Three-torus-causing mechanism in a third-order forced oscillator

• Published in: Progress of theoretical and experimental physics Vol. 2013; no. 9
• Main Authors: Itoh, Kaoru, Inaba, Naohiko, Sekikawa, Munehisa, Endo, Tetsuro
• Format: Journal Article
• Language: English
• Published: Oxford University Press 01.09.2013
• Subjects: A30
• ISSN: 2050-3911; 2050-3911
• DOI: 10.1093/ptep/ptt070

In this paper, we discuss the bifurcation of a limit cycle to a three-torus in a piecewise linear third-order forced oscillator. A three-torus cannot be generated in third-order autonomous oscillators; our dynamical model exhibits a three-torus of minimal dimension. We adopt a third-order piecewise linear oscillator that exhibits a two-torus and apply a periodic perturbation to this oscillator. First, appropriate parameter values are selected to induce a limit cycle in the oscillator. In addition, this limit cycle is synchronized to the periodic perturbation. When the angular frequency of the periodic perturbation decreases, the oscillator is desynchronized, and a two-torus appears via a saddle-node bifurcation. This was verified by tracking the fixed point corresponding to the limit cycle on the Poincaré map and calculating the eigenvalues of the fixed point. Furthermore, the variation of a bifurcation parameter results in the generation of a three-torus via a quasi-periodic Neimark-Sacker bifurcation. This bifurcation is identified as a quasi-periodic Neimark-Sacker bifurcation from the observation of the second and third degenerate negative Lyapunov exponents. It was confirmed that all of the three Lyapunov exponents become zero at the quasi-periodic Neimark-Sacker bifurcation point.