# American Institute of Mathematical Sciences

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, 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.

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##### 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.
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