August  2016, 9(4): 1119-1148. doi: 10.3934/dcdss.2016045

On the nonautonomous Hopf bifurcation problem

1. 

Università Politecnica delle Marche, Dipartimento di Ingegneria Industriale e Scienze Matematiche, Via Brecce Bianche, I-60131 Ancona, Italy

2. 

Dipartimento di Sistemi e Informatica, Università di Firenze, Facolta' di Ingegneria, Via di Santa Marta 3, 50139 Firenze

3. 

Universidad de Valladolid, Departamento de Matemática Aplicada, Esecuela de Ingegnerías Industriales, Paseo del Cauce 59, 47011 Valladolid, Spain

Received  July 2015 Revised  December 2015 Published  August 2016

Under well-known conditions, a one-parameter family of two-dimensional, autonomous ordinary differential equations admits a supercritical\break Andronov-Hopf bifurcation. Let such a family be perturbed by a non-autonomous term. We analyze the sense in which and some conditions under which the Andronov-Hopf pattern persists under such a perturbation.
Citation: 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
References:
[1]

J. Aliste-Prieto and T. Jäger, Almost periodic structures and the semi-conjugacy problem,, J. Differential Equations, 252 (2012), 4988. doi: 10.1016/j.jde.2012.01.030.

[2]

V. Anagnostopoulou, T. Jäger and G. Keller, A model for the non-autonomous Hopf bifurcation,, preprint, ().

[3]

L. Arnold, Random Dynamical Systems,, in Dynamical Systems, 1609 (1995), 1. doi: 10.1007/BFb0095238.

[4]

M. Bebutov, On dynamical systems in the space of continuous functions,, Bull. Moscow Univ. Matematica, (1941), 1.

[5]

K. Bjerkäv and R. Johnson, Minimal subsets of projective flows,, Discrete Contin. Dyn. Syst., 9 (2008), 493. doi: 10.3934/dcdsb.2008.9.493.

[6]

R. Botts, A. Homburg and T. Young, The Hopf bifurcation with bounded noise,, Discrete Contin. Dyn. Syst., 32 (2012), 2997. doi: 10.3934/dcds.2012.32.2997.

[7]

R. Bowen, Entropy for group endomorphisms and homogeneous spaces,, Trans. Amer. Math. Soc., 153 (1971), 401. doi: 10.1090/S0002-9947-1971-0274707-X.

[8]

B. Braaksma and H. Broer, On a quasi-periodic Hopf bifurcation,, Ann. Inst. H. Poincare Anal. Non Lineaire, 4 (1987), 115.

[9]

B. Braaksma, H. Broer and G. Huitema, Toward a quasi-periodic bifurcation theory,, Memoirs A.M.S., 83 (1990), 83.

[10]

H. Broer, G. Huitema and M. Sevryuk, Quasi-periodic Motions in Families of Dynamical Systems. Order Amidst Chaos,, Lecture Notes in Mathematics, (1645).

[11]

H. Broer, KAM-theory: Multiperiodicity in conservative and dissipative systems,, Nieuw Archief v. Wiskunde, 14 (1996), 65.

[12]

L. Chierchia and C. Falcolini, Compensations in small divisors problems,, Comm. Math. Phys., 175 (1996), 135. doi: 10.1007/BF02101627.

[13]

E. Coddington and N. Levinson, Theory of Ordinary Differential Equations,, Mc Graw Hill, (1955).

[14]

W. Coppel, Dichotomies in Stability Theory,, Lecture Notes in Mathematics 377, 377 (1978).

[15]

S. Diliberto, Perturbation theorems for periodic surfaces I,, Rend. Circ. Math. Palermo, 9 (1960), 265. doi: 10.1007/BF02851248.

[16]

S. Diliberto, New results in periodic surfaces and the averaging principle,, U.S.-Japanese seminar on Differential Equations, (1967), 49.

[17]

R. Ellis, Lectures on Topological Dynamics,, W.A. Benjamin Co., (1969).

[18]

B. Fayad, Weak mixing for reparametrized linear flows on the torus,, Ergodic Theory Dynam. Systems, 22 (2002), 187. doi: 10.1017/S0143385702000081.

