American Institute of Mathematical Sciences

Advanced
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, (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

[1]

Ryan T. Botts, Ale Jan Homburg, Todd R. Young. The Hopf bifurcation with bounded noise. Discrete & Continuous Dynamical Systems - A, 2012, 32 (8) : 2997-3007. doi: 10.3934/dcds.2012.32.2997
[2]

Matteo Franca, Russell Johnson, Victor Muñoz-Villarragut. On the nonautonomous Hopf bifurcation problem. Discrete & Continuous Dynamical Systems - S, 2016, 9 (4) : 1119-1148. doi: 10.3934/dcdss.2016045
[3]

John Guckenheimer, Hinke M. Osinga. The singular limit of a Hopf bifurcation. Discrete & Continuous Dynamical Systems - A, 2012, 32 (8) : 2805-2823. doi: 10.3934/dcds.2012.32.2805
[4]

Hooton Edward, Balanov Zalman, Krawcewicz Wieslaw, Rachinskii Dmitrii. Sliding Hopf bifurcation in interval systems. Discrete & Continuous Dynamical Systems - A, 2017, 37 (7) : 3545-3566. doi: 10.3934/dcds.2017152
[5]

Hongyong Zhao, Daiyong Wu. Point to point traveling wave and periodic traveling wave induced by Hopf bifurcation for a diffusive predator-prey system. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 0-0. doi: 10.3934/dcdss.2020129
[6]

Fernando Antoneli, Ana Paula S. Dias, Rui Paiva. Coupled cell networks: Hopf bifurcation and interior symmetry. Conference Publications, 2011, 2011 (Special) : 71-78. doi: 10.3934/proc.2011.2011.71
[7]

R. Ouifki, M. L. Hbid, O. Arino. Attractiveness and Hopf bifurcation for retarded differential equations. Communications on Pure & Applied Analysis, 2003, 2 (2) : 147-158. doi: 10.3934/cpaa.2003.2.147
[8]

Fatihcan M. Atay. Delayed feedback control near Hopf bifurcation. Discrete & Continuous Dynamical Systems - S, 2008, 1 (2) : 197-205. doi: 10.3934/dcdss.2008.1.197
[9]

Begoña Alarcón, Víctor Guíñez, Carlos Gutierrez. Hopf bifurcation at infinity for planar vector fields. Discrete & Continuous Dynamical Systems - A, 2007, 17 (2) : 247-258. doi: 10.3934/dcds.2007.17.247
[10]

Runxia Wang, Haihong Liu, Fang Yan, Xiaohui Wang. Hopf-pitchfork bifurcation analysis in a coupled FHN neurons model with delay. Discrete & Continuous Dynamical Systems - S, 2017, 10 (3) : 523-542. doi: 10.3934/dcdss.2017026
[11]

Fabien Crauste. Global Asymptotic Stability and Hopf Bifurcation for a Blood Cell Production Model. Mathematical Biosciences & Engineering, 2006, 3 (2) : 325-346. doi: 10.3934/mbe.2006.3.325
[12]

Zhihua Liu, Hui Tang, Pierre Magal. Hopf bifurcation for a spatially and age structured population dynamics model. Discrete & Continuous Dynamical Systems - B, 2015, 20 (6) : 1735-1757. doi: 10.3934/dcdsb.2015.20.1735
[13]

Hui Miao, Zhidong Teng, Chengjun Kang. Stability and Hopf bifurcation of an HIV infection model with saturation incidence and two delays. Discrete & Continuous Dynamical Systems - B, 2017, 22 (6) : 2365-2387. doi: 10.3934/dcdsb.2017121
[14]

Rebecca McKay, Theodore Kolokolnikov, Paul Muir. Interface oscillations in reaction-diffusion systems above the Hopf bifurcation. Discrete & Continuous Dynamical Systems - B, 2012, 17 (7) : 2523-2543. doi: 10.3934/dcdsb.2012.17.2523
[15]

Jaume Llibre, Claudio A. Buzzi, Paulo R. da Silva. 3-dimensional Hopf bifurcation via averaging theory. Discrete & Continuous Dynamical Systems - A, 2007, 17 (3) : 529-540. doi: 10.3934/dcds.2007.17.529
[16]

Pengmiao Hao, Xuechen Wang, Junjie Wei. Global Hopf bifurcation of a population model with stage structure and strong Allee effect. Discrete & Continuous Dynamical Systems - S, 2017, 10 (5) : 973-993. doi: 10.3934/dcdss.2017051
[17]

Jaume Llibre, Ernesto Pérez-Chavela. Zero-Hopf bifurcation for a class of Lorenz-type systems. Discrete & Continuous Dynamical Systems - B, 2014, 19 (6) : 1731-1736. doi: 10.3934/dcdsb.2014.19.1731
[18]

Jixun Chu, Pierre Magal. Hopf bifurcation for a size-structured model with resting phase. Discrete & Continuous Dynamical Systems - A, 2013, 33 (11&12) : 4891-4921. doi: 10.3934/dcds.2013.33.4891
[19]

Jisun Lim, Seongwon Lee, Yangjin Kim. Hopf bifurcation in a model of TGF-$\beta$ in regulation of the Th 17 phenotype. Discrete & Continuous Dynamical Systems - B, 2016, 21 (10) : 3575-3602. doi: 10.3934/dcdsb.2016111
[20]

Jaume Llibre, Amar Makhlouf, Sabrina Badi. $3$ - dimensional Hopf bifurcation via averaging theory of second order. Discrete & Continuous Dynamical Systems - A, 2009, 25 (4) : 1287-1295. doi: 10.3934/dcds.2009.25.1287

 Impact Factor: 

Tools

Metrics

  • PDF downloads (26)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences