Inverse problems for dynamical systems

Abstract

In this paper, we consider the problem of reconstructing a structurally stable vector field from its global phase portrait or from certain local pieces of the phase portrait. We show that for arbitrary finite sets of points and of simple closed curves, we can construct a structurally stable system having the elements of those sets as its critical points and limit cycles, and a prescribed topological type in a neighborhood of each of them. Given a global, structurally stable phase portrait, we can construct a structurally stable polynomial system which is topologically equivalent to it. Some of our results were originally announced in [ 28 1. The general theory of dynamical systems orginated in the study of solution curves for systems of ordinary differential equations in a phase space. This qualitative theory had its first major development in the work of Poincare on the topology of integral curves. A typical problem is to give a geometric description of the solutions of a particular system, or class of systems, with emphasis on the location of special features such as critical points and periodic orbits. An inverse problem, then, is the construction of a system of equations given geometric properties of the solutions which may be in the form of a “picture” of the flow (a schematic diagram of phase space) or a list of critical orbits defined analytically. A method for solving such problems is potentially useful in building models in the various fields in which the theory of dynamical systems is now being applied. It could provide a basic model to which statistical identification techniques [ 141 could then be applied. There is also an intrinsic interest in methods for constructing polynomial systems of prescribed type which is related to the lack of any progress toward a solution of Hilbert’s sixteenth problem. The recent appearance of a counterexample [27] to the Petrovsky-Landis conjecture for quadratic systems leaves the Problem completely open. One of the earliest results of this kind was obtained by Forster in 1938