[19]

B. Fayad, Analytic mixing reparametrizations of irrational flows,, Ergodic Theory Dynam. Systems, 22 (2002), 437. doi: 10.1017/S0143385702000214.

[20]

H. Furstenberg, Strict ergodicity and transformations of the torus,, Amer. Jour. Math., 83 (1961), 573. doi: 10.2307/2372899.

[21]

A. Gonzalez-Enriquez, A non-perturbative theorem on conjugation of torus diffeomorphisms to rigid rotations,, preprint, (2005).

[22]

A. Gonzalez-Enriquez and J. Vano, Estimate of smoothing and composition with applications to conjugation problems,, J. Dynam. Differential Equations, 20 (2008), 239. doi: 10.1007/s10884-006-9060-z.

[23]

W. Gottschalk and G. Hedlund, Topological Dynamics,, AMS Colloquium Publications 36, 36 (1955).

[24]

M. Hirsch, C. Pugh and M. Shub, Invariant Manifolds,, Lecture Notes in Math., 583 (1977).

[25]

W. Huang and Y. Yi, Almost periodically forced circle flows,, J. Funct. Anal., 257 (2009), 832. doi: 10.1016/j.jfa.2008.12.005.

[26]

G. Iooss, Bifurcation of Maps and Applications,, North Holland Math. Studies 36, 36 (1979).

[27]

R. Johnson, Concerning a theorem of Sell,, J. Differential Equations, 30 (1978), 324. doi: 10.1016/0022-0396(78)90004-9.

[28]

R. Johnson, P. Kloeden and R. Pavani, Two-step transition in nonautonomous bifurcations: an explanation,, Stoch. Dyn., 2 (2002), 67. doi: 10.1142/S0219493702000297.

[29]

R. Johnson and Y. Yi, Hopf bifurcation from non-periodic solutions of differential equations II,, J. Differential Equations, 107 (1994), 310. doi: 10.1006/jdeq.1994.1015.

[30]

J. Hale and H. Ko\ccak, Dynamics and Bifurcations,, Texts in Applied Mathematics, 3 (1991). doi: 10.1007/978-1-4612-4426-4.

[31]

A. Katok and B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems,, Cambridge Univ. Press, (1995). doi: 10.1017/CBO9780511809187.

[32]

N. Krylov and N. Bogoliubov, La théorie générale de la measure dans son application à l'étude des systémes dynamiques de la méchanique non linéare,, Ann. Math., 38 (1937), 65.

[33]

Y. Kuznetsov, Elements of Applied Bifurcation Theory,, Springer-Verlag, (1995). doi: 10.1007/978-1-4757-2421-9.

[34]

N. Levinson, Small periodic perturbations of an autonomous system with a stable orbit,, Ann. Math., 52 (1950), 727. doi: 10.2307/1969445.

[35]

Y. Neimark, On some cases of periodic motions depending on parameters,, Dokl. Akad. Nank. S.S.S.R., 129 (1959), 736.

[36]

V. Nemytskii and V. Stepanov, Qualitative Theory of Ordinary Differential Equations,, Princeton Univ. Press, (1960).

[37]

D. Ruelle and F. Takens, On the nature of turbulence,, Comm. Math. Phys., 20 (1971), 167. doi: 10.1007/BF01646553.

[38]

R. Sacker, On Invariant Surfaces and Bifurcation of Periodic Solutions of Ordinary Differential Equations,, Ph. D. thesis, (1964).

[39]

E. Stein, Singular Integrals and Differentiability Properties of Functions,, Princeton University Press, (1970).

[40]

Y. Yi, A generalized integral manifold theorem,, J. Differential Equations, 102 (1993), 153. doi: 10.1006/jdeq.1993.1026.

show all references

References:
[1]

J. Aliste-Prieto and T. Jäger, Almost periodic structures and the semi-conjugacy problem,, J. Differential Equations, 252 (2012), 4988. doi: 10.1016/j.jde.2012.01.030.

