Estimate for the amplitude of the limit cycle of the Liénard equation

We consider the nonlinear Liénard equation x ¨ ( t ) + f ( x ) x ˙ ( t ) + g ( x ) = 0 . Liénard obtained sufficient conditions on the functions f ( x ) and g ( x ) under which this equation has a unique stable limit cycle. Under additional conditions, we prove a theorem that permits one to estimate...

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Bibliographic Details
Published inDifferential equations Vol. 53; no. 3; pp. 302 - 310
Main Author Ignat’ev, A. O.
Format Journal Article
LanguageEnglish
Published Moscow Pleiades Publishing 01.03.2017
Springer Nature B.V
Subjects
Amplitudes
Difference and Functional Equations
Differential equations
Equilibrium
Estimates
Formulas (mathematics)
Mathematical analysis
Mathematics
Mathematics and Statistics
Nonlinear equations
Nonlinearity
Ordinary Differential Equations
Partial Differential Equations
Studies
Texts
Theorems
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Summary:We consider the nonlinear Liénard equation x ¨ ( t ) + f ( x ) x ˙ ( t ) + g ( x ) = 0 . Liénard obtained sufficient conditions on the functions f ( x ) and g ( x ) under which this equation has a unique stable limit cycle. Under additional conditions, we prove a theorem that permits one to estimate the amplitude (the maximum value of x ) of this limit cycle from above. The theorem is used to estimate the amplitude of the limit cycle of the van der Pol equation x ¨ ( t ) + μ [ x 2 ( t ) − 1 ] x ˙ ( t ) + x ( t ) = 0 .
Bibliography:ObjectType-Article-1
SourceType-Scholarly Journals-1
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content type line 23
ISSN:0012-2661
1608-3083
DOI:10.1134/S0012266117030028