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

Transcription

1 Bibliography [1] V.S. Afraimovich and L.P. Shil nikov, Invariant tori, their break-down and stochasticity, Amer. Math. Soc. Transl, 149 (1991), , Originally published in: Methods of qualitative theory of diff. eqs., Gorky State Univ. (1983), pp [2] A. Agliari, G.-I. Bischi, R. Dieci, and L. Gardini, Global bifurcations of closed invariant curves in two-dimensional maps: a computer assisted study, Int. J. Bif. Chaos 15 (2005), [3] E.L. Allgower and K. Georg, Numerical Continuation Methods: An Introduction, Springer-Verlag, Berlin, [4] V.I. Arnol d, Loss of stability of self-induced oscillations near resonance, and versal deformations of equivariant vector fields, Funkcional. Anal. i Priložen. 11 (1977), no. 2, 1 10, 95. [5], Geometrical Methods in the Theory of Ordinary Differential Equations, Springer-Verlag, New-York, [6] V.I. Arnol d, V.S. Afraimovich, Yu.S. Il yashenko, and L.P. Shil nikov, Bifurcation theory, Dynamical Systems V. Encyclopaedia of Mathematical Sciences (V.I. Arnol d, ed.), Springer-Verlag, New York, [7] D.G. Aronson, M.A. Chory, G.R. Hall, and R.P. McGehee, Bifurcations from an invariant circle for two-parameter families of maps of the plane: A computer-assisted study, Comm. Math. Phys. 83 (1982), [8] D. Arrowsmith and C. Place, An Introduction to Dynamical Systems, Cambridge University Press, Cambridge, [9] D.K. Arrowsmith, J.H.E. Cartwright, A.N. Lansbury, and C.M. Place, The Bogdanov map: bifurcations, mode locking, and chaos in a dissipative system, Int. J. Bif. Chaos 3 (1993), [10] A. Back, J. Guckenheimer, M. Myers, F. Wicklin, and P. Worfolk, DsTool: Computer assisted exploration of dynamical systems, Notices Amer. Math. 199

2 200 BIBLIOGRAPHY Soc. 39 (1992), [11] C. Baesens, J. Guckenheimer, S. Kim, and Mackay R.S., Three coupled oscillators: mode-locking, global bifurcations and toroidal chaos, Phys. D 49 (1991), [12] F.S. Berezovskaya and A.I. Khibnik, On the problem of bifurcations of auto-oscillation near resonance 1 : 4,[investigation of a model equation], Academy of Sciences of the USSR Scientific Centre of Biological Research, Pushchino, 1979, In Russian, with an English summary. [13], On the bifurcation of separatrices in the problem of stability loss of auto-oscillations near 1 : 4 resonance, J. Appl. Math. Mech. 44 (1980), , In Russian. [14] W.-J. Beyn, A. Champneys, E.J. Doedel, W. Govaerts, Yu.A. Kuznetsov, and B. Sandstede, Numerical continuation, and computation of normal forms, Handbook of Dynamical Systems (B. Fiedler, ed.), vol. 2, Elsevier Science, Amsterdam, 2002, pp [15] W.-J. Beyn and T. Hüls, Error estimates for approximating non-hyperbolic heteroclinic orbits of maps, Numer. Math. 99 (2004), no. 2, [16] W.-J. Beyn, T. Hüls, J.-M. Kleinkauf, and Y. Zou, Numerical analysis of degenerate connecting orbits for maps, Int. J. Bif. Chaos 14 (2004), no. 10, [17] W.-J. Beyn and J.-M. Kleinkauf, The numerical computation of homoclinic orbits for maps, SIAM J. Numer. Anal. 34 (1997), [18] W.-J. Beyn and W. Kleß, Numerical Taylor expansions of invariant manifolds in large dynamical systems, Numer. Math. 80 (1998), [19] V.S. Biragov, Bifurcations in two-parameter family of conservative mappings that are close to the Hénon map, Selecta Math. Sov. 9 (1990), , Originally published in: Methods of qualitative theory of diff. eqs., Gorky State Univ. (1987), pp [20] R.I. Bogdanov, Versal deformation of a singular point of a vector field on the plane in the case of zero eigenvalues, Funkcional Anal. i Priložen. 9 (1975), 63. [21], Bifurcations of a limit cycle of a certain family of vector fields on the plane, Trudy Sem. Petrovsk. Vyp. 2 (1976), [22], The versal deformation of a singular point of a vector field on the plane in the case of zero eigenvalues, Trudy Sem. Petrovsk. Vyp. 2 (1976), [23] M. Borland, L. Emery, H. Shang, and R. Soliday, User s Guide for SDDS Toolkit Version 1.30, (

