5. Let pi and p2 be repelling fixed points and suppose both points admit nondegenerate heteroclinic orbits connecting each other. That is, suppose there exists q; e W(Pi) and integers n1 and n2 such that loc fm (q1) = P2, fn² (42) = P1- Prove that f admits a hyperbolic invariant set on which the map is chaotic.

Q: Find f -6xydA over the triangular region D with vertices (0,0) (2,0), (0, 4) O68 -68 )-8 -16 16

A: Consider the integral

Q: 1. Sketch the z-domain root locus for the systems with following loop transfer functions (a) L(z) =…

A:

Q: b) Is there a linear map T: V → W, where dim V = 3, dim W = 4, rank T = 2, and dim ker T = 1?

A:

Q: Assume two consumers with utility functions of the form 1/2 1/2 UX (x₁, x₂) = x/²x/² and UX (y₁, 92)…

A: Given below clear explanation

Q: Prove that 10. Let f(0) = 20 be the map of S periodic points of f %3D are dense in S.

A: From the set of reals R, choose a subset P such that each and every point of R lies in P or if you…

Q: 5. Prove that the baker map 2x B(z) = 2z - 1 <z<1 is chaotic on (0,1).

A: Given the baker map, Bx=2x,0≤x≤122x-1,12≤x≤1

Q: Suppose {u1, Span{u;}nSpan{w;}={0}. Ur, W1, Ws} is a linearly independent subset of V. Show that •..…

A:

Q: (2) (a) Is the triangulated space ABD, BCD,CED, ABF, BCF, AEF, ADF, AED on a surface? (b) Let y be a…

A: 2 a) The triangulation of hexagon ABCDEF is given by Hence ABD,BCD,CED,ABF,BCF,AEF,ADF,AED forms a…

Q: Verify that the partial trace of a separable state gives: TrB =Pz(3)P%- EPz(z)p% ® o%

A: Given:

Q: 2. For n ≥ 2, define the n-dimensional hypercube Qn to have vertex set V(Qn) = {(1, .,.£n) = R¹" :…

A: Definition of Hamiltonian cycle :  A Hamiltonian cycle is  a closed loop on a graph where every node…

Q: 1. Show that the scaled symmetric random walk (W) (t): 0≤t≤T} is a martingale.

A:

Q: Problem 4: The parts (a) and (b) of this problem are independent of each other. G1 G2 6 (a) Prove…

A: we are given two graphs G1 and G2

Q: Extend the proof of the Schmidt decomposition to the case where two parties A and B may have state…

A: Suppose that A and B are two parties having state spaces of different dimensionally. The statement…

Q: nd the orthogonal trajectories of the family of confocal central conics given as: = 1 a² + à b² + à

A:

Q: Let G(x) = 4x(1 – x). Prove that for each positive integer k, there is an orbit of period k.

A: Given: G(x)=4x(1-x)

Q: Use powers of adjacency matrices to determine the number of paths of th e specified length between…

A: Given adjacency matrix

Q: 3. X = {m € N|1 ≤ m ≤ 20}. Y is the set of all nonempty subsets of X. An action of S20 on Y is given…

A: Given,       X=m∈ℕ | 1≤m≤20,       Y=Set of all non-empty subsets of X. and action of S20 on Y is…

Q: (b) Find the Wronskian W[y1,Y2](t) of y1 and y2-

A: W[y1,y2]= -t2et

Q: Use La Salle's Invariance Set Theorem to prove that the following system is asymptotically stable…

A: As per the question we are given the following dynamic system : x'' + x'3 + x = 0 And we have to use…

Q: Let x1 and x2 be binary variables, and let t be a continuous variable satisfying the constraints t ≥…

A: Let us consider the different possible values of , and accordingly determine  1.  The constraints…

Q: Theorem 7 For any values of the quotient -1, If A a2B1, a13 2 azB1, a1ß4 2 a481, a1B5 > a5,81, a233…

A: Explanation: In the theorem it is clearly mentioned that they have used positive equilibrium point…

Q: For n ≥ 2, define the n-dimensional hypercube Qn to have vertex set V(Qn) = {(x1,.,.,.,n) € R¹ : æ¡…

A:

Q: 2. Suppose that y, and y, have a nonzero Wronskian and that cik2 Y, = C1Y, + c2Y, and y, = kiy, +…

A:

Q: For such a system, there is an invariant curve that joins the saddle point p to itself. Then we say…

A: Note :- “Since you have asked multiple question, we will solve the first question for you. To get…

Q: 2. Let S,T C R" with S being a proper subset of T. If span(S) = span(T), then T is linearly…

A:

Q: 18: Consider the two linear maps T, and T2 on V, defined as T(X, X, x) = (0, xz x) and T,(x , x, x)…

A:

Q: 1. Let F(x) = x. Compute the first five points on the orbit of 1/2.

