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
Publication statusPublished - 1986
Externally publishedYes

Fingerprint

Nonlinear Ordinary Differential Equations
Ordinary differential equations
Integrability
Superposition
Riccati equation
differential equations
Riccati equations
Riccati Equation
System of Ordinary Differential Equations
Hamiltonians
Nonlinear dynamical systems
Nonlinear Dynamical Systems
Chaotic Behavior
dynamical systems
Algebra
Poincaré
Vector Field
Lie Algebra
algebra
Ordinary differential equation

ASJC Scopus subject areas

  • Organic Chemistry

Cite this

On the integrability of systems of nonlinear ordinary differential equations with superposition principles. / Bountis, T. C.; Papageorgiou, V.; Winternitz, P.

In: Journal of Mathematical Physics, Vol. 27, No. 5, 1986, p. 1215-1224.

Research output: Contribution to journalArticle

Bountis, T. C. ; Papageorgiou, V. ; Winternitz, P. / On the integrability of systems of nonlinear ordinary differential equations with superposition principles. In: Journal of Mathematical Physics. 1986 ; Vol. 27, No. 5. pp. 1215-1224.
@article{9aac1691e8954f94bb7649829a8696eb,
title = "On the integrability of systems of nonlinear ordinary differential equations with superposition principles",
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.",
author = "Bountis, {T. C.} and V. Papageorgiou and P. Winternitz",
year = "1986",
language = "English",
volume = "27",
pages = "1215--1224",
journal = "Journal of Mathematical Physics",
issn = "0022-2488",
publisher = "American Institute of Physics Publising LLC",
number = "5",

}

TY - JOUR

T1 - On the integrability of systems of nonlinear ordinary differential equations with superposition principles

AU - Bountis, T. C.

AU - Papageorgiou, V.

AU - Winternitz, P.

PY - 1986

Y1 - 1986

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=0011906176&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0011906176&partnerID=8YFLogxK

M3 - Article

VL - 27

SP - 1215

EP - 1224

JO - Journal of Mathematical Physics

JF - Journal of Mathematical Physics

SN - 0022-2488

IS - 5

ER -