[2]

V. Anagnostopoulou, T. Jäger and G. Keller, A model for the non-autonomous Hopf bifurcation,, preprint, ().

[3]

L. Arnold, Random Dynamical Systems,, in Dynamical Systems, 1609 (1995), 1. doi: 10.1007/BFb0095238.

[4]

M. Bebutov, On dynamical systems in the space of continuous functions,, Bull. Moscow Univ. Matematica, (1941), 1.

[5]

K. Bjerkäv and R. Johnson, Minimal subsets of projective flows,, Discrete Contin. Dyn. Syst., 9 (2008), 493. doi: 10.3934/dcdsb.2008.9.493.

[6]

R. Botts, A. Homburg and T. Young, The Hopf bifurcation with bounded noise,, Discrete Contin. Dyn. Syst., 32 (2012), 2997. doi: 10.3934/dcds.2012.32.2997.

[7]

R. Bowen, Entropy for group endomorphisms and homogeneous spaces,, Trans. Amer. Math. Soc., 153 (1971), 401. doi: 10.1090/S0002-9947-1971-0274707-X.

[8]

B. Braaksma and H. Broer, On a quasi-periodic Hopf bifurcation,, Ann. Inst. H. Poincare Anal. Non Lineaire, 4 (1987), 115.

[9]

B. Braaksma, H. Broer and G. Huitema, Toward a quasi-periodic bifurcation theory,, Memoirs A.M.S., 83 (1990), 83.

[10]

H. Broer, G. Huitema and M. Sevryuk, Quasi-periodic Motions in Families of Dynamical Systems. Order Amidst Chaos,, Lecture Notes in Mathematics, (1645).

[11]

H. Broer, KAM-theory: Multiperiodicity in conservative and dissipative systems,, Nieuw Archief v. Wiskunde, 14 (1996), 65.

[12]

L. Chierchia and C. Falcolini, Compensations in small divisors problems,, Comm. Math. Phys., 175 (1996), 135. doi: 10.1007/BF02101627.

[13]

E. Coddington and N. Levinson, Theory of Ordinary Differential Equations,, Mc Graw Hill, (1955).

[14]

W. Coppel, Dichotomies in Stability Theory,, Lecture Notes in Mathematics 377, 377 (1978).

[15]

S. Diliberto, Perturbation theorems for periodic surfaces I,, Rend. Circ. Math. Palermo, 9 (1960), 265. doi: 10.1007/BF02851248.

[16]

S. Diliberto, New results in periodic surfaces and the averaging principle,, U.S.-Japanese seminar on Differential Equations, (1967), 49.

[17]

R. Ellis, Lectures on Topological Dynamics,, W.A. Benjamin Co., (1969).

[18]

B. Fayad, Weak mixing for reparametrized linear flows on the torus,, Ergodic Theory Dynam. Systems, 22 (2002), 187. doi: 10.1017/S0143385702000081.

[19]

B. Fayad, Analytic mixing reparametrizations of irrational flows,, Ergodic Theory Dynam. Systems, 22 (2002), 437. doi: 10.1017/S0143385702000214.

[20]

H. Furstenberg, Strict ergodicity and transformations of the torus,, Amer. Jour. Math., 83 (1961), 573. doi: 10.2307/2372899.

[21]

A. Gonzalez-Enriquez, A non-perturbative theorem on conjugation of torus diffeomorphisms to rigid rotations,, preprint, (2005).

[22]

A. Gonzalez-Enriquez and J. Vano, Estimate of smoothing and composition with applications to conjugation problems,, J. Dynam. Differential Equations, 20 (2008), 239. doi: 10.1007/s10884-006-9060-z.

[23]

W. Gottschalk and G. Hedlund, Topological Dynamics,, AMS Colloquium Publications 36, 36 (1955).

[24]

M. Hirsch, C. Pugh and M. Shub, Invariant Manifolds,, Lecture Notes in Math., 583 (1977).

[25]

W. Huang and Y. Yi, Almost periodically forced circle flows,, J. Funct. Anal., 257 (2009), 832. doi: 10.1016/j.jfa.2008.12.005.

