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

T. C. Bountis, L. B. Drossos, M. Lakshmanan, S. Parthasarathy

Research output: Contribution to journalArticle

16 Citations (Scopus)

Abstract

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.

Original languageEnglish
Article number033
Pages (from-to)6927-6942
Number of pages16
JournalJournal of Physics A: Mathematical and General
Volume26
Issue number23
DOIs
Publication statusPublished - 1993
Externally publishedYes

Fingerprint

Non-integrability
Van Der Pol Oscillator
Duffing Oscillator
oscillators
International Space Station
Singularity
Hamiltonians
Single valued
Term
First order differential equation
series expansion
Series Expansion
Square root
quadratures
Quadrature
dynamical systems
Hamiltonian Systems
Nonexistence
Dynamical systems
Differential equations

ASJC Scopus subject areas

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

Cite this

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

In: Journal of Physics A: Mathematical and General, Vol. 26, No. 23, 033, 1993, p. 6927-6942.

Research output: Contribution to journalArticle

Bountis, TC, Drossos, LB, Lakshmanan, M & Parthasarathy, S 1993, 'On the non-integrability of a family of Duffing-van der Pol oscillators', 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
Bountis, T. C. ; Drossos, L. B. ; Lakshmanan, M. ; Parthasarathy, S. / On the non-integrability of a family of Duffing-van der Pol oscillators. In: Journal of Physics A: Mathematical and General. 1993 ; Vol. 26, No. 23. pp. 6927-6942.
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