Multistability in a three-dimensional oscillator: tori, resonant cycles and chaos


The emergence of multistability in a simple three-dimensional autonomous oscillator is investigated using numerical simulations, calculations of Lyapunov exponents and bifurcation analysis over a broad area of two-dimensional plane of control parameters. Using Neimark–Sacker bifurcation of 1:1 limit cycle as the starting regime, many parameter islands with the coexisting attractors were detected in the phase diagram, including the coexistence of torus, resonant limit cycles and chaos; and transitions between the regimes were considered in detail. The overlapping between resonant limit cycles of different winding numbers, torus and chaos forms the multistability.

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  1. 1.

    Here and further we use symbol: \(y=\dot{x}\).

  2. 2.

    Chart of dynamic regimes was constructed in the same way as the chart of Lyapunov exponents, except that for each point of the parameter plane the period of oscillations was determined by the number of fixed points in the Poincaré section by plane \(y=0\).

  3. 3.

    If we consider the bifurcation tree we cannot distinguish between quasiperiodic and chaotic oscillations, but using the charts of Lyapunov exponents in Fig. 2, we can see that for mentioned parameters there are quaisperiodic oscillations with two zero Lyapunov exponents.

  4. 4.

    n is a winding number of tongue.


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This work was supported by Russian Foundation for Basic Research (Grant No. 17-302-50014). Authors thank S.P. Kuznetsov for the helpful discussions and R. Volkova for the editorial efforts.

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Correspondence to Nataliya Stankevich.

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Stankevich, N., Volkov, E. Multistability in a three-dimensional oscillator: tori, resonant cycles and chaos. Nonlinear Dyn 94, 2455–2467 (2018).

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  • Multistability
  • Quasiperiodic oscillations
  • Chaos
  • Bifurcation analysis
  • Lyapunov exponents

Mathematics Subject Classification

  • 37C55
  • 37E45
  • 37E99