Chaos and hyperchaos via secondary Neimark–Sacker bifurcation in a model of radiophysical generator


Using an example of a radiophysical generator model, scenarios for the formation of various chaotic attractors are described, including chaos and hyperchaos. It is shown that as a result of a secondary Neimark–Sacker bifurcation, a hyperchaos with two positive Lyapunov exponents can occur in the system. A comparative analysis of chaotic attractors born as a result of loss of smoothness of an invariant curve, as a result of period-doubling bifurcations, and as a result of secondary Neimark–Sacker bifurcation was carried out.

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  1. 1.

    System (1) is four-dimensional and in fact it has four Lyapunov exponents, but the fourth exponent is always negative, and we did not add it to the plots.

  2. 2.

    In accordance with [40, 41], if for a flow dynamical system at variation of the parameter, the largest Lyapunov exponent is equal zero and two of the following are negative and equal to each other before bifurcation, and if two of the largest Lyapunov exponents are equal zero and the third is negative after bifurcation, it indicates the Neimark–Sacker bifurcation.

  3. 3.

    Also sometimes it is called adiabatic initial conditions, i.e., for each new value of the parameter, the initial conditions were chosen as the final state attained for the previous value of the parameter.


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Authors thank Igor Sataev, Alexey Kazakov and Serhiy Yanchuk for fruitful discussion of this problem.


The work was carried out with the financial support of the Russian Foundation of Basic Research, Grant No. 18-32-00285 (Introduction, Sects. 234.1) and Russian Science Foundation, Grant No. 17-12-01008 (Sect. 4.2). Analysis of multistability was carried out in the frame of project of the Russian Foundation of Basic Research, Grant No. 19-02-00610.

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Correspondence to Nataliya Stankevich.

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Stankevich, N., Kuznetsov, A., Popova, E. et al. Chaos and hyperchaos via secondary Neimark–Sacker bifurcation in a model of radiophysical generator. Nonlinear Dyn 97, 2355–2370 (2019).

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  • Hyperchaos
  • Secondary Neimark–Sacker bifurcation
  • Quasiperiodic oscillations
  • Multistability
  • Lyapunov exponents

Mathematics Subject Classification

  • 37C55
  • 37E45
  • 37E99