Limit Cycles

Abstract

For autonomous nonlinear and non-conservative systems described by the differential equations

$$ \frac{{dx}}{{dt}} = P(x,y),\quad \frac{{dy}}{{dt}} = Q(x,y) $$

(7.1)

a new kind of trajectory, the limit cycle, has been briefly encountered at various points in the preceding chapters. The Van der Pol (VdP) electronic oscillator with P(x, y) = y and Q(x, y) = ε(1 - x ²)y - x for example, made its debut in Chapter 2. In this chapter we would like to explore some of the more important properties of limit cycles in greater depth.

Excellent wretch Perdition, catch, my soul But I do love thee! and when I love thee not, Chaos has come again.

William Shakespeare (1564–1616), Othello, Act III, Scene iii