(1 point) Consider the discrete-time dynamical system x+1 = 2x,(1 - x). If x = 1,...
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...
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)
27.(a) State and prove Liapunov's theorem for a continuous-time dynamical system. (b) By finding a suitable Liapunov function show that the origin is a stable fixed point for the dynamical system What is the domain of stability? 27.(a) State and prove Liapunov's theorem for a continuous-time dynamical system. (b) By finding a suitable Liapunov function show that the origin is a stable fixed point for the dynamical system What is the domain of stability?
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...
Consider the discrete dynamical system given by the expression √1 + √1 + √1 + √1 + ⋯ where the " ⋯ " means the pattern continues forever. (A) Find a recurrence equation that models this pattern. (B) Instead of solving the recurrence equation, build a table of values from the recurrence equation through 10 iterations. (C) Find the nonnegative fixed point of this system and apply the Stability and Oscillation theorem to determine the system’s behavior around the fixed...
1) Let f(x) = 1 sin x, x E R , and consider the discrete-time dynamics given by xn+1 = f(xn), n = 0.1, 2, . . . How many fixed points are there? Stable or unstable?
Closed loop Controller - Dynamical System Consider the following continuous non-linear dynamical system: x1 = (11-2x1)ex1 2(2x1-4x2)e*z The system is driven by the following closed-loop controller: 1. For all values of K, find the equilibrium points of the closed loop system, i.e. find the equilibrium point as K varies between-co and +co 2. Consider the origin of the system. Determine the character of the origin for all values of the parameter K. Determine specifically for what values of K the...
1 1 Consider the function f(x.y,z) 2x y 2 the point P(3,0,1), and the unit vector u 0 Compute the gradient of f and evaluate it at P b. Find the unit vector in the direction of maximum increase of f at P c. Find the rate of change of the function in the direction of maximum increase at P d. Find the directional derivative at P in the direction of the given vector. a. 1 1 Consider the function...
FF3:38 3A23B 57% . Grades (1 point) Hassell's model is often used to study populations of insects. Suppose that the updating function for the population of a species of moth P in a sample plot is given by Problems 0.003P)2 Problem 1 Problem 2 Problem 3 Problem 4 Problem 5 Problem 6 Problem 7 Problem 8 Problem 9 Problem 10 Problem 11 Problem 12 Problem 13 Problem 14 Problem 15 Problem 16 Problem 17 Problem 18 Problem 19 Problem 20...
Let a be a positive real number. Consider a discrete-time echo system (called system 1) given by the difference equation y[n] = v[n] + av[n – 4). Here v is the input signal and y is the output signal. A. (1 mark) Determine the systems' transfer function H1 (2). B. (1 mark) What are the pole(s) of this system? Plot the pole(s) in the complex plane. C. (2 marks) Is this system stable? Explain your answer. D. (2 marks) Determine...