3 BIBLIOGRAPHY 201 [24] B.L.J. Braaksma and H.W. Broer, On a quasi-periodic Hopf bifurcation, Ann. Inst. H. Poincaré Anal. Non Linéaire 4 (1987), [25] F. Brauer and C. Castillo-Chávez, Mathematical Models in Population Biology and Epidemiology, Springer-Verlag, Berlin, [26] H.W. Broer, M. Golubitsky, and G. Vegter, The geometry of resonance tongues: a singularity theory approach, Nonlinearity 16 (2003), [27] H.W. Broer, G.B. Huitema, and M.B. Sevryuk, Quasi-periodic motions in families of dynamical systems. Order amidst chaos, vol. 1645, Springer- Verlag, Berlin, 1996, Lecture Notes in Mathematics. [28] H.W. Broer, G.B. Huitema, F. Takens, and B.L.J. Braaksma, Unfoldings and bifurcations of quasi-periodic tori, AMS, Providence, 1990, Mem. Amer. Math. Soc [29] H.W. Broer, R. Roussarie, and C. Simó, Invariant circles in the Bogdanov- Takens bifurcation for diffeomorphisms, Erg. Th. Dyn. Systems 16 (1996), [30] H.W. Broer, C. Simó, and J.C. Tatjer, Towards global models near homoclinic tangencies of dissipative diffeomorphisms, Nonlinearity 11 (1998), [31] H.W. Broer, C. Simó, and R. Vitolo, Bifurcations and strange attractors in the Lorenz-84 climate model with seasonal forcing, Nonlinearity 15 (2002), [32], The Hopf-Saddle-Node bifurcation for fixed points of 3D-diffeomorphisms, a dynamical inventory, preprint Groningen, [33], The Hopf-Saddle-Node bifurcation for fixed points of 3D-diffeomorphisms, analysis of a resonance bubble, preprint Groningen, [34] H.W. Broer, F. Takens, and F.O.O. Wagener, Integrable and nonintegrable deformations of the skew Hopf bifurcation, Regular and Chaotic Dynamics 4 (1999), [35] J.P. Carcassès and H. Kawakami, Existence of a cusp point on a fold bifurcation curve and stability of the associated fixed point. case of an n-dimensional map, Int. J. Bif. Chaos 5 (1999), [36] A.R. Champneys, J. Härterich, and B. Sandstede, A non-transverse homoclinic orbit to a saddle-node equilibrium, Ergodic Theory Dynam. Systems 16 (1996), [37] A. Chenciner, Bifurcations de points fixes elliptiques. I. Courbes invariantes, Inst. Hautes Etudes Sci. Publ. Math. 61 (1985), [38], Bifurcations de points fixes elliptiques. II. Orbites périodiques et ensembles de Cantor invariants, Invent. Math. 80 (1985),

