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

Dmitriy Yu.  Volkov

doi:10.3934/proc.2015.1098

Article Contents

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

This issue Previous Article Direct scattering of AKNS systems with $L^2$ potentials Next Article Blow-up of solutions to semilinear wave equations with non-zero initial data

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: September 2014

Revised: January 2015

Published: November 2015

Abstract / Introduction Related Papers Cited by

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.

Keywords:

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

Citation:

Related Papers

Cited by

References

[1]	S. Akhmanov and R. Khokhlov, Problems of Nonlinear Optics, Gordon and Breach, New-York, 1972.
[2]	B. L. J. Braaksma, H. R. Broer and G. B. Huitema, Unfolding and bifurcations of quasi-periodic tori. Toward a quasi-periodic bifurcation theory, Memoirs of the American Mathematical society, 83 (1990), 83-175.
[3]	N. Bussac, The Nonlinear three-wave system. Strange attractors and asymptotic solutions, Physica Scripta, T2/1 (1982), 110-118.
[4]	S. Chow, C. Li and D. Wang, Normal Forms and Bifurcations of Planar Vector Fields, Cambridge University Press, Cambridge, 2008.
[5]	S. N. Chow and J. K. Hale, Methods of bifurcation theory, Springer-Verlag, New York, 1982.
[6]	E. Dupuis, De L'Existence D'hypertores Pres D'Une Bifurcation de Hopf - Hopf avec resonance 1:2, Ph.D thesis Universitate d'Ottawa, 2000.
[7]	S. A. van Gils, M. Krupa and W. F. Langford, Hopf bifurcation with non-semisimple $1:1$ resonance, Nonlinearity, 3 (1990), 825-850.
[8]	J. Guckenheimer and P. Holmes, Nonlinear oscillations, dynamical systems, and bifurcations of vector fields, Springer-Verlag: New York, 1990.
[9]	B. D. Hassard, N. D. Kazarinoff and Y. H. Wan, Theory and applications of Hopf bifurcation, Cambridge University Press, Cambridge-New York, 1981.
[10]	D. W. Hughes and R. E. Proctor, Chaos and the effect of noise in a model of three-wave mode coupling, Phys. D, 46 (1990), no. 2, 163-176.
[11]	E. Knobloch and R. E. Proctor, The Dou-ble Hopf bi-fur-ca-tion with $2:1$ resonance, Proc. R. Soc. Lond. A., 415 (1988), 61-90.
[12]	Y. A. Kuznetsov, Elements of applied bifurcation theory, $3^{nd}$ edition, Springer-Verlag, New York, 2004.
[13]	V. G. LeBlanc and W. F. Langford, Classification and unfoldings of $1:2$ resonant Hopf bifurcation, Arch. Rational Mech. Anal., 136 (1996), 305-307.
[14]	V. G. LeBlanc, On some secondary bifurcations near resonant Hopf-Hopf interactions, Contin. Discrete Impuls. Systems, 7 (2000), 405-427.
[15]	O. Lopez-Rebollal and J. R. Sanmartin, A generic, hard transition to chaos, Phys. D, 89 (1995), no. 1-2, 204-221.
[16]	J. E. Marsden and M. McCracken, The Hopf bifurcation and its applications, Springer-Verlag New York, New York, 1976.
[17]	G. Revel, D. M. Alonso, and J. L. Moiola, Numerical semi-global analysis of a 1:2 resonant Hopf-Hopf bifurcation, Phys. D, 247 (2013), 40-53.
[18]	R. J. Sacker, On invariant surfaces and bifurcation of periodic solutions of ordinary differential equations, Ph.D thesis New York University, 1964.
[19]	J. M. Wersinger, J. M. Finn, and E. Ott, Bifurcation and "strange" behavior in instability saturation by nonlinear three-wave mode coupling, Phys. of Fluids, 23 (1980), no. 6, 1146-1164.
[20]	D. Yu. Volkov, The Andronov-Hopf Bifurcation with 2: 1 Resonance, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 300 (2003), Teor. Predst. Din. Sist. Spets. Vyp. 8, 259-265, 293; translation in J. Math. Sci.(N.Y.), 128(2) (2005), no. 2831-2834.