[26]

G. Iooss, Bifurcation of Maps and Applications,, North Holland Math. Studies 36, 36 (1979).

[27]

R. Johnson, Concerning a theorem of Sell,, J. Differential Equations, 30 (1978), 324. doi: 10.1016/0022-0396(78)90004-9.

[28]

R. Johnson, P. Kloeden and R. Pavani, Two-step transition in nonautonomous bifurcations: an explanation,, Stoch. Dyn., 2 (2002), 67. doi: 10.1142/S0219493702000297.

[29]

R. Johnson and Y. Yi, Hopf bifurcation from non-periodic solutions of differential equations II,, J. Differential Equations, 107 (1994), 310. doi: 10.1006/jdeq.1994.1015.

[30]

J. Hale and H. Ko\ccak, Dynamics and Bifurcations,, Texts in Applied Mathematics, 3 (1991). doi: 10.1007/978-1-4612-4426-4.

[31]

A. Katok and B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems,, Cambridge Univ. Press, (1995). doi: 10.1017/CBO9780511809187.

[32]

N. Krylov and N. Bogoliubov, La théorie générale de la measure dans son application à l'étude des systémes dynamiques de la méchanique non linéare,, Ann. Math., 38 (1937), 65.

[33]

Y. Kuznetsov, Elements of Applied Bifurcation Theory,, Springer-Verlag, (1995). doi: 10.1007/978-1-4757-2421-9.

[34]

N. Levinson, Small periodic perturbations of an autonomous system with a stable orbit,, Ann. Math., 52 (1950), 727. doi: 10.2307/1969445.

[35]

Y. Neimark, On some cases of periodic motions depending on parameters,, Dokl. Akad. Nank. S.S.S.R., 129 (1959), 736.

[36]

V. Nemytskii and V. Stepanov, Qualitative Theory of Ordinary Differential Equations,, Princeton Univ. Press, (1960).

[37]

D. Ruelle and F. Takens, On the nature of turbulence,, Comm. Math. Phys., 20 (1971), 167. doi: 10.1007/BF01646553.

[38]

R. Sacker, On Invariant Surfaces and Bifurcation of Periodic Solutions of Ordinary Differential Equations,, Ph. D. thesis, (1964).

[39]

E. Stein, Singular Integrals and Differentiability Properties of Functions,, Princeton University Press, (1970).

[40]

Y. Yi, A generalized integral manifold theorem,, J. Differential Equations, 102 (1993), 153. doi: 10.1006/jdeq.1993.1026.

[1]

Ling Xu, Jianhua Huang, Qiaozhen Ma. Upper semicontinuity of random attractors for the stochastic non-autonomous suspension bridge equation with memory. Discrete & Continuous Dynamical Systems - B, 2017, 22 (11) : 1-21. doi: 10.3934/dcdsb.2019115

[2]

Carmen Núñez, Rafael Obaya. A non-autonomous bifurcation theory for deterministic scalar differential equations. Discrete & Continuous Dynamical Systems - B, 2008, 9 (3&4, May) : 701-730. doi: 10.3934/dcdsb.2008.9.701

[3]

Felipe Rivero. Time dependent perturbation in a non-autonomous non-classical parabolic equation. Discrete & Continuous Dynamical Systems - B, 2013, 18 (1) : 209-221. doi: 10.3934/dcdsb.2013.18.209

[4]

Xiaoyue Li, Xuerong Mao. Population dynamical behavior of non-autonomous Lotka-Volterra competitive system with random perturbation. Discrete & Continuous Dynamical Systems - A, 2009, 24 (2) : 523-545. doi: 10.3934/dcds.2009.24.523

[5]

Alexandre N. Carvalho, José A. Langa, James C. Robinson. Non-autonomous dynamical systems. Discrete & Continuous Dynamical Systems - B, 2015, 20 (3) : 703-747. doi: 10.3934/dcdsb.2015.20.703

[6]

