Rights statement: 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
Final published version, 332 KB, PDF document
Available under license: CC BY
Research output: Contribution to Journal/Magazine › Journal article › peer-review
Research output: Contribution to Journal/Magazine › Journal article › peer-review
}
TY - JOUR
T1 - Maximum amplitude of limit cycles in Liénard systems
AU - Turner, Norman
AU - McClintock, Peter V. E.
AU - Stefanovska, Aneta
N1 - 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
PY - 2015/1
Y1 - 2015/1
N2 - 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.
AB - 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.
U2 - 10.1103/PhysRevE.91.012927
DO - 10.1103/PhysRevE.91.012927
M3 - Journal article
VL - 91
JO - Physical Review E
JF - Physical Review E
SN - 1539-3755
IS - 1
M1 - 012927
ER -