On the integrability of systems of nonlinear ordinary differential equations with superposition principles

T. C. Bountis, V. Papageorgiou, P. Winternitz

Research output: Contribution to journalArticle

87 Citations (Scopus)

Abstract

A new class of "solvable" nonlinear dynamical systems has been recently identified by the requirement that the ordinary differential equations (ODE's) describing each member of this class possess nonlinear superposition principles. These systems of ODE's are generally not derived from a Hamiltonian and are classified by associated pairs of Lie algebras of vector fields. In this paper, all such systems of n≤3 ODE's are integrated in a unified way by finding explicit integrals for them and relating them all to a "pivotal" member of their class: the projective Riccati equations. Moreover, by perturbing two parametrically driven projective Riccati equations (thus making them nonsolvable in the above sense) evidence is discovered of chaotic behavior on the Poincare surface of section - in the form of sensitive dependence on initial conditions - near a boundary separating bounded from unbounded motion.

Original languageEnglish
Pages (from-to)1215-1224
Number of pages10
JournalJournal of Mathematical Physics
Volume27
Issue number5
DOIs
Publication statusPublished - Jan 1 1986

    Fingerprint

ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • Mathematical Physics

Cite this