Nonlinear ODE Models

Abstract

In Chapter 1, phase-plane portraits were used to explore some simple nonlinear ODE models whose temporal evolution could not have been predicted, even qualitatively, before the portraits were numerically constructed. An example was the period-doubling route to chaos exhibited by the Duffing equation

$$ \ddot x + 2\gamma \dot x + \alpha x + \beta x^3 = F\cos \left( {\omega t} \right) $$

(1)

when the amplitude F of the driving force was increased, the other parameters being held fixed. If the nonlinear term, β x³, were not present, this “bizarre” period-doubling behavior would not even be possible. If we were to change the various coefficient values in (4.1), the response of the nonlinear system would in general be entirely different and not easily predicted on the basis of mathematical or physical intuition alone. To aid in the qualitative understanding of the behavior of nonlinear ODE systems such as this one, the concepts of fixed points and phase-plane analysis were discussed in Chapter 2.