- 3 Consider The Discrete Time Dynamical System On X R Given By The Iteration 0 1 Xn 1 Axn Where A 6 1 4 Mar 1 (67.66 KiB) Viewed 36 times
by the iteration
(a) Determine the stability (unstable, stable, asymptotically
stable or globally asymptotically stable) of the origin (0, 0).
(b) Determine the corresponding flow operator S nξ, n ∈ N0, ξ ∈
R 2 .
(c) Compute all equilibria (fixed points) and all 4-periodic
points. Show that the periodic points are dense and sketch the
dynamics in the phase space.
d) Show that the map has no sensitive dependence on initial
conditions. [8 marks] Hint: Show that ∥xn+1∥ = ∥xn∥, i.e. ∥Ax∥ =
∥x∥, where ∥ξ∥ = q ξ 2 1 + ξ 2 2 .
Is the map chaotic? Prove your answer.
3. Consider the discrete-time dynamical system on X = R², given by the iteration = 0-1 Xn+1 = Axn, where A = 6) 1 [4 marks] [10 marks] (a) Determine the stability (unstable, stable, asymptotically stable or globally asymp- totically stable) of the origin (0,0). (b) Determine the corresponding flow operator S„f, n € NO, & ER?. (c) Compute all equilibria (fixed points) and all 4-periodic points. Show that the peri- odic points are dense and sketch the dynamics in the phase space. (d) Show that the map has no sensitive dependence on initial conditions. Hint: Show that || 2n+1 || = ||x,0), i.e. ||A.x || = ||*||, where ||$|| = V8 +6. (e) Is the map chaotic? Prove your answer. [6 marks) [8 marks] [2 marks]