Ordinary Differential Equations with Strange Attractors

We study a third order system of nonlinear ordinary differential equations that may be thought of as describing the motion of a particle in a generic potential, V, which is a polynomial of order m in position $x(t)$. The coefficients of V are functions of a time-dependent parameter, λ, which is governed by a first-order linear inhomogeneous equation, with forcing proportional to a polynomial, g, in x. For a particular choice of g and for $m = 4$ the system reduces to one studied previously (Moore and Spiegel (1966)) in an attempt to describe overstable convection. We find that the aperiodicity found in Moore and Spiegel (1966) coincides with the appearance of a strange attractor in numerical solutions at the value of an order parameter indicated by the method of averaging (Baker, Moore and Spiegel (1971)). The location of the strange attractor in a particular phase space involves the catastrophe set of a “potential” of degree $m + 1$, as is implicit in Baker, Moore and Spiegel (1971). Similar results obtain when g is modified, as well as when, for m still = 4, the coefficients in V are chosen to make the system equivalent to that of Lorenz (1963). We suggest that the basic form of the system describes a family of strange atrractors and discuss the application of the results to overstable fluid dynamical systems.