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The point (x *, y*)T E lR 2 is a fixed poin t of I with x* and only if b x* + e y* = a , e x * + I y* Let us assu me t hat b] - ee Then x * = a I -de bI - e e and -I- 0 and y* -I- 0, if =d. > O. 20). 1 The Autonomous Case 29 For every (x , y)T E IR 2 the Jacobi matrix of f is given by Jj( x , y) = ( - h bx +1+h a - h b x - h ey - h ex ) - he y - h f y + 1+ h d- h e x - h f y which implies J j (x * , y *) = (1 - h b x* - h c x* ) _ h e y* 1 - h f y* . l1 ,2. l1 ,2) < O. This implies IA1,21< 1 (Exercise), if h > 0 is sufficiently small.
R (ej = j-t h unit vector) are linearly ind ependent. 26) is stable. 1 Definitions and Elementary Properties Let X be a metric space with metric d : X x X - t JR+ and let (fn)nEN be a sequence of cont inuous mappings f n : X - t X , n E N. Then the pair (X, (fn)n EN) is called a non-autonomous time-di scret e dynamical system . The dynamics in this system is defined by the sequence F = (Fn)n ENo of mappings Fn : X - t X given by Fn(x) = fn 0 fn-l and 0 ... 0 ft( x) for all x E X and n E N Fa = x for all x E X .
U(N - l)f (C N (xo, u(O), . . , u(N - 1)) - x) where ,\ > 0 and 1m is the m x m-unit matrix. (O) , . . (O) , . (N - 1)) <