Nonlinear Dynamics

, Volume 94, Issue 4, pp 2455–2467 | Cite as

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

  • Nataliya StankevichEmail author
  • Evgeny Volkov
Original Paper


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.


Multistability Quasiperiodic oscillations Chaos Bifurcation analysis Lyapunov exponents 

Mathematics Subject Classification

37C55 37E45 37E99 



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.

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.


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Copyright information

© Springer Nature B.V. 2018

Authors and Affiliations

  1. 1.Department of Applied CyberneticsSaint-Petersburg State UniversitySaint-PetersburgRussia
  2. 2.Department of Radioelectronics and TelecommunicationsYuri Gagarin State Technical University of SaratovSaratovRussia
  3. 3.Faculty of Information TechnologyUniversity of JyväskyläJyväskyläFinland
  4. 4.Department of Theoretical PhysicsLebedev Physical InstituteMoscowRussia

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