In some periodic vibratory systems, external energy is supplied to the system over part of a period and dissipated within the system in another part of the period. Such periodic oscillations are known as relaxation oscillations. Van der Pol [13.22] indicated several instances of occurrence of relaxation oscillations such as a pneumatic hammer, the scratching noise of a knife on a plate, the squeaking of a door, and the fluctuation of populations of animal species. Many relaxation oscillations are governed by van der Pol's equation:

a. Plot the phase-plane trajectories for three values of \(\alpha: \alpha=0.1, \alpha=1\), and \(\alpha=10\). Use the initial conditions (i) \(x(0)=0.5, \dot{x}(0)=0\) and (ii) \(x(0)=0, \dot{x}(0)=5\).

b. Solve Eq. (E.1) using the fourth-order Runge-Kutta method using the initial conditions stated in

(a) for \(\alpha=0.1, \alpha=1\), and \(\alpha=10\).

Transcribed Image Text:

xa(1 - x)x + x = 0 (E.1)