A: Given:    F(x) = x2   First five points on the orbit of (1/2)

Q: (b) Consider the discrete-time dynamical system in X = R given by the iteration of the map g(x) =…

A: Given: The discrete-time dynamical system in X=ℝ given by the iteration of the map gx=2x.   To…

Q: 3.9 Suppose that the second-order system i = f(x), with a locally Lipschitz f(x), has a limit cycle.…

A: Given that the second-order system x˙=f(x), with a locally Lipschitz f(x), has a limit cycle. To…

Q: If T1 and T2 are two topologies on the same space X, then * T1NT2 and T1UT2 are both topologies on X…

A:

Q: 2. For n ≥ 2, define the n-dimensional hypercube Qn to have vertex set T1,...,.*,) € R" : æ¡ € {0,…

A: Given: Let us consider the given n-dimensional hypercube Qn to have an vertex set,…

Q: Let :(;)-(::)(:) X1 mod 1. X2 1 X2 f is defined on the unit square in R?. Find the Lyapunov…

A: Given, We, known by Lyapunov exponents it tells the rate of divergence of nearby trajectory of key…

Q: Let X, and X, be two exponential rvs with common parameter 1= 1. 2 1) compute P(X, < 3) and P(X, <…

A: As requested i have answered 1st part using concept of exponential distribution.

Q: Let A = Consider the discrete dynamical system x+1 = Axg. If xo = what is x,? Enter it into x_2.…

A: Given discrete dynamical system is xi+1=Axi where matrix A is A=320073

Q: Prove that simple random walk on a finite, connected, and undirected graph is aperiodic if and only…

A: Given that: G=V,E is bipartite if V=V1⊔V2 with E⊆x,y:x∈V1, y∈V2. Equivalently, G is bipartite if it…

Q: A chow that the wsonskian d es will neves become zex0 to8 all spal

A: Here we use the fact ex&gt; 0 ,for all real x.

Q: (b) {T } is a sequence of bounded linear maps on a Banach space X and Tx = lim 1x exists for all xE…

A:

Q: 1. Consider the system ' = 3x + 6xy?, W =x² + 1. Are there any closed orbits? Explain your answer.

A: Given - Consider the system  x' = 3x + 6xy2, y' = x2 + 1. To find - Are there any closed orbits ?…

Q: 1. Suppose that y, and y, have a nonzero Wronskian and that c¡k2 – czk1 # 0. Show that Y, = C1Y, +…

A:

Q: The conditions for affine dependence are stronger than those for linear dependence, so an affinely…

A:

Q: 3.9 Suppose that the second-order system i = f(x), with a locally Lipschitz f(z) has a limit cycle.…

A: Given That :  The second order system is: x˙=f(x) with a locally Lipschitz f(x) has a limit cycle.…

Q: Compute I ds over the path C : F(t) = (t, 2t,ť²) from (0,0,0) to (4, 4, 4).

A: The answer is given below:

Q: consider the map given by F(x)=1-x² discuss the charactristics of the orbits starting at x=0

A: As per the question we are given the following map : F(x) = 1 - x2 And we have to explain the…

Q: Let X = (!,2) and T3} U = (1,2-)inel,2,U+ jult. %3D n+1 show that X is not T - space

A: This is a problem of Topology.

Q: The graph of f(x,y)=y2-x2 on XYZ space has a saddle point (a,b) if... choice: (choose one correct…

A: Objective is to find the condition for saddle point

Q: me with action space A = (a1, az) and parameter space e = {0, 02} has the following loss ma %3D P2…

A: Here the action space i A={a1,a2}

Q: 7. Show that the family of all solutions of the nonlinear ordinary differen- tial equation y" =y° is…

A: ANSWER :

Q: The work af the veeken fieled aleng the edge y the set Path one Hime in an onti -elockwise wes Ast 2…

A: Given : vector field F→x, y=x-y2, x+y2 and set D=x, y∈ℝ2 : 1≤x-32+y2≤4 , y≥0 To find : the work of…

Q: 4. Show that the distance between the two points z1 and z2 in an Argand diagram is |z1 – z2). In…

A: (6) z-1=2 Given that:The points z=x+iy satisfies the condition z-1=2 The complex number z=x+iy…