Maximum amplitude of limit cycles in Liénard systems

Turner, Norman and McClintock, Peter V. E. and Stefanovska, Aneta (2015) Maximum amplitude of limit cycles in Liénard systems. Physical Review E, 91 (1). ISSN 1539-3755

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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.

Item Type:
Journal Article
Journal or Publication Title:
Physical Review E
Additional Information:
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
Uncontrolled Keywords:
/dk/atira/pure/subjectarea/asjc/3100/3104
Subjects:
ID Code:
73360
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Deposited On:
09 Apr 2015 08:46
Refereed?:
Yes
Published?:
Published
Last Modified:
19 Sep 2020 03:17