2015, 2015(special): 1098-1104. doi: 10.3934/proc.2015.1098

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

1. 

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

Received  September 2014 Revised  January 2015 Published  November 2015

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.
Citation: Dmitriy Yu. Volkov. The Hopf -- Hopf bifurcation with 2:1 resonance: Periodic solutions and invariant tori. Conference Publications, 2015, 2015 (special) : 1098-1104. doi: 10.3934/proc.2015.1098
References:
[1]

S. Akhmanov and R. Khokhlov, Problems of Nonlinear Optics,, Gordon and Breach, (1972).   Google Scholar

[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.   Google Scholar

[3]

N. Bussac, The Nonlinear three-wave system. Strange attractors and asymptotic solutions,, Physica Scripta, T2/1 (1982), 110.   Google Scholar

[4]

S. Chow, C. Li and D. Wang, Normal Forms and Bifurcations of Planar Vector Fields,, Cambridge University Press, (2008).   Google Scholar

[5]

S. N. Chow and J. K. Hale, Methods of bifurcation theory,, Springer-Verlag, (1982).   Google Scholar

[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).   Google Scholar

[7]

S. A. van Gils, M. Krupa and W. F. Langford, Hopf bifurcation with non-semisimple $1:1$ resonance,, Nonlinearity, 3 (1990), 825.   Google Scholar

[8]

J. Guckenheimer and P. Holmes, Nonlinear oscillations, dynamical systems, and bifurcations of vector fields,, Springer-Verlag: New York, (1990).   Google Scholar

[9]

B. D. Hassard, N. D. Kazarinoff and Y. H. Wan, Theory and applications of Hopf bifurcation,, Cambridge University Press, (1981).   Google Scholar

[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), 163.   Google Scholar

[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.   Google Scholar

[12]

Y. A. Kuznetsov, Elements of applied bifurcation theory,, $3^{nd}$ edition, (2004).   Google Scholar

[13]

V. G. LeBlanc and W. F. Langford, Classification and unfoldings of $1:2$ resonant Hopf bifurcation,, Arch. Rational Mech. Anal., 136 (1996), 305.   Google Scholar

[14]

V. G. LeBlanc, On some secondary bifurcations near resonant Hopf-Hopf interactions,, Contin. Discrete Impuls. Systems, 7 (2000), 405.   Google Scholar

[15]

O. Lopez-Rebollal and J. R. Sanmartin, A generic, hard transition to chaos,, Phys. D, 89 (1995), 1.   Google Scholar

[16]

J. E. Marsden and M. McCracken, The Hopf bifurcation and its applications,, Springer-Verlag New York, (1976).   Google Scholar

[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.   Google Scholar

[18]

R. J. Sacker, On invariant surfaces and bifurcation of periodic solutions of ordinary differential equations,, Ph.D thesis New York University, (1964).   Google Scholar

[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), 1146.   Google Scholar

[20]

D. Yu. Volkov, The Andronov-Hopf Bifurcation with 2: 1 Resonance,, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 300 (2003), 300 (2003), 259.   Google Scholar

show all references

References:
[1]

S. Akhmanov and R. Khokhlov, Problems of Nonlinear Optics,, Gordon and Breach, (1972).   Google Scholar

[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.   Google Scholar

[3]

N. Bussac, The Nonlinear three-wave system. Strange attractors and asymptotic solutions,, Physica Scripta, T2/1 (1982), 110.   Google Scholar

[4]

S. Chow, C. Li and D. Wang, Normal Forms and Bifurcations of Planar Vector Fields,, Cambridge University Press, (2008).   Google Scholar

[5]

S. N. Chow and J. K. Hale, Methods of bifurcation theory,, Springer-Verlag, (1982).   Google Scholar

[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).   Google Scholar

[7]

S. A. van Gils, M. Krupa and W. F. Langford, Hopf bifurcation with non-semisimple $1:1$ resonance,, Nonlinearity, 3 (1990), 825.   Google Scholar

[8]

J. Guckenheimer and P. Holmes, Nonlinear oscillations, dynamical systems, and bifurcations of vector fields,, Springer-Verlag: New York, (1990).   Google Scholar

[9]

B. D. Hassard, N. D. Kazarinoff and Y. H. Wan, Theory and applications of Hopf bifurcation,, Cambridge University Press, (1981).   Google Scholar

[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), 163.   Google Scholar

[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.   Google Scholar

[12]

Y. A. Kuznetsov, Elements of applied bifurcation theory,, $3^{nd}$ edition, (2004).   Google Scholar

[13]

V. G. LeBlanc and W. F. Langford, Classification and unfoldings of $1:2$ resonant Hopf bifurcation,, Arch. Rational Mech. Anal., 136 (1996), 305.   Google Scholar

[14]

V. G. LeBlanc, On some secondary bifurcations near resonant Hopf-Hopf interactions,, Contin. Discrete Impuls. Systems, 7 (2000), 405.   Google Scholar

[15]

O. Lopez-Rebollal and J. R. Sanmartin, A generic, hard transition to chaos,, Phys. D, 89 (1995), 1.   Google Scholar

[16]

J. E. Marsden and M. McCracken, The Hopf bifurcation and its applications,, Springer-Verlag New York, (1976).   Google Scholar

[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.   Google Scholar

[18]

R. J. Sacker, On invariant surfaces and bifurcation of periodic solutions of ordinary differential equations,, Ph.D thesis New York University, (1964).   Google Scholar

[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), 1146.   Google Scholar

[20]

D. Yu. Volkov, The Andronov-Hopf Bifurcation with 2: 1 Resonance,, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 300 (2003), 300 (2003), 259.   Google Scholar

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