MT 328 - Non-Linear dynamical systems: Routes to Chaos Problem Sheet 3 Your solutions are to be submitted online via Moodle by Thurday 8.02.2024(up to 11pm). 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. 3.1 _ - Sketch the phase portrait for the linear system below and justify its trajectories and the stability and type of its critical point. You need to deduce the equations describing the system to justify your answers. dy=-2y-3y2 dt dy, =3y₁-2y2 dt 3.2 — Use the plane TrA/DetA to identify the Type and Stability of the critical points for the linear dx dt systems = AX, in each of the cases of matrix A, as given below; not to be done using the direct calculation of eigenvalues and eigenvectors of the matrices. 2 1 1 -4 2 1 1 2 (a)4={ } } };16)4={ } } });(0)4=| | 13 1): 004-(2, 3):10 - (223) ):(e)4={ 2 -3 2 8 -2 7 Justify your answers based on the values of Det A and TrA that you obtain directly from the matrices A and hence describe the eigenvalues for each system. (Help: Remember that when Det A<0, you don't need any other calculation as the CP is identified in the lower part of the plane TrA/DetA....) 3.3 For each of the linear dynamical systems - dx = A✗, with matrix A given below, find the dt differential equations describing the systems and describe the trajectories, stability, and type of their expected phase portraits and critical points. Sketch their phase portraits. It is crucial that you justify your answers: just drawing a phase portrait without justifying its features attracts no marks. 9 (a) A= 4 -9-3 (b) A= -21 02 (c) A= -12 4 -26-8 Below you can find relevant output from Mathematica to use in your solution. A = {{9,4}, {-9, -3}} Eigensystem[A] {{3,3}, {{-2, 3}, {0,0}}} A = {{-2, 1}, {0, 2}} Eigensystem[A] {{-2,2}, {{1,0}, {1,4}}} A = {{-12, 4}, {-26, -8}} Eigensystem[A] {{-10+10 i, -10-10 i }, {{1-5i, 13}, {1+5i, 13}}}