2. (a) Consider * = f(x), I ER", DEN, where :R" → R" is a continuously differentiable function Furthermore, let V:d(G) →

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2 A Consider F X I Er Den Where R R Is A Continuously Differentiable Function Furthermore Let V D G 1 (33.72 KiB) Viewed 20 times

2. (a) Consider * = f(x), I ER", DEN, where :R" → R" is a continuously differentiable function Furthermore, let V:d(G) →R a continously differentiable function where (G) is the cute of an open neighbourhood of the origin. Define the derivative of V with respect to time along the trajectories of (1), and then define what it is meant for V to be a Lynpuno function. (b) Let I = 0 be a stationary point of (1). (i) State Lyapuinov's first stability theorem. (ii) State Lyapuno's second stability theorem. (c) Consider i = -3 (2) (i) Show that the origin is a non-hyperbolic equilibrium (2). (ii) Find a b E R such that V(x,y) = x + boy? is a strict Lyapuno function for (2) (iii) Using the result from (ii) or otherwise, investigate the stability of the origin in (2). (11 marks) (d) Consider i = Ar, TER? where ER? Let A., Aye C be the eigenvalues of A. Derive a formula for the determinant and the trace of A in terms of and Iz, and then formulate a condition for asymptotie stability of the origin in terms of the determinant and the trace of A.