Question

(1 point) Consider the discrete-time dynamical system x+1 = 2x,(1 - x). If x = 1, Xr+1 = 0.5 Is x = an equilibrium for this system (yes/no)? yes What is the updating function f(x)? Compute the derivative: f'(x) = Evaluate the derivative at the equilibrium: Is the equilibruim stable, unstable or neither? stable

Answer 1

XH+ = 2 X+ (1 - 24t) - If xt = 1/2 x++) = 2X13 (1-2) - 1() = £ Aug If x= = - £ an equilibrium ? sol'nt on a discrete time dynamical system a point which doesn't depends upon time is called equilibrium point Let x* be the equilibrium point, so by Eq0 -x*= 2x*(1-x²) 2(x *) 2 -2x*+**=0 > x* (230* -1)=0 [x*=0, Ž Hence X+= 42 is equilibrium updating function! - The RoHS of Eq @ is the repdating function so f(x)=2x (1-x) Au of'(x) = 2(1-1) - 2x. equilibrium point is 211-25-2x=0 (: f'(21) =0) - 1-2x70 (x=1 so f '(x) at Hall f'(k)=0 a

stability of equilibrium - - we have two cases here (I when a</2 » foco f '(x) = 2(1-22] (1-2%) 70 = f'(x) >0 e (11 when a t & f Gl) = 2 (1-2x) (1-2 us <o » f (x) < 0 hear at x = 1 - we have phas diagram structure as > equilibrium point is stable x=2