Alternation of regular and chaotic dynamics in a simple two-degree-of-freedom system with nonlinear inertial coupling

G. G. Sigalov, O. V. Gendelman, M. A. AL-Shudeifat, L. I. Manevitch, Alexander F Vakakis, Lawrence Bergman

Research output: Contribution to journalArticle

Abstract

We show that nonlinear inertial coupling between a linear oscillator and an eccentric rotator can lead to very interesting interchanges between regular and chaotic dynamical behavior. Indeed, we show that this model demonstrates rather unusual behavior from the viewpoint of nonlinear dynamics. Specifically, at a discrete set of values of the total energy, the Hamiltonian system exhibits non-conventional nonlinear normal modes, whose shape is determined by phase locking of rotatory and oscillatory motions of the rotator at integer ratios of characteristic frequencies. Considering the weakly damped system, resonance capture of the dynamics into the vicinity of these modes brings about regular motion of the system. For energy levels far from these discrete values, the motion of the system is chaotic. Thus, the succession of resonance captures and escapes by a discrete set of the normal modes causes a sequence of transitions between regular and chaotic behavior, provided that the damping is sufficiently small. We begin from the Hamiltonian system and present a series of Poincaré sections manifesting the complex structure of the phase space of the considered system with inertial nonlinear coupling. Then an approximate analytical description is presented for the non-conventional nonlinear normal modes. We confirm the analytical results by numerical simulation and demonstrate the alternate transitions between regular and chaotic dynamics mentioned above. The origin of the chaotic behavior is also discussed.

Original languageEnglish (US)
Article number013118
JournalChaos
Volume22
Issue number1
DOIs
StatePublished - Jan 3 2012

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Alternation
alternations
Degrees of freedom (mechanics)
Chaotic Dynamics
Hamiltonians
Normal Modes
Chaotic Behavior
degrees of freedom
Degree of freedom
Hamiltonian Systems
Motion
Chaotic systems
Interchanges
Electron energy levels
Phase Locking
Mode Shape
modal response
eccentrics
Damping
Energy Levels

ASJC Scopus subject areas

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

Cite this

Sigalov, G. G., Gendelman, O. V., AL-Shudeifat, M. A., Manevitch, L. I., Vakakis, A. F., & Bergman, L. (2012). Alternation of regular and chaotic dynamics in a simple two-degree-of-freedom system with nonlinear inertial coupling. Chaos, 22(1), [013118]. https://doi.org/10.1063/1.3683480

Alternation of regular and chaotic dynamics in a simple two-degree-of-freedom system with nonlinear inertial coupling. / Sigalov, G. G.; Gendelman, O. V.; AL-Shudeifat, M. A.; Manevitch, L. I.; Vakakis, Alexander F; Bergman, Lawrence.

In: Chaos, Vol. 22, No. 1, 013118, 03.01.2012.

Research output: Contribution to journalArticle

Sigalov, G. G. ; Gendelman, O. V. ; AL-Shudeifat, M. A. ; Manevitch, L. I. ; Vakakis, Alexander F ; Bergman, Lawrence. / Alternation of regular and chaotic dynamics in a simple two-degree-of-freedom system with nonlinear inertial coupling. In: Chaos. 2012 ; Vol. 22, No. 1.
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