Citation for published version (APA): Holtman, S-J. (2009). Dynamics and geometry near resonant bifurcations Groningen: s.n.

Similar documents
DRIVEN and COUPLED OSCILLATORS. I Parametric forcing The pendulum in 1:2 resonance Santiago de Compostela

Geometry of Resonance Tongues

Resonance and fractal geometry

Resonance and fractal geometry

University of Groningen. Bifurcations in Hamiltonian systems Lunter, Gerard Anton

The Geometry of Resonance Tongues: A Singularity Theory Approach

Secular and oscillatory motions in dynamical systems. Henk Broer Johann Bernoulli Instituut voor Wiskunde en Informatica Rijksuniversiteit Groningen

Survey of strong normal-internal k : l resonances in quasi-periodically driven oscillators for l = 1, 2, 3.

Dynamics of Modified Leslie-Gower Predator-Prey Model with Predator Harvesting

Citation for published version (APA): Sok, R. M. (1994). Permeation of small molecules across a polymer membrane: a computer simulation study s.n.

Analysis of the Takens-Bogdanov bifurcation on m parameterized vector fields

In Arnold s Mathematical Methods of Classical Mechanics (1), it

Elements of Applied Bifurcation Theory

Schilder, F. (2005). Algorithms for Arnol'd tongues and quasi-periodic tori : a case study.

ME DYNAMICAL SYSTEMS SPRING SEMESTER 2009

DIFFERENTIAL EQUATIONS, DYNAMICAL SYSTEMS, AND AN INTRODUCTION TO CHAOS

Elements of Applied Bifurcation Theory

Introduction to Applied Nonlinear Dynamical Systems and Chaos

University of Groningen. Morphological design of Discrete-Time Cellular Neural Networks Brugge, Mark Harm ter

Elements of Applied Bifurcation Theory

Citation for published version (APA): Kooistra, F. B. (2007). Fullerenes for organic electronics [Groningen]: s.n.

Im(v2) Im(v3) Re(v2)

University of Groningen. Event-based simulation of quantum phenomena Zhao, Shuang

BIFURCATION PHENOMENA Lecture 4: Bifurcations in n-dimensional ODEs

Citation for published version (APA): Sarma Chandramouli, V. V. M. (2008). Renormalization and non-rigidity s.n.

AMADEU DELSHAMS AND RAFAEL RAMíREZ-ROS

Dynamical Systems. Pierre N.V. Tu. An Introduction with Applications in Economics and Biology Second Revised and Enlarged Edition.

University of Groningen. Extraction and transport of ion beams from an ECR ion source Saminathan, Suresh

DIFFERENTIAL EQUATIONS, DYNAMICAL SYSTEMS, AND AN INTRODUCTION TO CHAOS

Theoretical simulation of nonlinear spectroscopy in the liquid phase La Cour Jansen, Thomas

Resonance and Fractal Geometry

Global Analysis of Dynamical Systems Festschrift dedicated to Floris Takens for his 60th birthday

Citation for published version (APA): Fathi, K. (2004). Dynamics and morphology in the inner regions of spiral galaxies Groningen: s.n.

SD - Dynamical Systems

WHAT IS A CHAOTIC ATTRACTOR?

Superfluid helium and cryogenic noble gases as stopping media for ion catchers Purushothaman, Sivaji

University of Groningen. Statistical inference via fiducial methods Salomé, Diemer

Geometric approximation of curves and singularities of secant maps Ghosh, Sunayana

Citation for published version (APA): Shen, C. (2006). Wave Propagation through Photonic Crystal Slabs: Imaging and Localization. [S.l.]: s.n.

