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

On the nonautonomous Hopf bifurcation problem

Matteo Franca 1, , Russell Johnson 2, and  Victor Muñoz-Villarragut 3,

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.
Keywords: non-autonomous perturbation, circle extension, Andronov-Hopf bifurcation, suspension flow, integral manifold..
Mathematics Subject Classification: Primary: 37B55; Secondary: 34D35, 34C45, 34F1.
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), 4988-5001. doi: 10.1016/j.jde.2012.01.030.  Google Scholar

[2]

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

[3]

L. Arnold, Random Dynamical Systems, in Dynamical Systems, Fondazione C.I.M.E. 1994, ed. R. Johnson, Lecture Notes in Math., Springer-Verlag, Berlin, 1609 (1995), 1-43. doi: 10.1007/BFb0095238.  Google Scholar

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M. Bebutov, On dynamical systems in the space of continuous functions, Bull. Moscow Univ. Matematica, (1941), 1-52. Google Scholar

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K. Bjerkäv and R. Johnson, Minimal subsets of projective flows, Discrete Contin. Dyn. Syst., 9 (2008), 493-516. doi: 10.3934/dcdsb.2008.9.493.  Google Scholar

[6]

R. Botts, A. Homburg and T. Young, The Hopf bifurcation with bounded noise, Discrete Contin. Dyn. Syst., 32 (2012), 2997-3007. doi: 10.3934/dcds.2012.32.2997.  Google Scholar

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R. Bowen, Entropy for group endomorphisms and homogeneous spaces, Trans. Amer. Math. Soc., 153 (1971), 401-414. doi: 10.1090/S0002-9947-1971-0274707-X.  Google Scholar

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B. Braaksma and H. Broer, On a quasi-periodic Hopf bifurcation, Ann. Inst. H. Poincare Anal. Non Lineaire, 4 (1987), 115-168.  Google Scholar

[9]

B. Braaksma, H. Broer and G. Huitema, Toward a quasi-periodic bifurcation theory, Memoirs A.M.S., 83 (1990), 83-167. Google Scholar

[10]

H. Broer, G. Huitema and M. Sevryuk, Quasi-periodic Motions in Families of Dynamical Systems. Order Amidst Chaos, Lecture Notes in Mathematics, 1645. Springer-Verlag, Berlin, 1996.  Google Scholar

[11]

H. Broer, KAM-theory: Multiperiodicity in conservative and dissipative systems, Nieuw Archief v. Wiskunde, 14 (1996), 65-79.  Google Scholar

[12]

L. Chierchia and C. Falcolini, Compensations in small divisors problems, Comm. Math. Phys., 175 (1996), 135-160. doi: 10.1007/BF02101627.  Google Scholar

[13]

E. Coddington and N. Levinson, Theory of Ordinary Differential Equations, Mc Graw Hill, New York, 1955.  Google Scholar

[14]

W. Coppel, Dichotomies in Stability Theory, Lecture Notes in Mathematics 377, Springer-Verlag, Berlin, 1978.  Google Scholar

[15]

S. Diliberto, Perturbation theorems for periodic surfaces I, Rend. Circ. Math. Palermo, 9 (1960), 265-299. doi: 10.1007/BF02851248.  Google Scholar

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S. Diliberto, New results in periodic surfaces and the averaging principle, U.S.-Japanese seminar on Differential Equations, W.A. Benjamin Co., New York, 1967, 49-87.  Google Scholar

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R. Ellis, Lectures on Topological Dynamics, W.A. Benjamin Co., New York, 1969.  Google Scholar

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B. Fayad, Weak mixing for reparametrized linear flows on the torus, Ergodic Theory Dynam. Systems, 22 (2002), 187-201. doi: 10.1017/S0143385702000081.  Google Scholar

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B. Fayad, Analytic mixing reparametrizations of irrational flows, Ergodic Theory Dynam. Systems, 22 (2002), 437-468. doi: 10.1017/S0143385702000214.  Google Scholar

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H. Furstenberg, Strict ergodicity and transformations of the torus, Amer. Jour. Math., 83 (1961), 573-601. doi: 10.2307/2372899.  Google Scholar

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

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A. Gonzalez-Enriquez and J. Vano, Estimate of smoothing and composition with applications to conjugation problems, J. Dynam. Differential Equations, 20 (2008), 239-270. doi: 10.1007/s10884-006-9060-z.  Google Scholar

[23]

W. Gottschalk and G. Hedlund, Topological Dynamics, AMS Colloquium Publications 36, Amer. Math. Soc., Providence USA, 1955.  Google Scholar

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M. Hirsch, C. Pugh and M. Shub, Invariant Manifolds, Lecture Notes in Math., 583 Springer-Verlag, New York, 1977.  Google Scholar

