4

Find and characterize the stability of the fixed-points of the map Tn+l HTn (1 = Ta) where 0 < Tn < 1 for all n = 0.1, " and ( < ! < 4 points...

Question

Find and characterize the stability of the fixed-points of the map Tn+l HTn (1 = Ta) where 0 < Tn < 1 for all n = 0.1, " and ( < ! < 4 points

Find and characterize the stability of the fixed-points of the map Tn+l HTn (1 = Ta) where 0 < Tn < 1 for all n = 0.1, " and ( < ! < 4 points



Answers

Find all equilibrium points and determine their stability.
$$\frac{d y}{d x}=\left(4-y^{2}\right)(y+1)$$

Here we have the differential equation. D Y de acts equals why Tim's Y squared minus one. We just want to find its equilibrium points and determine whether each one is stable or unstable. Or I remember, um, to find equilibrium Points, if you have a differential equation of this form of the right hand side is only in terms of why then we confined the equilibrium points by just, um, setting this equal to zero, right? Actually finding where, um, the derivative of y with respect to x zero. Um, so if you want to find an equilibrium points, we just had this equal to zero. This is a product. This will only happen when y zero or when y squared minus one is zero. This last one left one just gives us y equals zero. Over here, we can rearrange it so y squared equals one. So why is it little plus or minus one? So our equilibrium points are y equals 01 and minus one. Now, for each of these, we need to determine their stability. And if you look in the section what this is is we make a number line and check whether do I d x is positive or negative on each interval, Um, cut up by these equilibrium points and then we see if, um if the function is moving toward, um, these a glittering points on both sides and it will be stable, Otherwise it's unstable. So to determine stability, let's let's first make our number line. And we only have three points that we care about our people over in points. We just need to figure out if, um do I t x is gonna be positive or negative on each of these. So when we are less than minus one, then the white term is negative in the Y squared minus one term is going to be positive for ex view, plugging like minus two. You square, this four minus one is three. So to the left over here is always gonna be negative in between minus one and zero The Y part. It's still gonna be negative. But the Y squared minus one is gonna be negative. Also is positive between zero and one. Um, the Y term is gonna be positive, and westward minus one is going to be negative. So we get negative, and if we are bigger than one. Then why is going to be positive? And Western medicine is also positive. Okay, that's what this means is that as we like, move along the graph. Um, since this thing is like decreasing it, it, like, moves away in this direction. Since it's increasing here, it moves to the right as we go to the right and we have somewhere things over here. And so then determine stability. The stable points are those where the arrow's air both kind of pointing towards it so that the function is gonna move towards this point. And so what are number line tells us is that why equal zero is stable, right? We have the arrows both pointing toward zero. That is, it increases to the left of zero in decreases to the right of zero so that it's sort of approaching it. And then the other two, um, both minus one and plus one or the opposite, right? Both errors are moving in the other direction, so these are both unstable

So here we're trying to find the equilibrium points and determine the stability for the differential equation. Do I d X equals C to the Y minus 12 y minus three. Let's remember to find our equilibrium points. What we do is we set this equals zero and we want to find the y values that make do I d act equal to zero. And since this is a product, this will only happen when one of these is zero. So over here we got That'll happen when y equals three here. If we move the one over and then taken out your longer both sides, we'll get why on the left hand side, enough for a lot of one is zero. And so our equilibrium points are just a zero and three and now to determine the stability. Remember, What we do is is like in cook one. We make our number line and we indicate our equilibrium points and we just want to check in each of these intervals whether do idea access, positive or negative. So when why is increasing or decreasing? So if we are to the left of zero on this left most interval then we're a negative number. And even the why is gonna be less than one, since it's one over e to something. And so the left products in D I. D. X is going to be negative. And why minus three new pregnant negative number is also gonna be negative. So they're both negative, and we get positives when they're in between, um, zero in three feet of the wise would be bigger than one. So the left term is positive, but the right term why minus three is gonna be negative, b negative. And then to the right of three. You know, the line minus one is gonna be positive. And now y minus three is also gonna be positive. So we get this and then you can think about as we let go along the curve. Since its has its increasing here as a positive slope, the function is like increasing, um, towards this point. But since this is negative here, um, the expression can wants to move this way, since that's like a neck of slow. And then we put a right arrow here for the positive slope on the last interval, and then remember, we can we can use this to determine which is stable and unstable. The stable ones are the ones where the errors are pointing towards it. That's why equal zero and the unstable are when the arrows are pointing away and there we go.

