### Abstract

We investigate the non-integrability of a family of Duffing-van der Pol oscillators x+ alpha x(x^{2}-1)+x+ beta x^{3}= gamma cos omega t by studying the analytic properties of the dynamics in complex time. We find that the solutions of (*) have no worse than algebraic singularities at t_{*}, with only (t-t_{*})^{1/2} terms present in their series expansions, unlike, for example, the alpha =0 Duffing case, where, typically, log(t-t_{*}) terms arise. Still, when integrating (*) around long enough contours, a remarkably intricate pattern of square root singularities emerges, on different sheets, which appears to prevent solutions from ever returning to the original sheet. Such evidence of infinitely-sheeted solutions, termed the ISS property, has also been observed in a number of Hamiltonian systems and is illustrated here on a simple example of a single, first-order differential equation. We suggest that the ISS property is a necessary condition for non-integrability, i.e. non-existence of a complete set of analytic, single-valued constants of the motion, which would permit the complete integration of a dynamical system in terms of quadratures.

Original language | English |
---|---|

Article number | 033 |

Pages (from-to) | 6927-6942 |

Number of pages | 16 |

Journal | Journal of Physics A: Mathematical and General |

Volume | 26 |

Issue number | 23 |

DOIs | |

Publication status | Published - 1993 |

Externally published | Yes |

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### ASJC Scopus subject areas

- Statistical and Nonlinear Physics
- Physics and Astronomy(all)
- Mathematical Physics

### Cite this

*Journal of Physics A: Mathematical and General*,

*26*(23), 6927-6942. [033]. https://doi.org/10.1088/0305-4470/26/23/033

**On the non-integrability of a family of Duffing-van der Pol oscillators.** / Bountis, T. C.; Drossos, L. B.; Lakshmanan, M.; Parthasarathy, S.

Research output: Contribution to journal › Article

*Journal of Physics A: Mathematical and General*, vol. 26, no. 23, 033, pp. 6927-6942. https://doi.org/10.1088/0305-4470/26/23/033

}

TY - JOUR

T1 - On the non-integrability of a family of Duffing-van der Pol oscillators

AU - Bountis, T. C.

AU - Drossos, L. B.

AU - Lakshmanan, M.

AU - Parthasarathy, S.

PY - 1993

Y1 - 1993

N2 - We investigate the non-integrability of a family of Duffing-van der Pol oscillators x+ alpha x(x2-1)+x+ beta x3= gamma cos omega t by studying the analytic properties of the dynamics in complex time. We find that the solutions of (*) have no worse than algebraic singularities at t*, with only (t-t*)1/2 terms present in their series expansions, unlike, for example, the alpha =0 Duffing case, where, typically, log(t-t*) terms arise. Still, when integrating (*) around long enough contours, a remarkably intricate pattern of square root singularities emerges, on different sheets, which appears to prevent solutions from ever returning to the original sheet. Such evidence of infinitely-sheeted solutions, termed the ISS property, has also been observed in a number of Hamiltonian systems and is illustrated here on a simple example of a single, first-order differential equation. We suggest that the ISS property is a necessary condition for non-integrability, i.e. non-existence of a complete set of analytic, single-valued constants of the motion, which would permit the complete integration of a dynamical system in terms of quadratures.

AB - We investigate the non-integrability of a family of Duffing-van der Pol oscillators x+ alpha x(x2-1)+x+ beta x3= gamma cos omega t by studying the analytic properties of the dynamics in complex time. We find that the solutions of (*) have no worse than algebraic singularities at t*, with only (t-t*)1/2 terms present in their series expansions, unlike, for example, the alpha =0 Duffing case, where, typically, log(t-t*) terms arise. Still, when integrating (*) around long enough contours, a remarkably intricate pattern of square root singularities emerges, on different sheets, which appears to prevent solutions from ever returning to the original sheet. Such evidence of infinitely-sheeted solutions, termed the ISS property, has also been observed in a number of Hamiltonian systems and is illustrated here on a simple example of a single, first-order differential equation. We suggest that the ISS property is a necessary condition for non-integrability, i.e. non-existence of a complete set of analytic, single-valued constants of the motion, which would permit the complete integration of a dynamical system in terms of quadratures.

UR - http://www.scopus.com/inward/record.url?scp=21344493664&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=21344493664&partnerID=8YFLogxK

U2 - 10.1088/0305-4470/26/23/033

DO - 10.1088/0305-4470/26/23/033

M3 - Article

AN - SCOPUS:21344493664

VL - 26

SP - 6927

EP - 6942

JO - Journal of Physics A: Mathematical and Theoretical

JF - Journal of Physics A: Mathematical and Theoretical

SN - 1751-8113

IS - 23

M1 - 033

ER -