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On the lack of analytic solutions of the Van der Pol oscillator
Author(s) -
Panayotounakos D.E.,
Panayotounakou N.D.,
Vakakis A.F.
Publication year - 2003
Publication title -
zamm ‐ journal of applied mathematics and mechanics / zeitschrift für angewandte mathematik und mechanik
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.449
H-Index - 51
eISSN - 1521-4001
pISSN - 0044-2267
DOI - 10.1002/zamm.200310040
Subject(s) - van der pol oscillator , relaxation oscillator , exact solutions in general relativity , mathematics , relaxation (psychology) , series (stratigraphy) , analytic function , mathematical physics , mathematical analysis , physics , quantum mechanics , nonlinear system , voltage controlled oscillator , psychology , social psychology , paleontology , voltage , biology
In this work it is shown that by a series of transformations the classical Van der Pol oscillator can be exactly reduced to Abel's equations of the second kind. The absence of exact analytic solutions in terms of known (tabulated) functions of the reduced equations leads to the conclusion that there are no exact solutions of the Van der Pol oscillator in terms of known (tabulated) functions. In the limits or small or large values of the parameter ϵ the reduced equations are amenable to asymptotic analysis. For the case of large values of the parameter (relaxation oscillations) an analytic solution to the problem is provided that is exact up to O (ϵ ‐2 ).