Difference Resonances in a controlled van der Pol-Duffing oscillator involving time. delay

DIFFERENTIAL GEOMETRY APPLIED TO DYNAMICAL SYSTEMS

University of Groningen. Laser Spectroscopy of Trapped Ra+ Ion Versolato, Oscar Oreste

System-theoretic properties of port-controlled Hamiltonian systems Maschke, B.M.; van der Schaft, Arjan

Stability and Bifurcation in the Hénon Map and its Generalizations

Invariant manifolds of the Bonhoeffer-van der Pol oscillator

Towards a Global Theory of Singularly Perturbed Dynamical Systems John Guckenheimer Cornell University

University of Groningen. Hollow-atom probing of surfaces Limburg, Johannes

A PROOF THAT S-UNIMODAL MAPS ARE COLLET-ECKMANN MAPS IN A SPECIFIC RANGE OF THEIR BIFURCATION PARAMETERS. Zeraoulia Elhadj and J. C.

Can a Hexapole magnet of an ECR Ion Source be too strong? Drentje, A. G.; Barzangy, F.; Kremers, Herman; Meyer, D.; Mulder, J.; Sijbring, J.

On Parametrized KAM Theory

NONLINEAR DYNAMICS PHYS 471 & PHYS 571

COMPLEX DYNAMICS AND CHAOS CONTROL IN DUFFING-VAN DER POL EQUATION WITH TWO EXTERNAL PERIODIC FORCING TERMS

arxiv: v1 [nlin.cd] 20 Jul 2010

SPLITTING OF SEPARATRICES FOR (FAST) QUASIPERIODIC FORCING. splitting, which now seems to be the main cause of the stochastic behavior in

Two models for the parametric forcing of a nonlinear oscillator

Citation for published version (APA): Kooistra, F. B. (2007). Fullerenes for organic electronics [Groningen]: s.n.

Determining the Global Topology of Resonance Surfaces for Periodically Forced Oscillator Families

Homoclinic tangles associated with closed invariant curves in families of 2D maps

vii Contents 7.5 Mathematica Commands in Text Format 7.6 Exercises

Citation for published version (APA): Hoekstra, S. (2005). Atom Trap Trace Analysis of Calcium Isotopes s.n.

SUPPLEMENTARY INFORMATION

Dynamical Systems with Applications using Mathematica

Citation for published version (APA): Andogah, G. (2010). Geographically constrained information retrieval Groningen: s.n.

Nonlinear Dynamics and Chaos Summer 2011

DYNAMICAL SYSTEMS. I Clark: Robinson. Stability, Symbolic Dynamics, and Chaos. CRC Press Boca Raton Ann Arbor London Tokyo

BIFURCATION PHENOMENA Lecture 1: Qualitative theory of planar ODEs

Bifurcations of phase portraits of pendulum with vibrating suspension point

On the periodic logistic equation

Differential Equations Dynamical Systems And An Introduction To Chaos Solutions Manual Pdf

On low speed travelling waves of the Kuramoto-Sivashinsky equation.

Invariant manifolds in dissipative dynamical systems

BIBLIOGRAPHY 201. [33], The Hopf-Saddle-Node bifurcation for fixed points of 3D-diffeomorphisms, a dynamical inventory, preprint Groningen, 2004.

UNIFORM SUBHARMONIC ORBITS FOR SITNIKOV PROBLEM

Computing 3D Bifurcation Diagrams

University of Groningen. The Parametrically Forced Pendulum Broer, Hendrik; Hoveijn, I.; Noort, M. van; Simó, C.; Vegter, Geert

QUASIPERIODIC RESPONSE TO PARAMETRIC EXCITATIONS

Period-doubling cascades of a Silnikov equation

Hopf bifurcation in coupled cell networks with abelian symmetry 1

Citation for published version (APA): Raimond, J. J. (1934). The coefficient of differential galactic absorption Groningen: s.n.

SANGRADO PAGINA 17CMX24CM. PhD Thesis. Splitting methods for autonomous and non-autonomous perturbed equations LOMO A AJUSTAR (AHORA 4CM)

Simple conservative, autonomous, second-order chaotic complex variable systems.

