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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 and 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.

[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, Fondazione C.I.M.E. 1994, ed. R. Johnson, Lecture Notes in Math., Springer-Verlag, Berlin, 1609 (1995), 1-43. doi: 10.1007/BFb0095238.

[4]

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

[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.

[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.

[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.

[8]

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

[9]

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

[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.

[11]

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

[12]

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

[13]

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

[14]

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

[15]

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

[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.

[17]

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

[18]

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

[19]

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

[20]

H. Furstenberg, Strict ergodicity and transformations of the torus, Amer. Jour. Math., 83 (1961), 573-601. 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), 239-270. doi: 10.1007/s10884-006-9060-z.

[23]

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

[24]

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

[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.

[26]

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

[27]

R. Johnson, Concerning a theorem of Sell, J. Differential Equations, 30 (1978), 324-339. 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), 67-92. 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), 310-340. doi: 10.1006/jdeq.1994.1015.

[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.

[31]

A. Katok and B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems, Cambridge Univ. Press, Cambridge UK, 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), 65-113.

[33]

Y. Kuznetsov, Elements of Applied Bifurcation Theory, Springer-Verlag, Berlin, 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), 727-738. 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), 736-739.

[36]

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

[37]

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

[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.

[39]

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

[40]

Y. Yi, A generalized integral manifold theorem, J. Differential Equations, 102 (1993), 153-187. 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), 4988-5001. 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, Fondazione C.I.M.E. 1994, ed. R. Johnson, Lecture Notes in Math., Springer-Verlag, Berlin, 1609 (1995), 1-43. doi: 10.1007/BFb0095238.

[4]

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

[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.

[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.

[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.

[8]

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

[9]

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

[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.

[11]

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

[12]

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

[13]

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

[14]

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

[15]

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

[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.

[17]

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

[18]

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

[19]

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

[20]

H. Furstenberg, Strict ergodicity and transformations of the torus, Amer. Jour. Math., 83 (1961), 573-601. 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), 239-270. doi: 10.1007/s10884-006-9060-z.

[23]

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

[24]

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

[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.

[26]

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

[27]

R. Johnson, Concerning a theorem of Sell, J. Differential Equations, 30 (1978), 324-339. 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), 67-92. 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), 310-340. doi: 10.1006/jdeq.1994.1015.

[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.

[31]

A. Katok and B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems, Cambridge Univ. Press, Cambridge UK, 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), 65-113.

[33]

Y. Kuznetsov, Elements of Applied Bifurcation Theory, Springer-Verlag, Berlin, 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), 727-738. 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), 736-739.

[36]

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

[37]

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

[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.

[39]

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

[40]

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

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