The Hopf -- Hopf bifurcation with 2:1 resonance: Periodic solutions and invariant tori

Dmitriy Yu.  Volkov

doi:10.3934/proc.2015.1098

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2015, Volume 2015: 1098-1104. Doi: 10.3934/proc.2015.1098 open access

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The Hopf -- Hopf bifurcation with 2:1 resonance: Periodic solutions and invariant tori

Dmitriy Yu. Volkov^1,

1.
Mathematics and Mechanics Faculty, St.Petersburg State University, 28, Universitetsky prospekt, Petergof, St.Petersburg, 199034

Received: August 31, 2014

Revised: December 31, 2014

Published: October 2015

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Abstract

A dissipative Hopf -- Hopf bifurcation with 2 :1 resonance are studied. A parameter dependent polynomial truncated normal form is derived. We study this truncated normal form. This system displays a large variety of behaviour both regular and chaotic solution. Existence of the periodic solutions and invariant tori of full system are proved. Analogy between dissipative Hopf - Hopf bifurcation with 2:1 resonance, generations of second harmonics in non-linear optics and resonant interaction of waves in a plasma is presented.

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Mathematics Subject Classification: Primary: 58F14; Secondary: 34C23, 34C25, 58C27.

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