Organising Centres in the Semi-global Analysis of Dynamical Systems

The notion of organising centre of a bifurcation diagram is used as an ordering principle in the analysis of nonlinear problems in pure and applied dynamical systems theory. When considering a given, generic n-parameter family of dynamical systems the codimension n bifurcations are isolated in the parameter space, generating a more global array of lower codimension bifurcations. It often makes sense to add one extra parameter to the system, e.g., by varying a ‘constant’ coefficient. In such cases, semi-global parts of the given n-dimensional bifurcation set can be understood as generic slices in versal unfoldings of codimension n + 1 singularities. This can give great insight in the structure of the given system, as we shall illustrate in two extensive case studies.

[1]  Hendrik Broer,et al.  Formally symmetric normal forms and genericity , 1989 .

[2]  F. Wagener.  Semi-local analysis of the k:1 and k:2 resonances in quasi-periodically forced systems , 2001 .

[3]  Andrey Shilnikov,et al.  Bifurcation and predictability analysis of a low-order atmospheric circulation model , 1995 .

[4]  Florian Wagener,et al.  Skiba points and heteroclinic bifurcations, with applications to the shallow lake system , 2003 .

[5]  J. Beddington,et al.  Mutual Interference Between Parasites or Predators and its Effect on Searching Efficiency , 1975 .

[6]  Bernd Krauskopf,et al.  Codimension-three unfoldings of reflectionally symmetric planar vector fields , 1997 .

[7]  George Huitema,et al.  Toward a quasi-periodic bifurcation theory , 1990 .

[8]  Carles Simó,et al.  Towards global models near homoclinic tangencies of dissipative diffeomorphisms , 1998 .

[9]  J. Cowan.  Nonlinear oscillations, dynamical systems, and bifurcations of vector fields : John Guckenheimer and Philip Holmes Springer-Verlag, New York, 1983, 453 pp., $32.00 , 1986 .

[10]  Bernd Krauskopf,et al.  Nonlinear Dynamics of Interacting Populations , 1998 .

[11]  H. Broer.  Survey on dissipative KAM theory including quasi-periodic bifurcation theory based on lectures by Henk Broer , 2005 .

[12]  George Huitema,et al.  Quasi-Periodic Motions in Families of Dynamical Systems: Order amidst Chaos , 2002 .

[13]  Àngel Jorba,et al.  On the Persistence of Lower Dimensional Invariant Tori under Quasi-Periodic Perturbations , 1997 .

[14]  Thomas F. Fairgrieve,et al.  AUTO 2000 : CONTINUATION AND BIFURCATION SOFTWARE FOR ORDINARY DIFFERENTIAL EQUATIONS (with HomCont) , 1997 .

[15]  H. Broer,et al.  Global Analysis of Dynamical Systems , 2001 .

[16]  G. F. Gause.  Experimental Studies on the Struggle for Existence: I. Mixed Population of Two Species of Yeast , 1932 .

[17]  Yuri A. Kuznetsov,et al.  Multiple attractors, catastrophes and chaos in seasonally perturbed predator-prey communities , 1993 .

[18]  J. Norris.  The closing of Arnol'd tongues for a periodically forced limit cycle , 1993 .

[19]  George Huitema,et al.  Unfoldings and Bifurcations of Quasi-Periodic Tori , 1990 .

[20]  C. S. Holling.  Some Characteristics of Simple Types of Predation and Parasitism , 1959, The Canadian Entomologist.

[21]  Huaiping Zhu,et al.  Bifurcation Analysis of a Predator-Prey System with Nonmonotonic Functional Response , 2003, SIAM J. Appl. Math..

[22]  M. L. Cartwright.  On non-linear differential equations of the second order , 1949 .

[23]  J. E. Littlewood,et al.  On Non‐Linear Differential Equations of the Second Order: I. the Equation y¨ − k(1‐y2)y˙ + y = bλk cos(λl + α), k Large , 1945 .

[24]  M. Deakin.  Catastrophe theory. , 1977, Science.

[25]  Robert Roussarie,et al.  Dynamics of a predator-prey model with non-monotonic response function , 2007 .

[26]  Bernd Krauskopf,et al.  Chaos in Periodically Driven Systems , 2000 .

[27]  J. Guckenheimer,et al.  Simple Resonance Regions of Torus Diffeomorphisms , 1991 .

[28]  Daan Lenstra,et al.  Fundamental issues of nonlinear laser dynamics. , 2000 .

[29]  Carles Simó,et al.  Bifurcations and strange attractors in the Lorenz-84 climate model with seasonal forcing , 2002 .

[30]  P. Holmes,et al.  Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields , 1983, Applied Mathematical Sciences.

[31]  T. W. Barrett,et al.  Catastrophe Theory, Selected Papers 1972-1977 , 1978, IEEE Transactions on Systems, Man, and Cybernetics.

[32]  H. Broer,et al.  Bifurcations of a predator-prey model with non-monotonic response function , 2005 .

[33]  David A. Rand,et al.  Bifurcations of the forced van der Pol oscillator , 1978 .

[34]  F. Wagener.  Structural analysis of optimal investment for firms with non-concave revenues , 2005 .

[35]  H. Broer,et al.  A Predator-Prey Model with Non-Monotonic Response Function , 2006 .

[36]  Floris Takens,et al.  Hyperbolicity and sensitive chaotic dynamics at homoclinic bifurcations : fractal dimensions and infinitely many attractors , 1993 .

[37]  Bernd Krauskopf,et al.  A planar model system for the saddle-node Hopf bifurcation with global reinjection , 2004 .

[38]  Freddy Dumortier,et al.  Generic 3-parameter families of vector fields on the plane, unfolding a singularity with nilpotent linear part. The cusp case of codimension 3 , 1987, Ergodic Theory and Dynamical Systems.

