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Existence, number, and stability of limit cycles in weakly dissipative, strongly nonlinear oscillators

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Abstract

Oscillators control many functions of electronic devices, but are subject to uncontrollable perturbations induced by the environment. As a consequence, the influence of perturbations on oscillators is a question of both theoretical and practical importance. In this paper, a method based on Abelian integrals is applied to determine the emergence of limit cycles from centers, in strongly nonlinear oscillators subject to weak dissipative perturbations. It is shown how Abelian integrals can be used to determine which terms of the perturbation are influent. An upper bound to the number of limit cycles is given as a function of the degree of a polynomial perturbation, and the stability of the emerging limit cycles is discussed. Formulas to determine numerically the exact number of limit cycles, their stability, shape and position are given.

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Authors and Affiliations

  1. Department of Electronics, Politecnico di Torino, 10129, Turin, Italy

    Michele Bonnin

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  1. Michele Bonnin
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Correspondence to Michele Bonnin.

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This work was partially supported by the Ministero dell’Istruzione, dell’Università e della Ricerca, under the FIRB project no. RBAU01LRKJ. The author thanks the Istituto Superiore Mario Boella and the regional government of Piedmont for financial support.

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Bonnin, M. Existence, number, and stability of limit cycles in weakly dissipative, strongly nonlinear oscillators. Nonlinear Dyn 62, 321–332 (2010). https://doi.org/10.1007/s11071-010-9719-1

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