little more complex there is a branch of mathematics called catastrophe theory which deals with this problem of unfolding these singularities and writing things in their normal form the minimal form and this is an elaborate there's an elaborate theory to this effect and I've just given a flavor of it a little glimpse of it in this in writing these forms now so there's a systematic way of doing this of unfolding these singularities and let me go on now to the idea of a limit cycle and let me introduce this as follows in our study of two dimensional systems we looked at various kinds of critical points these are points for the right hand sides of the two dynamical equations vanished one could ask are there sets of points continuous sets of points where you have some kind of equilibrium so you don't have point attractors or point repellers point point singularities in the vector field on the right hand side but can you have lines or can you have whole sets of continuous sets of points where you could have such behavior equilibrium kind of kind of equilibrium behavior but the answer is yes because let me do this again with the help of examples and then these are called limit cycles we're going to say a lot more about limit cycles as we go along because this is a typical feature of nonlinear systems and unlike the case of critical points these are much harder to detect and study so it makes them interesting and let's look once again at an example


We're out here on the tactical bay at the Jacksonville shooting experience and we were just doing some coach development training here and they figured, hey, why not have Ben teach us some more stuff so everybody can learn from it. Ben's going to show us how to move while shooting. Take it away, Ben. All right, so shooting on a move is basically the name of the game is to keep your muzzle as stable as possible as you move toward your target or side to side or whatever. So your shooting position and your grip is going to be key for this along with trigger control. Trigger control is always, always most important. Okay, notice how I had my knees bent. I was into the gun, knee bent into the gun, had the rifle tight and my feet went heel toe, heel toe, heel toe to keep a nice steady muzzle. I didn't stop to shoot. I shot through the wobble. I shot through the moving and that's how you get her done.