4 202 BIBLIOGRAPHY [39], Bifurcations de points fixes elliptiques. III. Orbites périodiques de petites périodes et élimination résonnante des couples de courbes invariantes, Inst. Hautes Etudes Sci. Publ. Math. 66 (1988), [40] C.Q. Cheng, Hopf bifurcations in nonautonomous systems at points of resonance, Sci. China Ser. A 33 (1990), no. 2, [41] C.Q. Cheng and Y.S. Sun, Metamorphoses of phase portraits of vector field in the case of symmetry of order 4, J. Differential Equations 95 (1992), no. 1, [42] S.-N. Chow, C. Li, and D. Wang, Normal Forms and Bifurcation of Planar Vector Fields, Cambridge University Press, Cambridge, [43] S.-N. Chow and D. Wang, Normal forms of bifurcating periodic orbits, Multiparameter bifurcation theory (Arcata, Calif., 1985), Contemp. Math., vol. 56, Amer. Math. Soc., Providence, RI, 1986, pp [44] P.M. Cincotta, C.M. Giordano, and C. Simó, Phase space structure of multi-dimensional systems by means of the mean exponential growth factor of nearby orbits, Phys. D 182 (2003), [45] P. H. Coullet and E. A. Spiegel, Amplitude equations for systems with competing instabilities, SIAM J. Appl. Math. 43 (1983), [46] N.V. Davydova, Old and Young. Can they coexist?, Ph.D. thesis, University of Utrecht, Netherlands, [47] A. Dhooge, W. Govaerts, and Yu.A. Kuznetsov, matcont: A matlab package for numerical bifurcation analysis of ODE s, ACM TOMS 29 (2003), , ( [48] A. Dhooge, W. Govaerts, Yu.A. Kuznetsov, H.G.E. Meijer, and B. Sautois, New features of the software matcont for bifurcation analysis of dynamical systems, Manuscript, [49] A. Dhooge, W. Govaerts, Yu.A. Kuznetsov, Mestrom W., and A.M. Riet, cl matcont: A continuation toolbox in matlab, Proceedings of the 2003 ACM symposium on applied computing, Melbourne, Florida, 2003, pp [50] O. Diekmann and J.A.P. Heesterbeek, Mathematical Epidemiology of Infectious Diseases, John Wiley & Sons, Ltd., Chichester, England, [51] W. Ding, J. Xie, and Q. Sun, Interaction of Hopf and period doubling bifurcations of a vibro-impact system, J. Sound Vibr. 275 (2004), [52] E.J. Doedel, A.R. Champneys, T.F. Fairgrieve, Yu.A. Kuznetsov, B. Sandstede, and X.-J. Wang, Continuation and Bifurcation Software for Ordinary Differential Equations (with HomCont), User s Guide, ( [53] E.J. Doedel, H.B. Keller, and J.P. Kernévez, Numerical analysis and con-

5 BIBLIOGRAPHY 203 trol of bifurcation problems: (II) Bifurcation in infinite dimensions, Int. J. Bif. Chaos 1 (1991), [54] J. Edmunds, J.M. Cushing, R.F. Costantino, S.M. Henson, B. Dennis, and R.A. Desharnais, Park s Tribolium competition experiments: a nonequilibrium species coexistence hypothesis, J. Animal Ecology 72 (2003), [55] C. Elphick, E. Tirapegui, M. Brachet, P. Coullet, and G. Iooss, A simple global characterization for normal forms of singular vector fields, Phys. D 32 (1987), [56] C.E. Frouzakis, R.A. Adomatis, and I.G. Kevrekidis, Resonance Phenomena in an adaptively-controlled system, Int. J. Bif. Chaos 1 (1991), [57] N.K. Gavrilov and L.P. Shil nikov, On three-dimensional systems close to systems with a structurally unstable homoclinic curve. part I, Math. USSR Sb. 17 (1972), [58], On three-dimensional systems close to systems with a structurally unstable homoclinic curve. part II, Math. USSR Sb. 19 (1973), [59] J. Gheiner, Codimension-two reflection and nonhyperbolic invariant lines, Nonlinearity 7 (1994), [60], Codimension n flips, Erg. Th. Dyn. Systems 18 (1998), [61] S.A. van Gils, On a formula for the direction of Hopf bifurcation, Technical Report TW/225, CWI, Amsterdam, [62] M.P. Golden and B.E. Ydstie, Bifurcation in model reference adaptive control systems, Systems Control Lett. 11 (1988), [63] S.V. Gonchenko and V.S. Gonchenko, On Andronov-Hopf bifurcations of two-dimensional diffeomorphisms with homoclinic tangencies, Preprint No. 556, WIAS, Berlin, [64] S.V. Gonchenko, V.S. Gonchenko, and J.C. Tatjer, Three-dimensional dissipative diffeomorphisms with codimension two homoclinic tangencies and generalized Hénon maps, Proc. of Int. Conf. Progress in Nonlinear Science dedicated to 100th Anniversary of A.A.Andronov, 2001, pp [65] S.V. Gonchenko and L.P. Shil nikov, On two-dimensional area-preserving maps with homoclinic tangencies that have infinitely many generic elliptic periodic points, Notes of St.-Petersburg Steklov Math. Inst. 300 (2003), [66] S.V. Gonchenko, L.P. Shil nikov, and O.V. Stenkin, On Newhouse regions with infinitely many stable and unstable tori, Proc. of Int. Conf. Progress in Nonlinear Science, dedicated to 100th Anniversary of A.A. Andronov, 2002, pp

