MT 328 - Non-Linear dynamical systems: Routes to Chaos Problem Sheet 2 Your solutions are to be submitted online via Moodle by Thursday 01.02.2024(up to 11:59pm). Please write your name

clearly at the top of each page, scan your work as a single PDF file and upload it onto Moodle, as per instructions you can find on this MT328 Moodle page. 2.1 You are given dynamical systems represented by the equation X A X .Take -8 13 4 (¯§ ¹3); (12/2¹) -5 8 (a) A = (a) A = = For each matrix A determine the Jordan form J and a matrix M that satisfies J= M ¹AM. Use Mathematica to find eigenvalues and eigenvectors and check your results for J and M. Write both dynamical systems in canonical form. 2.2 - Sketch phase portraits for the dynamical systems of equation X =A X, when A is given by: -1/2 0 0 (c) A = (33) (3) 0 9 (b) A = (b) A= -2 (²3) (d) A = 2.3 Consider a 2 × 2 matrix, A= = State what type of fixed point you have and where it is found in the phase-plane. Justify the phase portraits you found by referring to the shape of the trajectories and their stability properties. Be careful with part (d): is this a simple dynamical system? Where can you find the critical points? Practice question The next question will not be marked but it is a type of question that is often found in exam papers! (1) where a is a real parameter. Use pen and paper to solve 0 the equation of eigenvalues Det (A − λ1) = 0, with I the identity matrix, to find the eigenvalues of matrix A. Find the value of the parameter a for which A has repeated real eigenvalues. For values of a = a, state the type of eigenvalues of matrix A. Write the Jordan form of matrix A in each different case you found and write the dynamical system it represents in its canonical form. Discuss its critical point(s) and the corresponding stability properties for the different cases you found. Justify your answer.