University of Groningen. Dynamics amidst folding and twisting in 2-dimensional maps Garst, Swier

System theory and system identification of compartmental systems Hof, Jacoba Marchiena van den

Citation for published version (APA): Halbersma, R. S. (2002). Geometry of strings and branes. Groningen: s.n.

Hamiltonian Dynamics

The role of camp-dependent protein kinase A in bile canalicular plasma membrane biogenesis in hepatocytes Wojtal, Kacper Andrze

Maslov indices and monodromy

Resonance and Fractal Geometry

A Trivial Dynamics in 2-D Square Root Discrete Mapping

Dual photo- and redox- active molecular switches for smart surfaces Ivashenko, Oleksii

Renormalization and dimension for Kuperberg minimal sets

Peptide folding in non-aqueous environments investigated with molecular dynamics simulations Soto Becerra, Patricia

Stability Analysis of Uzawa-Lucas Endogenous Growth Model

Method of Averaging for Differential Equations on an Infinite Interval

University of Groningen. Atmospheric variability and the Atlantic multidecadal oscillation Sterk, Alef

Additive resonances of a controlled van der Pol-Duffing oscillator

CANARDS AND HORSESHOES IN THE FORCED VAN DER POL EQUATION

THE CENTER-FOCUS PROBLEM AND BIFURCATION OF LIMIT CYCLES IN A CLASS OF 7TH-DEGREE POLYNOMIAL SYSTEMS

University of Groningen. Enantioselective liquid-liquid extraction in microreactors Susanti, Susanti

Tobias Holck Colding: Publications. 1. T.H. Colding and W.P. Minicozzi II, Dynamics of closed singularities, preprint.

Transcription:

University of Groningen Dynamics and geometry near resonant bifurcations Holtman, Sijbo-Jan IMPORTANT NOTE: You are advised to consult the publisher's version (publisher's PDF) if you wish to cite from it. Please check the document version below. Document Version Publisher's PDF, also known as Version of record Publication date: 2009 Link to publication in University of Groningen/UMCG research database Citation for published version (APA): Holtman, S-J. (2009). Dynamics and geometry near resonant bifurcations Groningen: s.n. Copyright Other 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 the author(s) and/or copyright holder(s), unless the work is under an open content license (like Creative Commons). Take-down policy If you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediately and investigate your claim. Downloaded from the University of Groningen/UMCG research database (Pure): http://www.rug.nl/research/portal. For technical reasons the number of authors shown on this cover page is limited to 10 maximum. Download date: 05-04-2018

Bibliography [Arn82] [Arn93] [BCKV93] [BCKV95] V.I. Arnol d. Geometrical Methods in the Theory of Ordinary Differential Equations. Springer Berlin, 1982. V.I. Arnol d, editor. Dynamical systems VI: Singularity theory I, volume 6 of Ecyclopaedia of Mathematical Sciences. Springer-Verlag, Berlin, 1993. H.W. Broer, S.N. Chow, Y. Kim, and G. Vegter. A normally elliptic hamiltonian bifurcation. ZAMP, pages 389 432, 1993. H.W. Broer, S.N. Chow, Y. Kim, and G. Vegter. The hamiltonian doublezero eigenvalue. In W.F. Langford and W. Nagata, editors, Normal Forms and Homoclinic Chaos (Waterloo 1992), volume 4, pages 1 19. Fields Institute Communications, 1995. [BdBGV81] J.G. Blom, R. de Bruin, J. Grasman, and J.G. Verwer. Forced prey-predator oscillations. Journal of Mathematical Biology, 12:141 152, 1981. [BGV03] [BGV07] [BHLV98a] H.W. Broer, M. Golubitsky, and G. Vegter. The geometry of resonance tongues: A singularity approach. Nonlinearity, 16:1511 1538, 2003. H.W. Broer, M. Golubitsky, and G. Vegter. Geometry of resonance tongues. Singularity Theory. Proceedings of the 2005 Marseille Singularity School and Conference, pages 327 356, 2007. H.W. Broer, I. Hoveijn, G.A. Lunter, and G. Vegter. Equivariant singularity theory with distinguished parameters, two case studies of resonant Hamiltonian systems. Physica D, 112:64 80, 1998. [BHLV98b] H.W. Broer, I. Hoveijn, G.A. Lunter, and G. Vegter. Resonance in a springpendulum: algorithms for equivariant singularity theory. Nonlinearity, 11:1 37, 1998. [BHLV03] H.W. Broer, I. Hoveijn, G.A. Lunter, and G. Vegter. Bifurcations in Hamiltonian systems: Computing singularities by Gröbner bases. Springer LNM 1806, 2003. ISBN 3-540-00403-3.