6 204 BIBLIOGRAPHY [67] S.V. Gonchenko, L.P. Shil nikov, and D.V. Turaev, Dynamical phenomena in systems with structurally unstable Poincaré homoclinic orbits, Int. J. Bif. Chaos 6 (1996), [68] S.V. Gonchenko, L.P. Shil nikov, and D.V Turaev, On dynamical properties of diffeomorphisms with homoclinic tangencies, Preprint No. 795, WIAS, Berlin, [69] V.S. Gonchenko, On bifurcations of two-dimensional diffeomorphisms with a homoclinic tangency of manifolds of a neutral saddle, Proc. Steklov Inst. Math. 236, 2002, pp [70] V.S. Gonchenko and S.V. Gonchenko, On bifurcations of birth of closed invariant curves in the case of two-dimensional diffeomorphisms with a homoclinic tangencies, Proc. of Steklov Math. Inst., [71] V.S. Gonchenko, Yu.A. Kuznetsov, and H.G.E. Meijer, Generalized Hénon map and homoclinic tangencies, SIAM J. Appl. Dyn. Sys. 4 (2005), [72] V.S. Gonchenko and I.I. Ovsyannikov, On bifurcations of three-dimensional diffeomorphisms with a homoclinic tangency to a neutral saddle fixed point, Notes of St.-Petersburg Steklov Math. Inst. 300 (2003), [73] W. Govaerts, Yu.A. Kuznetsov, and B. Sijnave, Bifurcations of maps in the software package content, Computer Algebra in Scientific Computing CASC 99 (Munich), Springer, Berlin, 1999, pp [74] W.J.F. Govaerts, Numerical Methods for Bifurcations of Dynamical Equilibria, SIAM, Philadelphia, [75] A. Griewank, D. Juedes, H. Mitev, J. Utke, O. Vogel, and A. Walther, adol-c: a Package for the Automatic Differentiation of Algorithms Written in C/C++, ACM TOMS, Algor (1996), , ( [76] J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields, Springer-Verlag, New York, 1983, [77] J. Guckenheimer and B. Meloon, Computing Periodic Orbits and Their Bifurcations With Automatic Differentiation, SIAM J. Sci. Comput. 22 (2000), [78] B.D. Hassard, N.D. Kazarinoff, and Wan Y.-H., Theory and Applications of Hopf Bifurcation, Cambridge University Press, Cambridge, [79] M. Hénon, A two-dimensional mapping with a strange attractor, Comm. Math. Phys 50 (1976), [80] M.W. Hirsch, C.C. Pugh, and M. Shub, Invariant manifolds, Springer- Verlag, Berlin-New York, 1977, Lecture Notes in Mathematics, Vol. 583.

7 BIBLIOGRAPHY 205 [81] P. Holmes and D. Whitley, Bifurcations of one- and two-dimensional maps, Philos. Trans. Roy. Soc. London Ser. A 311 (1984), [82], Erratum: Bifurcations of one- and two-dimensional maps, Philos. Trans. Roy. Soc. London Ser. A 312 (1984), 601. [83] P.J. Holmes and D.A. Rand, Bifurcations of the forced van der Pol oscillator, Quart. Appl. Math. 35 (1978), [84] E. I. Horozov, Versal deformations of equivariant vector fields for cases of symmetry of order 2 and 3, Trudy Sem. Petrovsk. 5 (1979), [85] Maplesoft Inc., Maple 9, ( [86] The Mathworks Inc., Matlab 7.2, ( [87] G. Iooss, Bifurcation of maps and applications, North-Holland Mathematics Studies, vol. 36, North-Holland Publishing Co., Amsterdam, [88], Global characterization of the normal form for a vector field near a closed orbit, J. Diff. Eqs. 76 (1988), [89] G. Iooss and J.E. Los, Quasi-genericity of bifurcations to high-dimensional invariant tori for maps, Comm. Math. Phys. 119 (1988), [90] M. Ipsen, F. Hynne, and P.G. Sørensen, Systematic derivation of amplitude equations and normal forms for dynamical systems, Chaos 8 (1998), [91] H.B. Keller, Numerical solution of bifurcation and nonlinear eigenvalue problems, Applications of bifurcation theory (Proc. Advanced Sem., Univ. Wisconsin, Madison, Wis., 1976), Academic Press, New York, 1977, pp Publ. Math. Res. Center, No. 38. [92] A.I. Khibnik, Yu.A. Kuznetsov, V.V. Levitin, and E.V. Nikolaev, Continuation techniques and interactive software for bifurcation analysis of ODEs and iterated maps ( LOCBIF ), Physica D 62 (1993), [93] R. Khoshsiar Ghaziani and W. Govaerts, Numerical bifurcation analysis of a nonlinear stage structured cannibalism population model, Accepted in J. Diff. Eqs. Appl., [94] R. Khoshsiar Ghaziani, W. Govaerts, Yu.A. Kuznetsov, and H.G.E. Meijer, Bifurcation analysis of periodic orbits of maps in Matlab, Preprint nr. 1350, Department of Mathematics, Utrecht University, The Netherlands, Submitted, [95] J.-M. Kleinkauf, Numerische analyse tangentialer homokliner orbits, Ph.D. thesis, Universität Bielefeld, 1998, Shaker Verlag,Aachen. [96] B. Krauskopf, Bifurcation sequences at 1 : 4 resonance: an inventory, Nonlinearity 7 (1994), [97], The bifurcation set for the 1 : 4 resonance problem, Experiment. Math. 3 (1994),

