The research presented in this PhD thesis within the framework of nonlinear deterministic dynamical systems depending on parameters. The work is divided into four Chapters, where the first is a general introduction to the other three. Chapter two deals with the investigation of a time-periodic three-dimensional system of ordinary differential equations depending on three parameters, the Lorenz-84 model with seasonal forcing. The model is a variation on an autonomous system proposed in 1984 by the meteorologist E. Lorenz to describe general atmospheric circulation at mid latitude of the northern hemisphere. ...

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Vitolo, Renato,

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Vitolo, Renato

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   University of GroningenBifurcations of attractors in 3D diffeomorphismsVitolo, RenatoIMPORTANT NOTE: You are advised to consult the publisher's version (publisher's PDF) if you wish to cite fromit. Please check the document version below.Document VersionPublisher's PDF, also known as Version of recordPublication date:2003Link to publication in University of Groningen/UMCG research databaseCitation for published version (APA):Vitolo, R. (2003). Bifurcations of attractors in 3D diffeomorphisms: a study in experimental mathematics.s.n.CopyrightOther than for strictly personal use, it is not permitted to download or to forward/distribute the text or part of it without the consent of theauthor(s) and/or copyright holder(s), unless the work is under an open content license (like Creative Commons).The publication may also be distributed here under the terms of Article 25fa of the Dutch Copyright Act, indicated by the “Taverne” license.More information can be found on the University of Groningen website: https://www.rug.nl/library/open-access/self-archiving-pure/taverne-amendment.Take-down policyIf you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediatelyand investigate your claim.Downloaded from the University of Groningen/UMCG research database (Pure): http://www.rug.nl/research/portal. For technical reasons thenumber of authors shown on this cover page is limited to 10 maximum.Download date: 08-06-2022Bibliography[1] V.S. Afrajmovich, L.P. Shil′nikov: On invariant two-dimensional tori, their dis-integration and stochasticity, in Methods of the qualitative theory of differentialequations, Gor′kov. Gos. University, Gor′kij (1983), 3–26.[2] V.S. Anishchenko: Dynamical Chaos—Models and Experiments, World Scientific(1995).[3] V.S. Anishchenko, M.A. Safonova, U. Feudel, J. Kurths: Bifurcations and transi-tion to chaos through three-dimensional tori, IJBC 3(4) (1994), 595–607.[4] V.I. Arnol′d: Small denominators, I: Mappings of the circumference into itself,AMS Transl. (Ser. 2) 46 (1965), 213–284.[5] V.I. Arnol′d: Mathematical Methods of Classical Mechanics , Springer GTM 60(1980).[6] V.I. Arnol′d: Geometrical Methods in the Theory of Ordinary Differential Equa-tions (2nd ed.), Springer–Verlag (1988).[7] V.I. Arnol′d, V.S. Afrajmovich, Yu.S.Il′yashenko, L.P. Shil′nikov: I. BifurcationTheory in Dynamical Systems V (V.I. Arnol′d Ed.), Encyclopaedia of MathematicalSciences, Springer–Verlag (1982).[8] D.G. Aronson, M.A. Chory, G.R. Hall, R.P. McGehee: Bifurcations from an in-variant circle for two-parameter families of maps of the plane: a computer-assistedstudy, Comm. Math. Phys. 83(3) (1982), 303–354.[9] C. Baesens, J. Guckenheimer, S. Kim, R.S. McKay: Three coupled oscillators:mode-locking, global bifurcations and toroidal chaos, Phys. D 49(3) (1991), 387–475.[10] M. Benedicks, L. Carleson: On iterations of 1 − ax2 on (−1, 1), Ann. of Math.(2) 122(1) (1985), 1–25.[11] M. Benedicks, L. Carleson: The dynamics of the Hénon map, Ann. of Math. (2)133(1) (1991), 73–169.[12] G. Benettin, L. Galgani, A. Giorgilli, J.-M. Strelcyn: Lyapunov characteristicExponents for Smooth Dynamical Systems: A Method for Computing All of Them,Meccanica 15 (1980), 9–30.154 Bibliography[13] M. Bosch, C. Simó: Attractors in a Shil′nikov–Hopf scenario and a related one-dimensional map, Phys. D 62 (1993), 217–229.[14] B.L.J. Braaksma, H.W. Broer: On a quasi-periodic Hopf bifurcation, Ann. Inst.H. Poincaré Anal. Non Linéaire 4(2) (1987), 115–168.[15] W.L. Briggs, V.E. Henson: The DFT: An Owner’s Manual for the DiscreteFourier Transform, SIAM, Philadelphia (1995).[16] H.W. Broer: Formal normal forms for vector fields and some consequences forbifurcations in the volume preserving case, in Dynamical Systems and turbulence,Warwick 1980 (A. Dold and B. Eckmann Eds.), Springer LNM 898 (1980), 54–74.[17] H.W. Broer, G.B. Huitema, M.B. Sevryuk: Quasi-periodic Motions in Familiesof Dynamical Systems, Order amidst Chaos, Springer LNM 1645 (1996).[18] H.W. Broer, G.B. Huitema, F. Takens, B.L.J. Braaksma: Unfoldings and bifur-cations of quasi-periodic tori, Mem. AMS 83(421) (1990), 1–175.[19] H.W. Broer, H.M. Osinga, G. Vegter: Algorithms for computing normally hy-perbolic invariant manifolds, Z. angew. Math. Phys. 48 (1997), 480–524.[20] H.W. Broer, R. Roussarie: Exponential confinement of chaos in the bifurca-tion sets of real analytic diffeomorphisms, in Global Analysis of Dynamical Sys-tems, Festschrift dedicated to Floris Takens for his 60th birthday (H.W. Broer,B. Krauskopf, and G. Vegter Eds.), IOP Publish., Bristol (2001).[21] H.W. Broer, R. Roussarie, C. Simó: Invariant circles in the Bogdanov–Takensbifurcation for diffeomorphisms, ETDS 16 (1996), 1147–1172.[22] H.W. Broer, C. Simó: Hill’s equation with quasi-periodic forcing: resonancetongues, instability pockets and global phenomena, Bol. Soc. Bras. Mat 29 (1998),253–293.[23] H.W. Broer, C. Simó, J.C. Tatjer: Towards global models near homoclinic tan-gencies of dissipative diffeomorphisms, Nonlinearity 11 (1998), 667–770.[24] H.W. Broer, C. Simó, R. Vitolo: Bifurcations and strange attractors in theLorenz-84 climate model with seasonal forcing, Nonlinearity 15(4) (2002), 1205–1267.[25] H.W. Broer, C. Simó, R. Vitolo: Hénon-like strange attractors in a family ofmaps of the solid torus, preprint University of Groningen, submitted (2002).[26] H.W. Broer, C. Simó, R. Vitolo: The Hopf-saddle-node bifurcation for fixedpoints of diffeomorphisms, in preparation.[27] H.W. Broer, C. Simó, R. Vitolo: Quasi-periodic Hénon-like attractors in theLorenz-84 climate model with seasonal forcing, to appear in Proceedings Equadiff2003.Bibliography 155[28] H.W. Broer, S.J. van Strien: Infinitely many moduli of strong stability in di-vergence free unfoldings of singularities of vector fields, in Geometric Dynamics,Proceedings, Rio de Janeiro 1981 (J. Palis Ed.), Springer LNM 1007 (1983), 39–59.[29] H.W. Broer, F. Takens: Formally symmetric normal forms and genericity, Dy-namics Reported 2 (1989), 36–60.[30] H.W. Broer, F. 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Bifurcations of attractors in 3D diffeomorphisms:a study in experimental mathematics

