. a -bl Problem 16. Let A = be a transition matrix for a discrete dynamical...
Problem 4. (Discrete time dynamical system ). Consider the following discrete time dynamical system: Assume xo is given and 0.5 0.5 0.2 0.8 (a) Find eigenvalues of matrix A (b) For each eigenvalue find one eigenvector. (c) Let P be the matrix that has the eigenvectors as its columns. Find P-1 (d) Find P- AP (e) Use the answer from part (d) to find A" and xn-A"xo. (Your answers wl be in terms of n (f) Find xn and limn→ooXn...
Let Xn be a discrete Markov chain with transition matrix P . Show that the m-step transition probabilities are independent of the past. Hint: it is clear for m=1, apply mathematical induction on m
Problem 9. Consider the discrete time dynamical system (a) Determine stability of the origin. (b) Describe dynamics of the system. (c) Sketch the phase portrait. y(k + 1)「L 3y(A)-42(k) Problem 9. Consider the discrete time dynamical system (a) Determine stability of the origin. (b) Describe dynamics of the system. (c) Sketch the phase portrait. y(k + 1)「L 3y(A)-42(k)
number 12. 2.0. When will the value be between 8. +1 0.0 and 0.2? ider the linear discrete-time dynamical system y 1.0). For each of the following values of m, 1.0+m(),- a. Find the equilibrium. b. Graph and cobweb c. Compare your results with the stability condition. 10. m 1.5. 11, m=-0.5 13-16 IG . The following discrete-time dynamical systems have slope ekactly 1 at the equilibrium. Check this, and then iterate the librum to see 2.0. When will the...
Let A be an m × n matrix, let x Rn and let 0 be the zero vector in Rm. (a) Let u, v є Rn be any two solutions of Ax 0, and let c E R. Use the properties of matrix-vector multiplication to show that u+v and cu are also solutions of Ax O. (b) Extend the result of (a) to show that the linear combination cu + dv is a solution of Ax 0 for any c,d...
Let P be the transition probability matrix of a Markov chain. Show that if, for some positive integer r, Pr has all positive entries, then so does P", for all integers n 2 r
Problem 1 reference: 4. 110 points Consider the dynamical system given in Problem 1 (a)-ii. (a) 15 points] Verify that n(t)-2r sin(2t + θ), T1 z2(t)-r cos(2t + θ) is a solution with a proper choice of the constants r and θ. Suppose the initial condition is given by (0)-(1o,*20). Then, determine the constants r and 0 (expressed in io and 2)
Exercise 5.10. Let P be the transition matrix of a Markov chain (Xt)120 on a finite state space Ω. Show that the following statements are equivalent: (i) P is irreducible and aperiodic (ii) There exists an integer r 0 such that for all i,je Ω, (88) (ii) There exists an integer r 20 such that every entry of Pr is positive.
Intion of ants, do the following, being sure 2. Given the discrete time dynamical system (DTDS) describing a population to show all work A1 = 8A (2 - A), A0 = 1 (a) Find all equilibria. (b) Classify cach equilibrium for its stability using the Stability Criterion
(2.) A discrete-tim e Markov chan X, E {0,1,2) has the following transition probability matrix: 0.1 0.2 0.7 P-10.8 0.2 0 0.1 0.8 0.1 Suppose Pr(Xo = 0) = 0.3, Pr(X,-1) = 0.4, and Pr(Xo = 2) = 0.3. Compute the following. .lrn( (a) Pr (X0-0, X,-2, X2-1). (b) Pr(X2-iXoj) for all i,j