8 206 BIBLIOGRAPHY [98], Bifurcations at in a model for 1: 4 resonance, Erg. Th. Dyn. Systems 17 (1997), [99], Strong resonances and Takens s Utrecht preprint, Global analysis of dynamical systems, Inst. Phys., Bristol, 2001, pp [100] B. Krauskopf and B.E. Oldeman, A planar model system for the saddlenode Hopf bifurcation with global reinjection, Nonlinearity 17 (2004), [101] B. Krauskopf and H. Osinga, Globalizing two-dimensional unstable manifolds of maps, Int. J. Bif. Chaos 8 (1998), [102], Growing 1D and quasi-2d unstable manifolds of maps, J. Comp. Physics 146 (1998), [103] B. Krauskopf, H. M. Osinga, E. J. Doedel, M. E. Henderson, J. Guckenheimer, A. Vladimirsky, M. Dellnitz, and O. Junge, A survey of methods for computing (un)stable manifolds of vector fields, Int. J. Bif. Chaos 15 (2005), [104] L.G. Kurakin, Semi-invariant form of criteria for stability of fixed points of a mapping in critical cases, Izv. Severo-Kavkaz. Nauchn. Tsentra Vyssh. Shkoly Estestv. Nauk. 141 (1989), 31 35, In Russian. [105] Yu.A. Kuznetsov, Numerical normalization techniques for all codim 2 bifurcations of equilibria in ODE s, SIAM J. Numer. Anal. 36 (1999), [106], Elements of Applied Bifurcation Theory, Springer Verlag, Berlin, 2004, Third Edition. [107], Practical computation of normal forms on center manifolds at degenerate Bogdanov-Takens bifurcations, Int. J. Bif. Chaos 15 (2005), [108] Yu.A. Kuznetsov, W. Govaerts, E.J. Doedel, and A. Dhooge, Numerical periodic normalization for codim 1 bifurcations of limit cycles, SIAM J. Numer. Anal. 43 (2005), [109] Yu.A. Kuznetsov and V.V. Levitin, content: A multiplatform environment for analyzing dynamical systems, (ftp.cwi.nl/pub/content), [110] Yu.A. Kuznetsov and H.G.E. Meijer, Numerical normal forms for codim 2 bifurcations of maps with at most two critical eigenvalues, SIAM J. Sci. Comput. 26 (2005), [111], Remarks on interacting Neimark-Sacker bifurcations, Preprint nr. 1342, Department of Mathematics, Utrecht University, The Netherlands, To appear in J. Diff. Eqs. Appl., [112] Yu.A. Kuznetsov, H.G.E. Meijer, and L. van Veen, The fold-flip bifurca-