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W. Huang and Y. Yi, Almost periodically forced circle flows, J. Funct. Anal., 257 (2009), 832-902. doi: 10.1016/j.jfa.2008.12.005.  Google Scholar

[26]

G. Iooss, Bifurcation of Maps and Applications, North Holland Math. Studies 36, Amsterdam, 1979.  Google Scholar

[27]

R. Johnson, Concerning a theorem of Sell, J. Differential Equations, 30 (1978), 324-339. doi: 10.1016/0022-0396(78)90004-9.  Google Scholar

[28]

R. Johnson, P. Kloeden and R. Pavani, Two-step transition in nonautonomous bifurcations: an explanation, Stoch. Dyn., 2 (2002), 67-92. doi: 10.1142/S0219493702000297.  Google Scholar

[29]

R. Johnson and Y. Yi, Hopf bifurcation from non-periodic solutions of differential equations II, J. Differential Equations, 107 (1994), 310-340. doi: 10.1006/jdeq.1994.1015.  Google Scholar

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J. Hale and H. Ko\ccak, Dynamics and Bifurcations, Texts in Applied Mathematics, 3, Springer-Verlag, New York, 1991. doi: 10.1007/978-1-4612-4426-4.  Google Scholar

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

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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), 65-113. Google Scholar

[33]

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

[34]

N. Levinson, Small periodic perturbations of an autonomous system with a stable orbit, Ann. Math., 52 (1950), 727-738. doi: 10.2307/1969445.  Google Scholar

[35]

Y. Neimark, On some cases of periodic motions depending on parameters, Dokl. Akad. Nank. S.S.S.R., 129 (1959), 736-739.  Google Scholar

[36]

V. Nemytskii and V. Stepanov, Qualitative Theory of Ordinary Differential Equations, Princeton Univ. Press, Princeton USA, 1960.  Google Scholar

[37]

D. Ruelle and F. Takens, On the nature of turbulence, Comm. Math. Phys., 20 (1971), 167-192. doi: 10.1007/BF01646553.  Google Scholar

[38]

R. Sacker, On Invariant Surfaces and Bifurcation of Periodic Solutions of Ordinary Differential Equations, Ph. D. thesis, IMM-NYU 333, Courant Istitute, New York University, 1964.  Google Scholar

[39]

E. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton University Press, Princeton, N.J., 1970.  Google Scholar

[40]

Y. Yi, A generalized integral manifold theorem, J. Differential Equations, 102 (1993), 153-187. doi: 10.1006/jdeq.1993.1026.  Google Scholar

show all references

References:
[1]

J. Aliste-Prieto and T. Jäger, Almost periodic structures and the semi-conjugacy problem, J. Differential Equations, 252 (2012), 4988-5001. doi: 10.1016/j.jde.2012.01.030.  Google Scholar

[2]

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

[3]

L. Arnold, Random Dynamical Systems, in Dynamical Systems, Fondazione C.I.M.E. 1994, ed. R. Johnson, Lecture Notes in Math., Springer-Verlag, Berlin, 1609 (1995), 1-43. doi: 10.1007/BFb0095238.  Google Scholar

[4]

M. Bebutov, On dynamical systems in the space of continuous functions, Bull. Moscow Univ. Matematica, (1941), 1-52. Google Scholar

[5]

K. Bjerkäv and R. Johnson, Minimal subsets of projective flows, Discrete Contin. Dyn. Syst., 9 (2008), 493-516. doi: 10.3934/dcdsb.2008.9.493.  Google Scholar

[6]

R. Botts, A. Homburg and T. Young, The Hopf bifurcation with bounded noise, Discrete Contin. Dyn. Syst., 32 (2012), 2997-3007. doi: 10.3934/dcds.2012.32.2997.  Google Scholar

[7]

R. Bowen, Entropy for group endomorphisms and homogeneous spaces, Trans. Amer. Math. Soc., 153 (1971), 401-414. doi: 10.1090/S0002-9947-1971-0274707-X.  Google Scholar

[8]

B. Braaksma and H. Broer, On a quasi-periodic Hopf bifurcation, Ann. Inst. H. Poincare Anal. Non Lineaire, 4 (1987), 115-168.  Google Scholar

[9]

B. Braaksma, H. Broer and G. Huitema, Toward a quasi-periodic bifurcation theory, Memoirs A.M.S., 83 (1990), 83-167. Google Scholar

[10]

H. Broer, G. Huitema and M. Sevryuk, Quasi-periodic Motions in Families of Dynamical Systems. Order Amidst Chaos, Lecture Notes in Mathematics, 1645. Springer-Verlag, Berlin, 1996.  Google Scholar

[11]

H. Broer, KAM-theory: Multiperiodicity in conservative and dissipative systems, Nieuw Archief v. Wiskunde, 14 (1996), 65-79.  Google Scholar

[12]

