Question 3. Consider the discrete-time dynamical system on X = R2, given by the iteration 0 Xn+1 = Axn, where A = -6). 1 (a) Determine the stability (unstable, stable, asymptotically stable or globally asymp- totically stable) of the origin (0,0). (b) Determine the corresponding flow operator S,&,n e No. 6 € R (c) Compute all equilibria (fixed points) and all 3. Consider the discrete-time dynamical system on X = R2, given by the iteration 0 Xn+1 = Axn, where A = -6). 1 (a) Determine the stability (unstable, stable, asymptotically stable or globally asymp- totically stable) of the origin (0,0). (b) Determine the corresponding flow operator S,&,n e No. 6 € R (c) Compute all equilibria (fixed points) and all 4-periodic points. Show that the peri- odic points are dense and sketch the dynamics in the phase space. (d) Show that the map has no sensitive dependence on initial conditions. Hint: Show that || X +i || = |xll . i.e. ||AX|| = ||xl|, where (14|| (e) Is the map chaotic? Prove your answer. I

Solution:(a)A=\left[\begin{array}{cc}0 & -1 \\1 & 0\end{array}\right]Find Eigen values of \operatorname{det}(A-d I)=0\begin{array}{c}A-\lambda I=\left[\begin{array}{cc}0 & -1 \\1 & 0\end{array}\right]-\left[\begin{array}{ll}\lambda & 0 \\0 & \lambda\end{array}\right] \\A-\lambda I=\left[\begin{array}{cc}0-\lambda & -1 \\1 & 0-\lambda\end{array}\right] \\\operatorname{det}(A-\lambda I)=0\left[\begin{array}{cc}0-\lambda & -1 \\1 & 0-\lambda\end{array} \mid=0\right. \\(0-\lambda)(0-\lambda)+1=0 \\\lambda^{2}+1=0 \\\lambda^{2}=-1 \Rightarrow \lambda=\sqrt{-1} \\\lambda= \pm i\end{array}Eigen values ore \lambda= \pm iFind Eigen noefer corres ponding Eigen value \lambda= \pm \grave{l}, \lambda=i\begin{array}{c}{\left[\begin{array}{cc}0-\lambda & -1 \\1 & 0-\lambda\end{array}\right]\left[\begin{array}{l}x_{1} \\x_{2}\end{array}\right]=\left[\begin{array}{l}0 \\0\end{array}\right]} \\{\left[\begin{array}{cc}0-i & -1 \\1 & 0-i\end{array}\right]\left[\begin{array}{l}x_{1} \\x_{2}\end{array}\right]=\left[\begin{array}{l}0 \\0\end{array}\right]} \\-x_{1} i-x_{2}=0 \\x_{2}=-i x_{1}\end{array}Eigen veetar v_{1}=\left[\begin{array}{r}1 \\ -i\end{array}\right]CS Scanned with CamScannerلy\begin{array}{l}\lambda= \pm i, \quad V=\left[\begin{array}{r}1 \\-i\end{array}\right] \\{\left[\begin{array}{cc}0 & -1 \\1 & 0\end{array}\right]\left[\begin{array}{l}1 \\0\end{array}\right]=\left[\begin{array}{c}0 \\-1\end{array}\right]}\end{array}contrsEquillibrium = contreCS Scanned with CamScanner ...