1. (a) Use the definition of a continuous-time dynamical system to show that the flow operator Six= - (1) *+ el(1-x) def

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1 A Use The Definition Of A Continuous Time Dynamical System To Show That The Flow Operator Six 1 El 1 X Def 1 (39.63 KiB) Viewed 40 times

1. (a) Use the definition of a continuous-time dynamical system to show that the flow operator Six= - (1) *+ el(1-x) defines a continuous-time dynamical system on X = R1 = (x ER | x2 01. [6 marks] (b) Compute Sex for each of the four points x = 0,1,1,2. Use this to sketch the dynamics in the phase space. 16 marks (c) Determine the following w-limit sets of (1): 18 marks (i) ({0}). (ii) ((0,1)), (iii) ([1,2]). (d) Consider the continuous-time dynamical system given by the solutions of the initial value problem with the ordinary differential equation * = -x sinx. Determine all equilibria and their stability (unstable, stable or asymptotically stable). [10 marks] Hint: For one equilibrium you might need to consider a Lyapunov function such as V(x) = x2