ISSN:
1573-269X
Keywords:
homoclinic bifurcations criteria
;
elliptic Lindstedt–Poincaré method
;
Melnikov function
Source:
Springer Online Journal Archives 1860-2000
Topics:
Mathematics
Notes:
Abstract A criterion to predict bifurcation of homoclinic orbits instrongly nonlinear self-excited one-degree-of-freedom oscillator $$\ddot x + c_1 x + c_2 f(x) = \varepsilon g(\mu ,x,\dot x),$$ is presented. TheLindstedt–Poincaré perturbation method is combined formally withthe Jacobian elliptic functions to determine an approximation of thelimit cycles near homoclinicity. We then apply a criterion forpredicting homoclinic orbits, based on the collision of the bifurcatinglimit cycle with the saddle equilibrium. In particular we show that thiscriterion leads to the same results, formally and to leading order, asthe standard Melnikov technique. Explicit applications of this criterionto quadratic or cubic nonlinearities f(x) are included.
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1023/A:1008316010341
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