128 Bibliography [BHV08] [BL95] [BNR + 07] [BNRS06] [BR01] [Brö75] [Bro09] [BS98] [BS00] [BST98] [BSV02] [BSV08] [BV92] H.W. Broer, S.J. Holtman, and G. Vegter. Recognition of the bifurcation type of resonance in mildly degenerate Hopf-Neĭmark-Sacker families. Nonlinearity, 21:2463 2482, 2008. H.W. Broer and M. Levi. Geometrical aspects of stability theory for Hill s equations. Arch. Rational Mech. Anal., 131:225 240, 1995. H.W. Broer, V. Naudot, R. Roussarie, K. Saleh, and F.O.O. Wagener. Organising centres in the semi-global analysis of dynamical systems. Int. J. Appl. M. Stat., 12(D7), 2007. H.W. Broer, V. Naudot, R. Roussarie, and K. Saleh. A predator-prey model with non-monotonic response function. Regular and Chaotic Dynamics, 11:155 165, 2006. H.W. Broer and R. Roussarie. Exponential confinement of chaos in the bifurcation set of real analytic diffeomorphisms. In H.W. Broer, B. Krauskopf, and G. Vegter, editors, Global Analysis of Dynamical systems, Festschrift dedicated to Floris Takens for his 60th birthday, pages 167 210. IOP, Bristol and Philadelphia, 2001. T. Bröcker. Differential Germs and Catastrophes. London Mathematical Society Lecture Note Series. Cambridge University Press, Cambridge, 1975. H.W. Broer. Normal forms in perturbation theory. In R. Meyers, editor, Encyclopædia of Complexity and System Science. Springer-Verlag, 2009. H.W. Broer and C. Simo. Hill s equation with quasi-periodic forcing: resonance tongues, instability pockets and global phenomena. Bol. Soc. Bras. Mat., 29:253 293, 1998. H.W. Broer and C. Simo. Resonance tongues in Hill s equations: a geometric approach. J. Diff. Eqns., 166:290 327, 2000. H.W. Broer, C. Simó, and J.C. Tatjer. Towards global models near homoclinic tangencies of dissipative diffeomorphisms. Nonlinearity, 11:667 770, 1998. H.W. Broer, C. Simó, and R. Vitolo. Bifurcations and strange attractors in the lorenz-84 climate model with seasonal forcing. Nonlinearity, 15(4):1205 1267, 2002. H.W. Broer, C. Simó, and R. Vitolo. The Hopf-saddle-node bifurcation for fixed points of 3D-diffeomorphisms: analysis of a resonance bubble. Phys. D, 237(13):1773 1799, 2008. H.W. Broer and G. Vegter. Bifurcational aspects of parametric resonance. In Dynamics Reported, volume 1 of New Series, pages 1 51. Springer-Verlag, 1992.

