### 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 language | English |
---|---|

Pages (from-to) | 1215-1224 |

Number of pages | 10 |

Journal | Journal of Mathematical Physics |

Volume | 27 |

Issue number | 5 |

Publication status | Published - 1986 |

Externally published | Yes |

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### ASJC Scopus subject areas

- Organic Chemistry

### Cite this

*Journal of Mathematical Physics*,

*27*(5), 1215-1224.

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

Research output: Contribution to journal › Article

*Journal of Mathematical Physics*, vol. 27, no. 5, pp. 1215-1224.

}

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.

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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 -