9 BIBLIOGRAPHY 207 tion, Int. J. Bif. Chaos 14 (2004), [113] Yu.A. Kuznetsov, S. Muratori, and S. Rinaldi, Bifurcations and chaos in a periodic predator-prey model, Int. J. Bif. Chaos 2 (1992), [114] Yu.A. Kuznetsov and C. Piccardi, Bifurcation analysis of periodic SEIR and SIR epidemic models, J. Math. Biol. 32 (1994), [115] J. Laskar, C. Froeschlé, and A. Celletti, The measure of chaos by the numerical analysis of the fundamental frequencies. Application to the standard mapping, Phys. D 56 (1992), [116] P.H. Leslie and J.C. Gower, The properties of a stochastic model for two competing species, Biometrika 45 (1958), [117] P.H. Leslie, T. Park, and D.B. Mertz, The effect of varying the initial numbers on the outcome of competition between two Tribolium species, J. Animal Ecology 37 (1968), [118] E. Lindtner, A. Steindl, and H. Troger, Generic one-parameter bifurcations in the motion of a simple robot, J. Comput. Appl. Math. 26 (1989), [119] E.N. Lorenz, Irregularity: a fundamental property of the atmosphere, Tellus 36A (1984), [120] J.E. Los, Small divisor phenomena in an invariant curve doubling bifurcation on a cylinder, Ann. Inst. H. Poincaré Anal. Non Linéaire 5 (1988), [121], Non-normally hyperbolic invariant curves for maps in R 3 and doubling bifurcation, Nonlinearity 2 (1989), [122] R. S. MacKay, Renormalisation in area-preserving maps, Advanced Series in Nonlinear Dynamics, vol. 6, World Scientific Publishing Co. Inc., River Edge, NJ, [123] H.G.E. Meijer, A transcritical-flip bifurcation in a model for a robot-arm, Symmetry and perturbation theory (Cala Gonone, 2004), World Sci. Publishing, River Edge, NJ, 2004, pp [124] V.K. Mel nikov, Qualitative description of strong resonance in a nonlinear system, Dokl. Akad. Nauk SSSR 148 (1963), [125] C. Mira, Chaotic Dynamics, World Scientific, Singapore, [126], Some historical aspects of nonlinear dynamics - Possible trends for the future, J. Franklin Inst. 334B (1997), [127] J. Murdock, Normal Forms and Unfoldings of Local Dynamical Systems, Springer-Verlag, New-York, [128] J.I. Neĭmark, Some cases of the dependence of periodic motions on parameters, Dokl. Akad. Nauk SSSR 129 (1959), [129] A.I. Neĭshtadt, Bifurcations of the phase pattern of an equation system

10 208 BIBLIOGRAPHY arising in the problem of stability loss of self-oscillations close to 1 : 4 resonance, J. Appl. Math. Mech. 42 (1978), [130] S. Newhouse, J. Palis, and F. Takens, Bifurcations and stability of families of diffeomorphisms, Inst. Hautes Etudes Sci. Publ. Math. 57 (1983), [131] H. Nusse and J. Yorke, Dynamics: Numerical Explorations, Springer Verlag, New York, 1998, Second Edition. [132] J. Palis and F. Takens, Hyperbolicity and Sensitive Chaotic Dynamics at Homoclinic Bifurcations: Fractal Dimensions and Infinitely Many Attractors, Cambridge Studies in Advanced Mathematics, vol. 35, Cambridge University Press, Cambridge, [133] T.N. Palmer, The influence of north-west Atlantic sea surface temperature: An unplanned experiment, Weather 30 (1995), [134] B.B. Peckham and I.G. Kevrekidis, Period doubling with higher-order degeneracies, SIAM J. Math. Anal. 22 (1991), [135], Lighting Arnol d flames: resonance in doubly forced periodic oscillators, Nonlinearity 15 (2002), [136] D. Roose and V. Hlavaçek, A direct method for the computation of Hopf bifurcation points, SIAM J. Appl. Math 45 (1985), [137] C. Rousseau, Codimension 1 and 2 bifurcations of fixed points of diffeomorphisms and periodic solutions of vector fields, Ann. Sci. Math. Québec 13 (1990), [138] R.J. Sacker, A new approach to the perturbation theory of invariant surfaces, Comm. Pure Appl. Math. 18 (1965), [139] K. Saleh, Organising centres in the semi-global analysis of dynamical systems, Ph.D. thesis, University of Groningen, Netherlands, [140] K. Saleh and F.O.O. Wagener, Semi-global analysis of periodic and quasiperiodic k : 1 and k : 2 resonances, CeNDEF Working paper University of Amsterdam, [141] J.A. Sanders and F. Verhulst, Averaging Methods in Nonlinear Dynamical Systems, Springer-Verlag, New-York, [142] A.L. Shil nikov, G. Nicolis, and C. Nicolis, Bifurcation and predictability analysis of a low-order atmospheric circulation model, Int. J. Bif. Chaos 5 (1995), [143] A.N. Shoshitaishvili, The bifurcation of the topological type of the singular points of vector fields that depend on parameters, Trudy Sem. Petrovsk. (1975), no. Vyp. 1, [144] C. Simó, On the analytical and numerical approximation of invariant manifolds, Modern Methods in Celestial Mechanics, Proceedings of the 13th spring school on astrophysics in Goutelas (D. Benest and C. Froeschlé,