Bibliography 129 [BV08] [Che85a] [Che85b] [Che88] [Dev89] [DRSZ91] [DV94] [GH83] [GS88] [GSS85] H.W. Broer and G. Vegter. Generic Hopf-Neĭmarck-Sacker bifurcations in feedforward systems. Nonlinearity, 21:1547 1578, 2008. A. Chenciner. Bifurcations de points fixes elliptiques, i. courbes invariantes. Publ. Math. IHES, 61:67 127, 1985. A. Chenciner. Bifurcations de points fixes elliptiques, ii. orbites périodiques et ensembles de cantor invariants. Invent. Math., 80:81 106, 1985. A. Chenciner. Bifurcations de points fixes elliptiques, iii. orbites périodiques de petites périodes et elimination résonnantes des couples de courbes invariantes. Publ. Math. IHES, 66:5 91, 1988. R.L. Devaney. An introduction to chaotic dynamical systems. Addison-Wesley, 1989. F. Dumortier, R. Rousarie, J. Sotomayor, and H. Zoladek. Bifurcations on planar vector fields. Springer-Verlag, 1991. A. Doelman and F. Verhulst. Bifurcations and strongly non-linear self-exited oscillations. Mathematical Methods in the Applied Sciences, 17:189 207, 1994. J. Guckenheimer and P. Holmes. Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields. Springer-Verlag, 1983. M. Golubitsky and D.G. Schaeffer. Singularities and Groups in Bifurcation Theory: Vol. II, volume 69 of Applied Mathematical Sciences. Springer-Verlag, New-York, 1988. M. Golubitsky, I.N. Stewart, and D.G. Schaeffer. Singularities and Groups in Bifurcation Theory: Vol. I, volume 51 of Applied Mathematical Sciences. Springer-Verlag, New-York, 1985. [GWdPL76] C.G. Gibson, K. Wirthmüller, A.A. du Plessis, and E.J.N. Looijenga. Topological Stability of Smooth Mappings. Springer-Verlag, Berlin, 1976. Lecture Notes in Mathematics, Vol. 552. [Hal80] J. Hale. Ordinary Differential Equations. Krieger, 1980. [HH81] [Hol] [Hum79] C. Holmes and P. Holmes. Second order averaging and bifurcations of subharmonics in Duffing s equation. Journal of Sound and Vibration, 78(2):161 174, 1981. S.J. Holtman. Homepage. http://www.math.rug.nl/~sijbo. A. Hummel. Bifurcations of periodic points. PhD thesis, University of Groningen, 1979.

130 Bibliography [Ily08] [IY95] [JZ05] [Kif08] [Kra94] [Kuz95] [LM09] Y. Ilyashenko. Some open problems in real and complex dynamical systems. Nonlinearity, 21:T101 T107, 2008. Y. Ilyashenko and S. Yakovenko. Finite cyclicity of elementary polycycles in generic families. In Concerning the Hilbert 16th problem, volume 165 of Amer. Math. Soc. Transl. Ser. 2, pages 21 95. Amer. Math. Soc., Providence, RI, 1995. A. Jorba and M. Zou. A software package for the numerical integration of ODEs by means of high-order Taylor methods. Experiment. Math., 14(1):99 117, 2005. Y. Kifer. Convergence, nonconvergence and adiabatic transitions in fully coupled averaging. Nonlinearity, 21:T27 T32, 2008. B. Krauskopf. Bifurcation sequences at 1:4 resonance: an inventory. Nonlinearity, 7:1073 1091, 1994. Y.A. Kuznetsov. Elements of Applied Bifurcation Theory, volume 112 of Applied Mathematical Sciences. Springer-Verlag, Berlin and New-York, 1995. H.E. Lomeli and J.D. Meiss. Resonance zones and lobe volumes for exact volume-preserving maps. Nonlinearity, 22:1761 1789, 2009. [Mou91] A. Mourtada. Cyclicité finie des polycycles hyperboliques de champs de vecteurs du plan. Algorithme de finitude. Ann. Inst. Fourier (Grenoble), 41(3):719 753, 1991. [MP95] [MP96] R.P. McGehee and B.B. Peckham. Determining the global topology of resonance surfaces for periodically forced oscillator families. In W.F. Langford and W. Nagata, editors, Normal Forms and Homoclinic Chaos, volume 4 of Fields Institute Communications, pages 233 254. AMS, 1995. R.P. McGehee and B.B. Peckham. Arnol d flames and resonance surface folds. Int. J. Bifurcations and Chaos, 6:315 336, 1996. [Mun99] J.R. Munkres. Topology. Prentice Hall, 1999. [NPT83] [PFK95] [PK91] S.E. Newhouse, J. Palis, and F. Takens. Bifurcation and stability of families of diffeomorphisms. Publ.Math.IHES, 57:1 71, 1983. B.B. Peckham, C.E. Frouzakis, and I.G. Kevrekidis. Bananas and banana splits: a parametric degeneracy in the Hopf bifurcation for maps. SIAM J. Math. Anal., 26:190 217, 1995. B.B. Peckham and I.G. Kevrekidis. Period doubling with higher-order degeneracies. SIAM J. Math. Anal., 22:1552 1574, 1991.

