This Dynamical systems
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b. F(x)=x+x2+, =-1
to find the fixed points of F ,solve F(x)=x for x.to get the solutions x=sqrt(-) and x= -sqrt (-).in order to determine the behavior of these fixed points,we need to evaluate the equation F'(x)=2x+1 at sqrt(-)and -sqrt (-).for between -1 and 0, it is easy to see that F'(-sqrt(-)) falls between -1 and 1, which implies sqrt (-) is an attracting fixed point.on the other hand sqrt(-) is repelling fixed point for all less than 0 ,since F'(sqrt(-)) is greater than 1. at = -1 ,the attracting fixed point -sqrt(-(-1))= -1 becomes neutral ,with F' (-1)=2(-1)+ 1= -1.this implies that F(x) x+x2+.
undergoes a period - doubling bifurcation at =1
c. G(x) =x +x3 ,=-1
first set G(x) =x and solve for x.the solutions are x=0 , x=sqrt(1 - ) and x= - sqrt (1-) . for is greater than or equal to 1 , there is only one reveal value fixed point at the origin , and there are three for 1. From the equation
G'(x) = +3x2 , we see that G'(0)=, which implies that the fixed point at the origin is attracting for between -1 and 1 .evaluating G'(x) =at the other two fixed points ,sqrt(1 -) and - sqrt (1-), yields the same response of 3 -2 . This implies that both fixed points are attracting for between 1 and 2 . so , when = -1 , we have G'(0) = -1 and the derivative at the other two points is greater than one ,indicating a period doubling bifurcation at the origin with the other two fixed points repelling.
e . S(x) = sin x , =1
S(x) has a fixed point at 0 for all values of . for between -1 and 1, this is the only fixed point of the function.
S' (x) =cos x indicates that at the origin we have S'(x) =. this implies that for between -1 and 1 , the origin is repelling.The origin becomes neutral at =1 ,and is repelling for values of greater than 1. at the bifurcation point,the function gives rise to two other fixed points ,one less than 0 and one greater than 0. this can easily seen by graphing the function .This can easily seen by graphing the function .These two fixed points will be attracting for 1.
i .E(x) = (ex-1), = 1
E(x) has a fixed point at the origin for all values of .Since E'(x) = ex, E'(0) =.Therefore , when the absolute value of is less than one , the origin is attracting . At the bifurcation point of =1, the fixed point is neutral.and when the absolute value of is greater than 1,0 is repelling. Also , for values of greater than one , another fixed point appears.
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This Dynamical systems do 1abefhi write down explanations and writedown neatly In Chaotic Dynamical Systems.pdf -...
Requesting the solution to the problem below from Ordinary Differential Equations and Dynamical Systems, Gerald Teschl. Thanks. Additional materials: Problem 7.2 (Volterra principle). Show that for any orbit of the Volterra- Lotka system (7.3), the time average over one period 1 1 T | (0)2 = 1, T | g(t)dt =1 is independent of the orbit. (Hint: Integrate log(r(t)) over one period.) 7.1. Examples from ecology In this section we want to consider a model from ecology. It describes two...
9. For each of the following pairs of functions, determine the highest order of contact between the two functions at the indicated point ro: (a) f,g :R-R given by f(x)-cos a and g(xo0 (b) f,g RR given by f(x) coS and g(x) 1 - j; ro0. (c) f,g:R-R given by f(x)-sinr and g(x0 (d) f,g:R-R given by f(x) sinz and ga)0. (e) f, g : R → R given by f(x)-e*2 and g(x)-1+z?: zo = 0. (f) f, g :...
1. True/False a. F(x)=x2+2 is chaotic on the interval [-2,2). b.F(z)= z2-1 has a connected filled Julia set. C. F(x) kx(1-x) has a saddle node bifurcation at k-1. d. The subset of the sequence space that consists of all sequences that contain infinitely many 0's is dense. e. If a continuous function on the real line has a periodic point of period 60, then it also ha a periodic point of period 40. 1. True/False a. F(x)=x2+2 is chaotic on...
1. True/False a. F(x)=x2+2 is chaotic on the interval [-2,2). b.F(z)= z2-1 has a connected filled Julia set. C. F(x) kx(1-x) has a saddle node bifurcation at k-1. d. The subset of the sequence space that consists of all sequences that contain infinitely many 0's is dense. e. If a continuous function on the real line has a periodic point of period 60, then it also ha a periodic point of period 40. 1. True/False a. F(x)=x2+2 is chaotic on...
#19 all parts Problems 17 through 19 deal with competitive systems much like those in Examples 1 and 2 except that some coefficients depend on a parameter a. In each of these problems, assume that x, y, and a are always nonnegative. In each of Problems 17 through 19: (a) Sketch the nullclines in the first quadrant, as in Figure 9.4.5. For different ranges of a your sketch may resemble different parts of Figure 9.4.5 (b) Find the critical points...
2. Let X be a continuous random variable with pdf f(x) = { cr", [w] <1, f() = 0. Otherwise, where the parameter c is constant (with respect to x). (a) Find the constant c. (b) Compute the cumulative distribution function F(2) of X. (c) Use F(2) (from b) to determine P(X > 1/2). (d) Find E(X) and V(X).
Let X be exponentially distributed with parameter 3. a) Compute P(X > 6 | X > 2). b) Compute E(7e-12x+8+ 5). c) Let Y be independent from X. Suppose the PDF for Y is f(x) = 2x for 0 ≤ x ≤ 1 (and 0 else). Find the PDF of X + Y.
You only need to do Q2 (a)'s (i) and (ii). No need to do part B 2. (a) Let X be a random variable with a continuous distribution F. (i) Show that the Random Variable Y = F(X) is uniformly distributed over (0,1). (Hint: Al- though F is the distribution of X, regard it simply as a function satisfying certain properties required to make it a CDF ! (ii) Now, given that Y = y, a random variable Z is...
5. Let X have a uniform distribution on the interval (0,1). Given X = x, let Y have a uniform distribution on (0, 2). (a) The conditional pdf of Y, given that X = x, is fyıx(ylx) = 1 for 0 < y < x, since Y|X ~U(0, X). Show that the mean of this (conditional) distribution is E(Y|X) = , and hence, show that Ex{E(Y|X)} = i. (Hint: what is the mean of ?) (b) Noting that fr\x(y|x) =...
Other × D 442 WS.1 Spring 201 9pdf do product costs equal total ma × | , G Let XX be a random vanable wc × 1 + ile i file:///C/UsersOwner/Downloads/442%20ws%20%2319620Spring%202019.pdf e io s e) Find E(2x +3). f) Given EX 2 (and you need not verify this), find: i) VarX and ii) Var(2x +3) kx(x -1), x3,-2,2,3 2. Lst X be a random variable with probabae 2. Let X be a random variable with otherwise Do the following a)...