11 BIBLIOGRAPHY 209 eds.), Edition Frontières, Gift-sur Yvette, France, 1989, Available via [145], On the use of Lyapunov exponents to detect global properties of the dynamics, EQUADIFF 2003, Proceedings of the international Conference on Differential Equations Hasselt, Belgium July 2003 (F. Dumortier, H. Broer, J. Mawhin, A. Vanderbauwhede, and S. Verduyn Lunel, eds.), World-Scientific, Singapore, [146] C. Simó, H.W. Broer, and R. Roussarie, A numerical survey on the Takens-Bogdanov bifurcation for diffeomorphisms, European Conference on Iteration Theory (Batschuns, 1989) (C. Mira, N. Netzer, and C. Simó, eds.), World-Scientific, Singapore, [147] Y.L. Song, The Juvenile/Adult Leslie Gower Competition Model, Manuscript, [148] A. Steindl, Bifurcation of codimension 2 for a discrete map, Continuation and bifurcations: numerical techniques and applications (Leuven, 1989) (D. Roose, B. De Dier, and A. Spence, eds.), NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., vol. 313, Kluwer Acad. Publ., Dordrecht, 1990, pp [149] F. Takens, Forced oscillations and bifurcations, Comm. Math. Inst. Rijksuniv. Utrecht, (Reprinted in Global analysis of dynamical systems, Broer, B. Krauskopf and G. Vegter. Institute of Physics, Bristol, 2001), [150] F. Takens and F.O.O. Wagener, Resonances in skew and reducible quasiperiodic Hopf bifurcations, Nonlinearity 13 (2000), [151] H. Troger and A. Steindl, Nonlinear stability and bifurcation theory, Springer-Verlag, Vienna, [152] D.V. Turaev, On dimension of non-local bifurcational problems, Int. J. Bif. Chaos 5 (1996), [153] L. van Veen, Baroclinic Flow and the Lorenz-84 model, Int. J. Bif. Chaos 13 (2003), [154] A. Vanderbauwhede, Local bifurcation and symmetry, vol. 75, Pitman, Boston, 1982, Research Notes in Mathematics. [155], Center manifolds and their basic properties. An introduction, Delft Progr. Rep. 12 (1988), [156] A. Vanderbauwhede and S. A. van Gils, Center manifolds and contractions on a scale of Banach spaces, J. Funct. Anal. 72 (1987), [157] R. Vitolo, Bifurcations in 3D diffeomorphisms - A study in experimental mathematics, Ph.D. thesis, University of Groningen, Netherlands, [158] Y.H. Wan, Bifurcation into invariant tori at points of resonance, Arch. Rational Mech. Anal. 68 (1978),

12 210 BIBLIOGRAPHY [159] G.-L. Wen, Q.G. Wang, and M.-S. Chiu, Delay feedback control for interaction of Hopf and period doubling bifurcations in discrete-time systems, Int. J. Bif. Chaos 16 (2006), [160] G.-L. Wen and D. Xu, Feedback control of Hopf-Hopf interaction bifurcation with development of torus solutions in high-dimensional maps, Phys. Let. A 321 (2004), [161], Implicit criteria of eigenvalue assignment and transversality of bifurcation control in four-dimensional maps, Int. J. Bif. Chaos 14 (2004), [162] S. Wieczorek, B. Krauskopf, and D. Lenstra, Unnested islands of period doublings in an injected semiconductor laser, Phys. Rev. E 64 (2001), , 9. [163] Wolfram, Mathematica 5.2, ( [164] J. Xie and W. Ding, Hopf-Hopf bifurcation and invariant torus T 2 of a vibro-impact system, Int. J. Non-Lin. Mech. 40 (2005), [165] A. Zegeling, Equivariant unfoldings in the case of symmetry of order 4, Serdica 19 (1993), no. 1, [166] K. Zholondek, Versality of a family of symmetric vector fields on the plane, Mat. Sb. (N.S.) 120(162) (1983),