Bibliography 131 [PK02] [PS96] [PT87] B.B. Peckham and I.G. Kevrekidis. Lighting Arnol d flames: Resonance in doubly forced periodic oscillators. Nonlinearity, 15:405 428, 2002. T. Poston and I. Stewart. Catastrophe theory and its applications. Dover publications, Dover, 1996. J. Palis and F. Takens. Hyperbolicity and the creation of homoclinic orbits. Ann. Math., 125:337 374, 1987. [Rey80] J. W. Reyn. Generation of limit cycles from separatrix polygons in the phase plane. In Geometrical approaches to differential equations (Proc. Fourth Scheveningen Conf., Scheveningen, 1979), volume 810 of Lecture Notes in Math., pages 264 289. Springer, Berlin, 1980. [Rou86] [Sim90] [SKT04] [SVM07] [Tak74a] R. Roussarie. On the number of limit cycles which appear by perturbation of separatrix loop of planar vector fields. Bol. Soc. Brasil. Mat., 17(2):67 101, 1986. C. Simó. On the analytical and numerical approximation of invariant manifolds. In Daniel Benest and Claude Froeschlé, editors, Méthodes Modernes de la Mecánique Céleste (Course given at Goutelas, France, 1989), pages 285 329. Editions Frontières, 1990. M. Siewe, F.M. Kameni, and C. Tchawona. Resonant oscillation and homoclinic bifurcation in a Φ 6 -van der pol oscillator. Chaos, Solitons and Fractals, 24(4):841 853, 2004. J.A. Sanders, F. Verhulst, and J. Murdock. Averaging Methods in Nonlinear Dynamical Systems, volume 59 of Applied Mathematical Sciences. Springer, 2007. F. Takens. Forced oscillations and bifurcations. In Applications of Global Analysis I, volume 3, pages 1 59. Communications of the Mathematical Institute of the University of Utrecht, 1974. Reprinted in: H.W. Broer, B. Krauskopf and G. Vegter, editors, Global Analysis of Dynamical Systems. Festschrift Dedicated to Floris Takens for his 60th Birthday (Leiden, 2001), Inst. Physics, Bristol (2001), 1-61. [Tak74b] F. Takens. Singularities of vector fields. Publ. Math. IHES, 42:48 100, 1974. [Tho75] [Van92] R. Thom. Structural Stability and Morphogenesis. Benjamin-Addison Wesley, New-York, 1975. A. Vanderbauwhede. Branching of periodic solutions in time-reversible systems. In H.W. Broer and F. Takens, editors, Geometry and Analysis in Non-linear Dynamics, volume 222 of Pitman Research Notes in Mathematics, pages 97 113. Pitman London, 1992.

132 Bibliography [vdpvdm27] B. van der Pol and J. van der Mark. Frequency demultiplication. Nature, 120:363 364, 1927. [Wol08] [Yag96] Wolfram Research, Inc. Mathematica, Version 7.0. Wolfram Research, Inc., Champaign, Illinois, 2008. K. Yagasaki. Second-order averaging and Melnikov analyses for forced nonlinear oscillators. Journal of Sound and Vibration, 190(4):587 609, 1996.