Bifurcational Mechanism of Multistability Formation and Frequency Entrainment in a van der Pol Oscillator with an Additional Oscillatory Circuit

Sergey Astakhov1 Oleg Astakhov2 Vladimir Astakhov3 Jürgen Kurths4 1Information Security of Automated Systems Dept., Yuri Gagarin Technical University of Saratov, Politekhnicheskaya st. 77, Saratov 410054, Russian Federation 2Department of Dynamic Modeling and Biomedical Engineering, Saratov State University, Astrakhanskaya st. 83, Saratov 410012, Russian Federation 3Radioelectronics and Telecommunications Dept., Yuri Gagarin Technical University of Saratov, Politekhnicheskaya st. 77, Saratov 410054, Russian Federation 4Potsdam Institute for Climate Impact Research (PIK), Potsdam 14473, Germany Received: February 14, 2016 In this paper, the bifurcational mechanism of frequency entrainment in a van der Pol oscillator coupled with an additional oscillatory circuit is studied. It is shown that bistability observed in the system is based on two bifurcations: a supercritical Andronov–Hopf bifurcation and a sub-critical Neimark–Sacker bifurcation. The attracting basin boundaries are determined by stable and unstable invariant manifolds of a saddle two-dimensional torus.