So the differential equation is a wide crime is equal to five. Why? Times two e two The power off negative white negative life minus one. So we know that why Hat is an equilibrium if y crimes equal to zero at my hat. So if we set this equal to zero, if we set this equal to zero, we end up with a white hat, has to eat the power of negative lifetime. Just one is equal to zero. So we know for sure that I had is equal to zero or to eat about negative. Why hat is equal to one solving this out We end up with why two hat is equal to 0.69 Suite 12 jacket. Wie hat is locally stable. Equilibrium or not, we find the derivative off. This'd equation right here. So differentiating with respect to why gives us a G prime off y is equal to 10 e to the power of native Why times of one minus y minus fire. So plugging in Jeez, equal to zero. We get g prime off. Why had one is, um greater than zero which implies that why I had one is equal to zero is a locally unstable equilibrium, plugging in G prime off my hat to we end up with a number that is less than zero to negative number. So this implies that why two is a locally stable equilibrium.


Similar Solved Questions

5 answers
An L~C circuit has induciance 360 H and a ~capacilance 0.300 uF Dunng the curent oscillations Maxmum curentin #ne inducorPart 2Whatis the maximum energystoredthe capacitarany time duringthe current cacillations?Express outanerbt4oMASVlew Avallablc Hlnt(s)AZdSubmitPart BHow many bmas per second does Ihe cadaco conlain the amounrgy {cund part 4?Express your anstet tlmts per second:View Available Hint(s)Azd
An L~C circuit has induciance 360 H and a ~capacilance 0.300 uF Dunng the curent oscillations Maxmum curentin #ne inducor Part 2 Whatis the maximum energy stored the capacitar any time duringthe current cacillations? Express outanerbt 4oMAS Vlew Avallablc Hlnt(s) AZd Submit Part B How many bmas per ...
5 answers
Find the general solution of the given system_ dx 3x -y dt dx 9x 3y dt(x(t) , y(t)) C1(1,3) + C2[ (1,3)t+ (1,2) ]
Find the general solution of the given system_ dx 3x -y dt dx 9x 3y dt (x(t) , y(t)) C1(1,3) + C2[ (1,3)t+ (1,2) ]...
5 answers
H Eana aenusdu Ucilen Hnatmn # cusec 1 1 tD
H Eana aenusdu Ucilen Hnatmn # cusec 1 1 tD...
5 answers
QUESTION 1From an ecological perspective (using specific terms and information that we learned in class) explain the rationale behind Thanos' (and Malthus) stance: What does population size have t0 do with disease hunger, war, economic collapse, and other sources of "misery"
QUESTION 1 From an ecological perspective (using specific terms and information that we learned in class) explain the rationale behind Thanos' (and Malthus) stance: What does population size have t0 do with disease hunger, war, economic collapse, and other sources of "misery"...
5 answers
(c) (e) slimne 2 (@) 4j14M all answers W molds tree are following correct 8 SLSIU are more visibl conspicuous diploids than as haploids
(c) (e) slimne 2 (@) 4j14M all answers W molds tree are following correct 8 SLSIU are more visibl conspicuous diploids than as haploids...
5 answers
An electric motor is rotating with angular speed of 250 5 rad/s: Its angular speed In revlmin is: (1 rev = 2 # rad)Ca 2387.32 revlminb 1500 revlmin41.67 revlmin4.26.18 revtmin
