ELSEVIER Physica A 228 ( 1996) 236-244 Order and the ubiquitous occurrence of chaos A.S. Fokas ‘sb, T. Bountis ’ ” Department of Mathematics. Imperial College of Science, Technology, und Medicine. Lxmdon SW72B.Z UK h Institute jiw Nonlinear Studies. Clarkson Universit)~, Potsdum, NY 13699.5815. USA ’ Depclrtment of Muthemutics, University of Patrus. 261 IO. Putrus, Greece zyxwvutsrqponmlkjihgfedc Abstract For a large class of ODE’s, which includes the Van der Pol equation, we determine analytically the asymptotic location of the singularities in the complex r-plane. By integrating these ODE’s numerically we show that if the singularities are dense, which is the generic case, the solution is chaotic, in the sense of sensitive dependence on initial conditions. In the exceptional case that the singularities are not dense, the solution exhibits order (taxis). Chaos is ubiquitous even for jfirs/ order ODE’s in complex t. A great number of problems arising in various branches of applied sciences can be modelled by nonlinear ODES. These equations have been the object of intense study for several centuries. Until the early 197Os, nonlinear ODES were divided into two classes: integrable, if they possessed a sufficient number of single-valued integrals, and non- integrable otherwise. An example of a non-integrable equation is the well-known Van der Pol equation w = 0, a! > 0. In the last twenty five years remarkable new developments have occurred: (a) Firstly, the notion of integrability has been extended substantially, using the so- called inverse spectral method [ 11. In the case that sufficiently many integrals can be given explicitly, this method provides a powerful tool for implementing the Liouville integration scheme (it gives an algorithm for parametrizing the Liouville torus in terms of the Jacobian torus) [2]. In the case that the underlying integrals cannot be given explicitly, this method reduces the solution of the nonlinear ODE to the solution of a linear integral equation [ 31. Examples of these two cases are the Kowalevskaya top [ 41 and the first PainlevC transcendent [S], respectively. 0378.4371/96/$15.00 @ 1996 Elsevw Science B.V. All rights reserved SSDlO378.4371(95)00435-I