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12 1. 6. ±iv of a matrix A E gl(d,JR). With D := [ >. svt -sinvt] and 1 := [ 1 O] 11 >. ' smvt cosvt ' 0 1 ' one obtains for R tR e_R 2! _R 2! I D tR R In other words, for xo =[xi, yi, ... , Xm, Ym]T E JR 2m the solution of x = Jx is given with j = 1, ... 3b) m tk-j k=j m J L (k _ ")! (xk cos vt - Yk sinvt), tk-j L (k _ J")! (xk sin vt + Yk cos vt). 3. 7. 4. Then the k-dimensional subspace generated by the first k canonical basis vectors (1, 0, ... , 0) T, ... , (0, ... 0, 1, 0, ... , 0) T, 1 ~ k ~ m, is invariant under eJt.

By definition, a solution with negative Lyapunov exponent tends to the origin and a solution with positive Lyapunov exponent becomes unbounded. It is appropriate to formulate the relevant stability concepts not just for linear differential equations, but for general nonlinear difference equations of the form Xn+I = f(xn), where f : ~d ---+ ~d. In general, the solutions

5 for hyperbolic systems also holds in the discrete-time case, with an analogous proof. Additionally, one has to take into account that the subset of contractions in Gl(d, R) has exactly two path connected components determined by det A < 0 and det A > 0, respectively. 6]. 4. 1. 9: Leth: X ~ X be a topological conjugacy for dynamical systems cI>, '11 : R x X -----+ X on a metric state space X. Then (i) the point p EX is a fixed point of cI> if and only if h(p) is a fixed point of '11; (ii) the solution cI>(·,p) is periodic with period T if and only if '11(·, h(p)) is periodic with period T.

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A Problem Book in Algebra by V. A. Krechmar

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