An electric motor is rotating with angular speed of 250 5 rad/s: Its angular speed In revlmin is: (1 rev = 2 # rad) Ca 2387.32 revlmin b 1500 revlmin 41.67 revlmin 4.26.18 revtmin...
5 answers
Determine la carga en el condensador de 40jF mostrada en la Figura de abajo.25uFSov40uF60pF
Determine la carga en el condensador de 40jF mostrada en la Figura de abajo. 25uF Sov 40uF 60pF...
5 answers
V9 -x2 Use Polar Coordinates to evaluate J $ (x3 + xy )dydx V9-xzMaximum number of characters (including HTML tags added by text editor): 32.000 Show Rich lex Edkolonddanger Zue}
V9 -x2 Use Polar Coordinates to evaluate J $ (x3 + xy )dydx V9-xz Maximum number of characters (including HTML tags added by text editor): 32.000 Show Rich lex Edkolonddanger Zue}...
5 answers
What is the hydronium ion concentration in a solution whose pH is 10.0$?$
What is the hydronium ion concentration in a solution whose pH is 10.0$?$...
5 answers
Evaluate 3 dA wherethe region enclosed byI+X and 1 hounded
Evaluate 3 dA where the region enclosed by I+X and 1 hounded...
5 answers
In the circuit shown below. calculate; the total current of the circuit H the current through resistor Rz iii) the voltage through Ra2 Marks 2 Marks MarkR;" 5010412 V_R; hoo50
In the circuit shown below. calculate; the total current of the circuit H the current through resistor Rz iii) the voltage through Ra 2 Marks 2 Marks Mark R;" 50 104 12 V_ R; hoo 50...
5 answers
A satellite in a circular orbit of radius R around planet X hasan orbital period T. If Planet X instead had one fourth as muchmass, the orbital period of this satellite in an orbit of the sameradius would be. (Please Explain)a. 4T.b. 2T.c. T√2d. T/2.e. T/4
A satellite in a circular orbit of radius R around planet X has an orbital period T. If Planet X instead had one fourth as much mass, the orbital period of this satellite in an orbit of the same radius would be. (Please Explain) a. 4T. b. 2T. c. T√2 d. T/2. e. T/4...
5 answers
Caeeiceelat Uapaeta tnaltaTean #"a"t 57n0 IcCO a e l Welrete JLAmd eu?Nojtat 1156pnoHualcnDouTNaeeEranotc dr 4tettouotywrj"olsccn #acnllaLoEcDendi IRcund o 0ona 0iMovneaeteenlTaconcl(RuoDiennetteicio'CeeaaleaA #CenDJrOTcecnlniasnet
Caeeiceelat Uapaeta tnaltaTean #"a"t 57n0 IcCO a e l Welrete JLAmd eu? Nojtat 1156pno Hualcn DouT NaeeEr anotc dr 4tettouotywrj" olsccn #acnllaLoEcDendi IRcund o 0ona 0i MovneaeteenlTaconcl (Ruo Diennettei cio' CeeaaleaA #Cen DJrOT cecnlniasnet...
5 answers
Unlformly charged Insulating semicirclelength 16.0 cm Dent Into the snapesemicirclesnownfigure DelouTota cnarge453 VC, Flndthe eiectrc potentlacencer
unlformly charged Insulating semicircle length 16.0 cm Dent Into the snape semicircle snown figure Delou Tota cnarge 453 VC, Flndthe eiectrc potentla cencer...
5 answers
The Fourier coefficient #, in the Fourier series of f(x) -Ix2 ((-1)" _1)(-1)
The Fourier coefficient #, in the Fourier series of f(x) -Ix 2 ((-1)" _1) (-1)...
5 answers
5.9 - 6.7 6.8 - 7.6 7.7 - 8.5 8.6 ~ 9.4181993
5.9 - 6.7 6.8 - 7.6 7.7 - 8.5 8.6 ~ 9.4 18 19 9 3...
4 answers
Consider the sukxparespan(HEE)) pints) Convert $ to relation forin What relation(s) are nerxled to have e5? Japints) Find basis forSn+wz + ws + W4
Consider the sukxpare span (HEE)) pints) Convert $ to relation forin What relation(s) are nerxled to have e5? Ja pints) Find basis for Sn +wz + ws + W4...

-- 0.024707--