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A three-parameter study of a degenerate case of the Hopf-pitchfork bifurcation

A Algaba-+, E Freire++, E Gamero++ and A J Rodríguez-Luis++

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Recommended by A I Neishtadt

A codimension-three unfolding for the BbbZ2-symmetric Hopf-pitchfork bifurcation, in the presence of an additional nonlinear degeneracy, is analysed.

Up to ten distinct topological equivalence classes for the unfolding are found. A rich variety of dynamical and bifurcation behaviours is pointed out. Beyond the bifurcations present in the nondegenerate case, we show that the following bifurcations appear locally: Takens - Bogdanov of periodic orbits, degenerate pitchfork of periodic orbits, and global connections involving equilibria and/or periodic orbits.

The local results achieved, extended by means of numerical continuation methods, are used to understand the dynamics of a modified van der Pol - Duffing electronic oscillator, for a certain range of the parameters.


PACS

02.30.Oz Bifurcation theory

02.30.Hq Ordinary differential equations

MSC

37G15 Bifurcations of limit cycles and periodic orbits

34C37 Homoclinic and heteroclinic solutions

34C23 Bifurcation (See mainly 37Gxx)

Subjects

Mathematical physics

Dates

Issue 4 (July 1999)

Received 1 April 1998, in final form 7 April 1999



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