Bibliography [1] V.S. Afraimovich and L.P. Shil'nikov, Invariant tori, their break-down and stochasticity, Amer. Math. Soc. Transl, 149 (1991), 201{211, Originally published in: Methods of qualitative theory of diff. eqs., Gorky State Univ. (1983), pp. 3-26. [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), 1285{1328. [3] E.L. Allgower and K. Georg, Numerical Continuation Methods: An Intro- duction, Springer-Verlag, Berlin, 1990. [4] V.I. Arnol'd, Loss of stability of self-induced oscillations near resonance, and versal deformations of equivariant vector fields, Funkcional. Anal. i Priloˇzen. 11 (1977), no. 2, 1{10, 95. [5] , Geometrical Methods in the Theory of Ordinary Differential Equations, Springer-Verlag, New-York, 1983. [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, 1994. [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), 303{354. [8] D. Arrowsmith and C. Place, An Introduction to Dynamical Systems, Cambridge University Press, Cambridge, 1990. [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), 803{842. [10] A. Back, J. Guckenheimer, M. Myers, F. Wicklin, and P. Worfolk, DsTool: Computer assisted exploration of dynamical systems, Notices Amer. Math. 199 200 BIBLIOGRAPHY Soc. 39 (1992), 303{309. [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), 387{475. [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), 938{943, 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. 149{219. [15] W.-J. Beyn and T. Huls,¨ Error estimates for approximating non-hyperbolic heteroclinic orbits of maps, Numer. Math. 99 (2004), no. 2, 289{323. [16] W.-J. Beyn, T. Huls,¨ J.-M. Kleinkauf, and Y. Zou, Numerical analysis of degenerate connecting orbits for maps, Int. J. Bif. Chaos 14 (2004), no. 10, 3385{3407. [17] W.-J. Beyn and J.-M. Kleinkauf, The numerical computation of homo- clinic orbits for maps, SIAM J. Numer. Anal. 34 (1997), 1207{1236. [18] W.-J. Beyn and W. Kleß, Numerical Taylor expansions of invariant man- ifolds in large dynamical systems, Numer. Math. 80 (1998), 1{38. [19] V.S. Biragov, Bifurcations in two-parameter family of conservative map- pings that are close to the H´enon map, Selecta Math. Sov. 9 (1990), 273{ 282, Originally published in: Methods of qualitative theory of diff. eqs., Gorky State Univ. (1987), pp. 10-24. [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 Priloˇzen. 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), 23{35. [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), 37{65. [23] M. Borland, L. Emery, H. Shang, and R. Soli- day, User's Guide for SDDS Toolkit Version 1.30, (http://www.aps.anl.gov/asd/oag/software.shtml), 2005. BIBLIOGRAPHY 201 [24] B.L.J. Braaksma and H.W. Broer, On a quasi-periodic Hopf bifurcation, Ann. Inst. H. Poincar´e Anal. Non Lin´eaire 4 (1987), 115{168. [25] F. Brauer and C. Castillo-Ch´avez, Mathematical Models in Population Biology and Epidemiology, Springer-Verlag, Berlin, 2000. [26] H.W. Broer, M. Golubitsky, and G. Vegter, The geometry of resonance tongues: a singularity theory approach, Nonlinearity 16 (2003), 1511{ 1538. [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, Unfold- ings and bifurcations of quasi-periodic tori, AMS, Providence, 1990, Mem. Amer. Math. Soc. 421. [29] H.W. Broer, R. Roussarie, and C. Sim´o, Invariant circles in the Bogdanov- Takens bifurcation for diffeomorphisms, Erg. Th. Dyn. Systems 16 (1996), 1147{1172. [30] H.W. Broer, C. Sim´o, and J.C. Tatjer, Towards global models near homo- clinic tangencies of dissipative diffeomorphisms, Nonlinearity 11 (1998), 667{770. [31] H.W. Broer, C. Sim´o, and R. Vitolo, Bifurcations