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   University of GroningenBifurcations of attractors in 3D diffeomorphismsVitolo, RenatoIMPORTANT NOTE: You are advised to consult the publisher's version (publisher's PDF) if you wish to cite fromit. Please check the document version below.Document VersionPublisher's PDF, also known as Version of recordPublication date:2003Link to publication in University of Groningen/UMCG research databaseCitation for published version (APA):Vitolo, R. (2003). Bifurcations of attractors in 3D diffeomorphisms: a study in experimental mathematics.Groningen: s.n.CopyrightOther than for strictly personal use, it is not permitted to download or to forward/distribute the text or part of it without the consent of theauthor(s) and/or copyright holder(s), unless the work is under an open content license (like Creative Commons).Take-down policyIf you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediatelyand investigate your claim.Downloaded from the University of Groningen/UMCG research database (Pure): http://www.rug.nl/research/portal. For technical reasons thenumber of authors shown on this cover page is limited to 10 maximum.Download date: 12-11-2019Bibliography[1] V.S. Afrajmovich, L.P. Shil′nikov: On invariant two-dimensional tori, their dis-integration and stochasticity, in Methods of the qualitative theory of differentialequations, Gor′kov. Gos. University, Gor′kij (1983), 3–26.[2] V.S. Anishchenko: Dynamical Chaos—Models and Experiments, World Scientific(1995).[3] V.S. Anishchenko, M.A. Safonova, U. Feudel, J. Kurths: Bifurcations and transi-tion to chaos through three-dimensional tori, IJBC 3(4) (1994), 595–607.[4] V.I. Arnol′d: Small denominators, I: Mappings of the circumference into itself,AMS Transl. (Ser. 2) 46 (1965), 213–284.[5] V.I. Arnol′d: Mathematical Methods of Classical Mechanics , Springer GTM 60(1980).[6] V.I. Arnol′d: Geometrical Methods in the Theory of Ordinary Differential Equa-tions (2nd ed.), Springer–Verlag (1988).[7] V.I. Arnol′d, V.S. Afrajmovich, Yu.S.Il′yashenko, L.P. Shil′nikov: I. BifurcationTheory in Dynamical Systems V (V.I. Arnol′d Ed.), Encyclopaedia of MathematicalSciences, Springer–Verlag (1982).[8] D.G. Aronson, M.A. Chory, G.R. Hall, R.P. McGehee: Bifurcations from an in-variant circle for two-parameter families of maps of the plane: a computer-assistedstudy, Comm. Math. Phys. 83(3) (1982), 303–354.[9] C. Baesens, J. Guckenheimer, S. Kim, R.S. McKay: Three coupled oscillators:mode-locking, global bifurcations and toroidal chaos, Phys. D 49(3) (1991), 387–475.[10] M. Benedicks, L. Carleson: On iterations of 1 − ax2 on (−1, 1), Ann. of Math.(2) 122(1) (1985), 1–25.[11] M. Benedicks, L. Carleson: The dynamics of the He´non map, Ann. of Math. (2)133(1) (1991), 73–169.[12] G. Benettin, L. Galgani, A. Giorgilli, J.-M. Strelcyn: Lyapunov characteristicExponents for Smooth Dynamical Systems: A Method for Computing All of Them,Meccanica 15 (1980), 9–30.154 Bibliography[13] M. Bosch, C. Simo´: Attractors in a Shil′nikov–Hopf scenario and a related one-dimensional map, Phys. D 62 (1993), 217–229.[14] B.L.J. Braaksma, H.W. Broer: On a quasi-periodic Hopf bifurcation, Ann. Inst.H. Poincare´ Anal. Non Line´aire 4(2) (1987), 115–168.[15] W.L. Briggs, V.E. Henson: The DFT: An Owner’s Manual for the DiscreteFourier Transform, SIAM, Philadelphia (1995).[16] H.W. 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Simo´: Invariant circles in the Bogdanov–Takensbifurcation for diffeomorphisms, ETDS 16 (1996), 1147–1172.[22] H.W. Broer, C. Simo´: Hill’s equation with quasi-periodic forcing: resonancetongues, instability pockets and global phenomena, Bol. Soc. Bras. Mat 29 (1998),253–293.[23] H.W. Broer, C. Simo´, J.C. Tatjer: Towards global models near homoclinic tan-gencies of dissipative diffeomorphisms, Nonlinearity 11 (1998), 667–770.[24] H.W. Broer, C. Simo´, R. Vitolo: Bifurcations and strange attractors in theLorenz-84 climate model with seasonal forcing, Nonlinearity 15(4) (2002), 1205–1267.[25] H.W. Broer, C. Simo´, R. Vitolo: He´non-like strange attractors in a family ofmaps of the solid torus, preprint University of Groningen, submitted (2002).[26] H.W. Broer, C. Simo´, R. Vitolo: The Hopf-saddle-node bifurcation for fixedpoints of diffeomorphisms, in preparation.[27] H.W. Broer, C. Simo´, R. 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A unifying view of bifurcations in a semiconductor laser subject to optical injection,