[39]  H. Broer,et al.  On a quasi-periodic Hopf bifurcation , 1987 .

[40]  René Thom,et al.  Structural stability and morphogenesis - an outline of a general theory of models , 1977, Advanced book classics.

[41]  Bernd Krauskopf,et al.  Bifurcation sequences at 1:4 resonance: an inventory , 1994 .

[42]  Heinz Hanßmann,et al.  Normal-internal resonances in quasi-periodically forced oscillators: a conservative approach , 2003 .

[43]  高橋 利衛,et al.  〔47〕非線型振動論 : 〔Nonlinear Vibrations in Mechanical and Electrical Systems, J.J.Stoker, Interscience Pub.N.Y., 1950,本文264ページ, 5$〕(1.應用力學および機械力學) , 1951 .

[44]  Marcelo Viana,et al.  Abundance of strange attractors , 1993 .

[45]  M. Golubitsky,et al.  Singularities and groups in bifurcation theory , 1985 .

[46]  Robert Roussarie,et al.  Bifurcations of planar vector fields , 1990 .

[47]  Ralph Abraham,et al.  Foundations Of Mechanics , 2019 .

[48]  Hassler Whitney,et al.  Singularities of Mappings of Euclidean Spaces , 1992 .

[49]  Willy Govaerts,et al.  Numerical Continuation of Bifurcations of Limit Cycles in MATLAB , 2005, SIAM J. Sci. Comput..

[50]  W. Brock,et al.  Heterogeneous beliefs and routes to chaos in a simple asset pricing model , 1998 .

[51]  R. Arditi,et al.  Coupling in predator-prey dynamics: Ratio-Dependence , 1989 .

[52]  L. V. Veen,et al.  Time scale interaction in low-order climate models , 2002 .

[53]  Florian Wagener,et al.  Bifurcation Routes to Volatility Clustering , 2000 .

[54]  H. Broer,et al.  Catastrophe theory in Dulac unfoldings , 2006, Ergodic Theory and Dynamical Systems.

[55]  Hendrik Broer,et al.  Hopf saddle-node bifurcation for fixed points of 3D-diffeomorphisms: Analysis of a resonance ‘bubble’ , 2008 .

[56]  Bernd Krauskopf,et al.  Strong resonances and Takens’s Utrecht preprint , 2001 .

[57]  C. Simó,et al.  On quasi-periodic perturbations of elliptic equilibrium points , 1996 .

[58]  A. J. Lotka.  Elements of Physical Biology. , 1925, Nature.

[59]  J. Gambaudo.  Perturbation of a Hopf Bifurcation by an External Time- Periodic Forcing , 1985 .

[60]  Floris Takens,et al.  Singularities of vector fields , 1974 .

[61]  R. Vitolo.  Bifurcations of attractors in 3D diffeomorphisms : a study in experimental mathematics , 2003 .

[62]  J. Moser.  Combination tones for Duffing's equation , 1965 .

[63]  M. Solomon.  The Natural Control of Animal Populations , 1949 .

[64]  Santiago Ibáñez,et al.  Singularities of vector fields on , 1998 .

[65]  M. Čadek.  Singularities and groups in bifurcation theory, volume I , 1990 .

[66]  H. Hanßmann.  The Quasi-Periodic Centre-Saddle Bifurcation , 1998 .

[67]  H. Broer.  KAM theory: The legacy of Kolmogorov’s 1954 paper , 2004 .

[68]  J. Palis,et al.  Geometric theory of dynamical systems , 1982 .

[69]  Alain Chenciner,et al.  Bifurcations de points fixes elliptiques. II. Orbites periodiques et ensembles de Cantor invariants , 1985 .

[70]  H. Whitney.  On Singularities of Mappings of Euclidean Spaces. I. Mappings of the Plane Into the Plane , 1955 .

[71]  V. Arnold.  BIFURCATIONS AND SINGULARITIES IN MATHEMATICS AND MECHANICS , 1989 .

[72]  Gennady Denisov,et al.  The influence of predator saturation effect and competition among predators on predator-prey system dynamics , 1981 .

[73]  Gert Vegter,et al.  Bifurcational Aspects of Parametric Resonance , 1992 .

[74]  Stephen Wolfram,et al.  The Mathematica Book , 1996 .

[75]  F. Takens.  Forced oscillations and bifurcations , 2001 .

[76]  Martin Golubitsky,et al.  The geometry of resonance tongues: a singularity theory approach , 2003 .

[77]  F. Verhulst,et al.  Averaging Methods in Nonlinear Dynamical Systems , 1985 .

[78]  Alain Chenciner,et al.  Bifurcations De Points Fixes Elliptiques , 1987 .

[79]  V. Arnold.  Mathematical Methods of Classical Mechanics , 1974 .

[80]  Vladimir Igorevich Arnold,et al.  Geometrical Methods in the Theory of Ordinary Differential Equations , 1983 .

[81]  F. Wagener.  On the quasi-periodic d-fold degenerate bifurcation , 2005 .

[82]  J. Ross,et al.  A detailed study of a forced chemical oscillator: Arnol'd tongues and bifurcation sets , 1989 .

[83]  Edward N. Lorenz,et al.  Irregularity: a fundamental property of the atmosphere* , 1984 .

[84]  J. Munkres,et al.  Elementary Differential Topology. , 1967 .

[85]  Duane C. Hanselman,et al.  Mastering MATLAB , 2004 .

[86]  H. Broer,et al.  The Hopf-saddle-node bifurcation for fixed points of 3D-diffeomorphisms , 2007 .

[87]  Theodor Brocker,et al.  Mappings of the plane into the plane , 1975 .