Thorsten Hüls, Yongkui Zou. On computing heteroclinic trajectories of non-autonomous maps. Discrete & Continuous Dynamical Systems - B, 2012, 17 (1) : 79-99. doi: 10.3934/dcdsb.2012.17.79

[7]

Thorsten Hüls. A model function for non-autonomous bifurcations of maps. Discrete & Continuous Dynamical Systems - B, 2007, 7 (2) : 351-363. doi: 10.3934/dcdsb.2007.7.351

[8]

Maciej J. Capiński, Piotr Zgliczyński. Covering relations and non-autonomous perturbations of ODEs. Discrete & Continuous Dynamical Systems - A, 2006, 14 (2) : 281-293. doi: 10.3934/dcds.2006.14.281

[9]

Mark Comerford, Todd Woodard. Orbit portraits in non-autonomous iteration. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 2253-2277. doi: 10.3934/dcdss.2019144

[10]

Xinyuan Liao, Caidi Zhao, Shengfan Zhou. Compact uniform attractors for dissipative non-autonomous lattice dynamical systems. Communications on Pure & Applied Analysis, 2007, 6 (4) : 1087-1111. doi: 10.3934/cpaa.2007.6.1087

[11]

Joaquim P. Mateus, Paulo Rebelo, Silvério Rosa, César M. Silva, Delfim F. M. Torres. Optimal control of non-autonomous SEIRS models with vaccination and treatment. Discrete & Continuous Dynamical Systems - S, 2018, 11 (6) : 1179-1199. doi: 10.3934/dcdss.2018067

[12]

Cung The Anh, Tang Quoc Bao. Dynamics of non-autonomous nonclassical diffusion equations on $R^n$. Communications on Pure & Applied Analysis, 2012, 11 (3) : 1231-1252. doi: 10.3934/cpaa.2012.11.1231

[13]

Tomás Caraballo, P.E. Kloeden. Non-autonomous attractors for integro-differential evolution equations. Discrete & Continuous Dynamical Systems - S, 2009, 2 (1) : 17-36. doi: 10.3934/dcdss.2009.2.17

[14]

Chunyou Sun, Daomin Cao, Jinqiao Duan. Non-autonomous wave dynamics with memory --- asymptotic regularity and uniform attractor. Discrete & Continuous Dynamical Systems - B, 2008, 9 (3&4, May) : 743-761. doi: 10.3934/dcdsb.2008.9.743

[15]

Xiaolin Jia, Caidi Zhao, Juan Cao. Uniform attractor of the non-autonomous discrete Selkov model. Discrete & Continuous Dynamical Systems - A, 2014, 34 (1) : 229-248. doi: 10.3934/dcds.2014.34.229

[16]

Olivier Goubet, Wided Kechiche. Uniform attractor for non-autonomous nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, 2011, 10 (2) : 639-651. doi: 10.3934/cpaa.2011.10.639

[17]

Grzegorz Łukaszewicz, James C. Robinson. Invariant measures for non-autonomous dissipative dynamical systems. Discrete & Continuous Dynamical Systems - A, 2014, 34 (10) : 4211-4222. doi: 10.3934/dcds.2014.34.4211

[18]

Bixiang Wang. Random attractors for non-autonomous stochastic wave equations with multiplicative noise. Discrete & Continuous Dynamical Systems - A, 2014, 34 (1) : 269-300. doi: 10.3934/dcds.2014.34.269

[19]

Wen Tan, Chunyou Sun. Dynamics for a non-autonomous reaction diffusion model with the fractional diffusion. Discrete & Continuous Dynamical Systems - A, 2017, 37 (12) : 6035-6067. doi: 10.3934/dcds.2017260

[20]

Barbara Bianconi, Francesca Papalini. Non-autonomous boundary value problems on the real line. Discrete & Continuous Dynamical Systems - A, 2006, 15 (3) : 759-776. doi: 10.3934/dcds.2006.15.759

2018 Impact Factor: 0.545

Metrics

  • PDF downloads (15)
  • HTML views (0)
  • Cited by (0)

[Back to Top]