and strange attractors in the Lorenz-84 climate model with seasonal forcing, Nonlinearity 15 (2002), 1205{1267. [32] , The Hopf-Saddle-Node bifurcation for fixed points of 3D-diffeo- morphisms, a dynamical inventory, preprint Groningen, 2004. [33] , The Hopf-Saddle-Node bifurcation for fixed points of 3D-diffeo- morphisms, analysis of a resonance `bubble', preprint Groningen, 2004. [34] H.W. Broer, F. Takens, and F.O.O. Wagener, Integrable and non- integrable deformations of the skew Hopf bifurcation, Regular and Chaotic Dynamics 4 (1999), 17{43. [35] J.P. Carcass`es 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), 875{894. [36] A.R. Champneys, J. H¨arterich, and B. Sandstede, A non-transverse homo- clinic orbit to a saddle-node equilibrium, Ergodic Theory Dynam. Systems 16 (1996), 431{450. [37] A. Chenciner, Bifurcations de points fixes elliptiques. I. Courbes invari- antes, Inst. Hautes Etudes Sci. Publ. Math. 61 (1985), 67{127. [38] , Bifurcations de points fixes elliptiques. II. Orbites p´eriodiques et ensembles de Cantor invariants, Invent. Math. 80 (1985), 81{106. 202 BIBLIOGRAPHY [39] , Bifurcations de points fixes elliptiques. III. Orbites p´eriodiques de `petites' p´eriodes et ´elimination r´esonnante des couples de courbes invari- antes, Inst. Hautes Etudes Sci. Publ. Math. 66 (1988), 5{91. [40] C.Q. Cheng, Hopf bifurcations in nonautonomous systems at points of resonance, Sci. China Ser. A 33 (1990), no. 2, 206{219. [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, 130{139. [42] S.-N. Chow, C. Li, and D. Wang, Normal Forms and Bifurcation of Planar Vector Fields, Cambridge University Press, Cambridge, 1994. [43] S.-N. Chow and D. Wang, Normal forms of bifurcating periodic or- bits, Multiparameter bifurcation theory (Arcata, Calif., 1985), Contemp. Math., vol. 56, Amer. Math. Soc., Providence, RI, 1986, pp. 9{18. [44] P.M. Cincotta, C.M. Giordano, and C. Sim´o, Phase space structure of multi-dimensional systems by means of the mean exponential growth factor of nearby orbits, Phys. D 182 (2003), 151{178. [45] P. H. Coullet and E. A. Spiegel, Amplitude equations for systems with competing instabilities, SIAM J. Appl. Math. 43 (1983), 776{821. [46] N.V. Davydova, Old and Young. Can they coexist?, Ph.D. thesis, Univer- sity of Utrecht, Netherlands, 2004. [47] A. Dhooge, W. Govaerts, and Yu.A. Kuznetsov, matcont: A matlab package for numerical bifurcation analysis of ODE's, ACM TOMS 29 (2003), 141{164, (http://www.matcont.ugent.be). [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 dynam- ical systems, Manuscript, 2006. [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. 161{166. [50] O. Diekmann and J.A.P. Heesterbeek, Mathematical Epidemiology of In- fectious Diseases, John Wiley & Sons, Ltd., Chichester, England, 2000. [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), 27{45. [52] E.J. Doedel, A.R. Champneys, T.F. Fairgrieve, Yu.A. Kuznetsov, B. Sandstede, and X.-J. Wang, Continuation and Bifurcation Soft- ware for Ordinary Differential Equations (with HomCont), User's Guide, (http://indy.cs.concordia.ca), 1997-2000. [53] E.J. Doedel, H.B. Keller, and J.P. Kern´evez, Numerical analysis and con- BIBLIOGRAPHY 203 trol of bifurcation problems: (II) Bifurcation in infinite dimensions, Int. J. Bif. Chaos 1 (1991), 745{772. [54] J. Edmunds, J.M. Cushing, R.F. Costantino, S.M. Henson, B. Dennis, and R.A. Desharnais, Park's Tribolium competition experiments: a non- equilibrium species coexistence hypothesis, J. Animal Ecology 72 (2003), 703{712. [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), 95{127.