Question

Determine all periodic solutions, all limit cycles, and determine their stability charac- teristics for the following system dr dt -1

Answer 1

Given that dr/dt = r^2 (1 - r^2), d theta/dt = 1 therefore Equilibrium solution for dr/dt = r^2 (1 - r^2) is dr/dt = 0 r^2 (1 - r^2) = 0 r = plus-minus 1 r = 1, -1 are critical points Therefore agree 2 periodic trajectories namely circle with radius 1 and -1 centered at origin. dr/dt > 0 For 1 < r < -1 dr/dt < 0 For all other t > 0 For r = 1, the trajectori on both sides more away from r = 1, thus it is not a limit circle, there for the trajectories from atleast one of its sides needs to spiral towards r = 1, it is thus unstable points solution d theta/dt = 1 theta = integral 1 = t + t_0 which is unstable.