American Institute of Mathematical Sciences

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

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, 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.
Keywords: Hopf - Hopf bifurcation, invariant tori., periodic solution.
Mathematics Subject Classification: Primary: 58F14; Secondary: 34C23, 34C25, 58C2.
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, New-York, 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), 83-175.  Google Scholar

[3]

N. Bussac, The Nonlinear three-wave system. Strange attractors and asymptotic solutions, Physica Scripta, T2/1 (1982), 110-118. Google Scholar

[4]

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

[5]

S. N. Chow and J. K. Hale, Methods of bifurcation theory, Springer-Verlag, New York, 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), 825-850.  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, Cambridge-New York, 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), no. 2, 163-176.  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), 61-90.  Google Scholar

[12]

Y. A. Kuznetsov, Elements of applied bifurcation theory, $3^{nd}$ edition, Springer-Verlag, New York, 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), 305-307.  Google Scholar

[14]

V. G. LeBlanc, On some secondary bifurcations near resonant Hopf-Hopf interactions, Contin. Discrete Impuls. Systems, 7 (2000), 405-427.  Google Scholar

[15]

O. Lopez-Rebollal and J. R. Sanmartin, A generic, hard transition to chaos, Phys. D, 89 (1995), no. 1-2, 204-221.  Google Scholar

[16]

J. E. Marsden and M. McCracken, The Hopf bifurcation and its applications, Springer-Verlag New York, 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), 40-53.  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), no. 6, 1146-1164.  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), Teor. Predst. Din. Sist. Spets. Vyp. 8, 259-265, 293; translation in J. Math. Sci.(N.Y.), 128(2) (2005), no. 2831-2834.  Google Scholar

show all references

References:
[1]

S. Akhmanov and R. Khokhlov, Problems of Nonlinear Optics, Gordon and Breach, New-York, 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), 83-175.  Google Scholar

[3]

N. Bussac, The Nonlinear three-wave system. Strange attractors and asymptotic solutions, Physica Scripta, T2/1 (1982), 110-118. Google Scholar

[4]

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

[5]

S. N. Chow and J. K. Hale, Methods of bifurcation theory, Springer-Verlag, New York, 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), 825-850.  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, Cambridge-New York, 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), no. 2, 163-176.  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), 61-90.  Google Scholar

[12]

Y. A. Kuznetsov, Elements of applied bifurcation theory, $3^{nd}$ edition, Springer-Verlag, New York, 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), 305-307.  Google Scholar

[14]

V. G. LeBlanc, On some secondary bifurcations near resonant Hopf-Hopf interactions, Contin. Discrete Impuls. Systems, 7 (2000), 405-427.  Google Scholar

[15]

O. Lopez-Rebollal and J. R. Sanmartin, A generic, hard transition to chaos, Phys. D, 89 (1995), no. 1-2, 204-221.  Google Scholar

[16]

J. E. Marsden and M. McCracken, The Hopf bifurcation and its applications, Springer-Verlag New York, 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), 40-53.  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), no. 6, 1146-1164.  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), Teor. Predst. Din. Sist. Spets. Vyp. 8, 259-265, 293; translation in J. Math. Sci.(N.Y.), 128(2) (2005), no. 2831-2834.  Google Scholar

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