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Analytics of heteroclinic bifurcation in a 3:1 subharmonic resonance

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Abstract

Analytical approximation of heteroclinic bifurcation in a 3:1 subharmonic resonance is given in this paper. The system we consider that produces this bifurcation is a harmonically forced and self-excited nonlinear oscillator. This bifurcation mechanism, resulting from the disappearance of a stable slow flow limit cycle at the bifurcation point, gives rise to a synchronization phenomenon near the 3:1 resonance. The analytical approach used in this study is based on the collision criterion between the slow flow limit cycle and the three saddles involved in the bifurcation. The amplitudes of the 3:1 subharmonic response and of the slow flow limit cycle are approximated and the collision criterion is applied leading to an explicit analytical condition of heteroclinic connection. Numerical simulations are performed and compared to the analytical finding for validation.

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Authors and Affiliations

  1. Laboratory of Mechanics, University Hassan II-Casablanca, Casablanca, Morocco

    Mohamed Belhaq

  2. FSTM, University Hassan II-Mohammadia, Mohammadia, Morocco

    Abdelhak Fahsi

Authors
  1. Mohamed Belhaq
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  2. Abdelhak Fahsi
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Correspondence to Mohamed Belhaq.

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Belhaq, M., Fahsi, A. Analytics of heteroclinic bifurcation in a 3:1 subharmonic resonance. Nonlinear Dyn 62, 1001–1008 (2010). https://doi.org/10.1007/s11071-010-9780-9

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  • DOI: https://doi.org/10.1007/s11071-010-9780-9

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