L. Chierchia and C. Falcolini, Compensations in small divisors problems, Comm. Math. Phys., 175 (1996), 135-160. doi: 10.1007/BF02101627.  Google Scholar

[13]

E. Coddington and N. Levinson, Theory of Ordinary Differential Equations, Mc Graw Hill, New York, 1955.  Google Scholar

[14]

W. Coppel, Dichotomies in Stability Theory, Lecture Notes in Mathematics 377, Springer-Verlag, Berlin, 1978.  Google Scholar

[15]

S. Diliberto, Perturbation theorems for periodic surfaces I, Rend. Circ. Math. Palermo, 9 (1960), 265-299. doi: 10.1007/BF02851248.  Google Scholar

[16]

S. Diliberto, New results in periodic surfaces and the averaging principle, U.S.-Japanese seminar on Differential Equations, W.A. Benjamin Co., New York, 1967, 49-87.  Google Scholar

[17]

R. Ellis, Lectures on Topological Dynamics, W.A. Benjamin Co., New York, 1969.  Google Scholar

[18]

B. Fayad, Weak mixing for reparametrized linear flows on the torus, Ergodic Theory Dynam. Systems, 22 (2002), 187-201. doi: 10.1017/S0143385702000081.  Google Scholar

[19]

B. Fayad, Analytic mixing reparametrizations of irrational flows, Ergodic Theory Dynam. Systems, 22 (2002), 437-468. doi: 10.1017/S0143385702000214.  Google Scholar

[20]

H. Furstenberg, Strict ergodicity and transformations of the torus, Amer. Jour. Math., 83 (1961), 573-601. doi: 10.2307/2372899.  Google Scholar

[21]

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

[22]

A. Gonzalez-Enriquez and J. Vano, Estimate of smoothing and composition with applications to conjugation problems, J. Dynam. Differential Equations, 20 (2008), 239-270. doi: 10.1007/s10884-006-9060-z.  Google Scholar

[23]

W. Gottschalk and G. Hedlund, Topological Dynamics, AMS Colloquium Publications 36, Amer. Math. Soc., Providence USA, 1955.  Google Scholar

[24]

M. Hirsch, C. Pugh and M. Shub, Invariant Manifolds, Lecture Notes in Math., 583 Springer-Verlag, New York, 1977.  Google Scholar

[25]

W. Huang and Y. Yi, Almost periodically forced circle flows, J. Funct. Anal., 257 (2009), 832-902. doi: 10.1016/j.jfa.2008.12.005.  Google Scholar

[26]

G. Iooss, Bifurcation of Maps and Applications, North Holland Math. Studies 36, Amsterdam, 1979.  Google Scholar

[27]

R. Johnson, Concerning a theorem of Sell, J. Differential Equations, 30 (1978), 324-339. doi: 10.1016/0022-0396(78)90004-9.  Google Scholar

[28]

R. Johnson, P. Kloeden and R. Pavani, Two-step transition in nonautonomous bifurcations: an explanation, Stoch. Dyn., 2 (2002), 67-92. doi: 10.1142/S0219493702000297.  Google Scholar

[29]

R. Johnson and Y. Yi, Hopf bifurcation from non-periodic solutions of differential equations II, J. Differential Equations, 107 (1994), 310-340. doi: 10.1006/jdeq.1994.1015.  Google Scholar

[30]

J. Hale and H. Ko\ccak, Dynamics and Bifurcations, Texts in Applied Mathematics, 3, Springer-Verlag, New York, 1991. doi: 10.1007/978-1-4612-4426-4.  Google Scholar

[31]

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

[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), 65-113. Google Scholar

[33]

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

[34]

N. Levinson, Small periodic perturbations of an autonomous system with a stable orbit, Ann. Math., 52 (1950), 727-738. doi: 10.2307/1969445.  Google Scholar

[35]

Y. Neimark, On some cases of periodic motions depending on parameters, Dokl. Akad. Nank. S.S.S.R., 129 (1959), 736-739.  Google Scholar

[36]

V. Nemytskii and V. Stepanov, Qualitative Theory of Ordinary Differential Equations, Princeton Univ. Press, Princeton USA, 1960.  Google Scholar

[37]

D. Ruelle and F. Takens, On the nature of turbulence, Comm. Math. Phys., 20 (1971), 167-192. doi: 10.1007/BF01646553.  Google Scholar

[38]

R. Sacker, On Invariant Surfaces and Bifurcation of Periodic Solutions of Ordinary Differential Equations, Ph. D. thesis, IMM-NYU 333, Courant Istitute, New York University, 1964.  Google Scholar

[39]

E. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton University Press, Princeton, N.J., 1970.  Google Scholar

[40]

Y. Yi, A generalized integral manifold theorem, J. Differential Equations, 102 (1993), 153-187. doi: 10.1006/jdeq.1993.1026.  Google Scholar

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