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Maximum amplitude of limit cycles in Liénard systems

Research output: Contribution to journalJournal article

Published
Article number012927
<mark>Journal publication date</mark>01/2015
<mark>Journal</mark>Physical Review E
Issue number1
Volume91
Number of pages13
Publication statusPublished
Early online date30/01/15
Original languageEnglish

Abstract

We establish sufficient criteria for the existence of a limit cycle in the Lienard system ˙x = y − εF(x),˙y = −x, where F(x) is odd. In their simplest form the criteria lead to the result that, for all finite nonzero ε, the amplitude of the limit cycle is less than ρ and 0 a ρ u, where F(a) = 0 and integral from 0 to u of F(x)dx = 0. We take the van der Pol oscillator as a specific example and establish that for all finite, nonzero ε, the amplitude of its limit cycle is less than 2.0672, a value whose precision is limited by the capacity of our symbolic computation software package. We show how the criterion for the upper bound can be extended to establish a bound on the amplitude of a limit cycle in systems where F(x) contains both odd and even components. We also show how the criteria can be used to establish bounds for bifurcation sets.

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This article is available under the terms of the Creative Commons Attribution 3.0 License. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI. ©2015 American Physical Society