Active and passive ocean regimes in a low-order climate model, Tellus 53A

Alligood: Cascades of period-doubling bifurcations: a prerequisite for horseshoes,

Lakshmanan: Interruption of torus doubling bifurcation and genesis of strange nonchaotic attractors in a quasiperiodically forced map: Mechanisms and their characterizations,

Multidimensional nonhyperbolic attractors,

Positive Lyapunov exponent for generic one-parameter families of unimodal maps,

Semi-local analysis of the k:1 and k:2 resonances in quasiperiodically forced systems, in Global Analysis of Dynamical Systems, Festschrift dedicated to Floris Takens for his 60th birthday

Stochastic dynamics of deterministic systems,

Strange attractors in higher dimensions,

Takens: Forced oscillations and bifurcations, in Applications of Global Analysis

Tatjer: Three dimensional dissipative diﬀeomorphisms with homoclinic tangencies,

The dynamical complexity of optically injected semiconductor lasers,

Tsujii: A simple proof for monotonicity of entropy in the quadratic family,

Varadarajan: Lie Groups, Lie Algebras, and Their Representations,

Veen: Time scale interaction in low-order climate models,

Bifurcations of attractors in 3D diffeomorphisms : a study in experimental mathematics

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Bifurcations of attractors in 3D diffeomorphisms: a study in experimental mathematics

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Bifurcations of attractors in 3D diffeomorphisms : a study in experimental mathematics

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