References
Published online by Cambridge University Press: 15 March 2019
- Type
- Chapter
- Information
- Numerical Bifurcation Analysis of MapsFrom Theory to Software, pp. 389 - 399Publisher: Cambridge University PressPrint publication year: 2019
References
Abraham, R., Gardini, L., and Mira, C. 1997. Chaos in Discrete Dynamical Systems: A Visual Introduction in 2 Dimensions. Springer-Verlag, New York.Google Scholar
Afraimovich, V. S., and Shilnikov, L. P. 1983. Invariant two-dimensional tori, their breakdown and stochasticity. Pages 3–26 of: Methods of the Qualitative Theory of Differential Equations. Gorky State University, Gorky (in Russian).Google Scholar
Agiza, H. N. 1999. On the analysis of stability, bifurcation, chaos and chaos control of Kopel map. Chaos Solitons Fractals, 10(11), 1909–1916.Google Scholar
Al-Hdaibat, B., Govaerts, W., van Kekem, D. L., and Kuznetsov, Yu. A. 2018. Remarks on homoclinic structure in Bogdanov–Takens map. J. Difference Equ. Appl., 24(4), 575–587.Google Scholar
Allgower, E. L., and Georg, K. 1990. Numerical Continuation Methods: An Introduction. Springer-Verlag, Berlin.Google Scholar
Anderson, D. R., Myran, N. G., and White, D.L. 2005. Basins of attraction in a Cournot duopoly model of Kopel. J. Difference Equ. Appl., 11(10), 879–887.CrossRefGoogle Scholar
Andronov, A. A., Leontovich, E. A., Gordon, I. I., and Maier, A. G. 1973. Theory of Bifurcations of Dynamical Systems on a Plane. Israel Program for Scientific Translations, Jerusalem.Google Scholar
Arnold, V. I. 1977. Loss of stability of self-induced oscillations near resonance, and versal deformations of equivariant vector fields. Functional Anal. Appl, 11(2), 85–92.Google Scholar
Arnold, V. I. 1983. Geometrical Methods in the Theory of Ordinary Differential Equations. Springer-Verlag, New York.Google Scholar
Arnold, V. I., Afraimovich, V. S., Ilyashenko, Yu. S., and Shilnikov, L. P. 1994. Bifurcation Theory, Dynamical Systems V. Springer-Verlag, New York.Google Scholar
Aronson, D. G., Chory, M. A., Hall, G. R., and McGehee, R. P. 1982. Bifurcations from an invariant circle for two-parameter families of maps of the plane: a computer-assisted study. Comm. Math. Phys., 83(3), 303–354.Google Scholar
Arrowsmith, D. K., and Place, C. M. 1990. An Introduction to Dynamical Systems. Cambridge University Press, Cambridge.Google Scholar
Back, A., Guckenheimer, J., Myers, M., Wicklin, F., and Worfolk, P. 1992. DsTool: computer assisted exploration of dynamical systems. Notices Amer. Math. Soc., 39, 303–309.Google Scholar
Baesens, C., and MacKay, R. S. 2007. Resonances for weak coupling of the unfolding of a saddle-node periodic orbit with an oscillator. Nonlinearity, 20(5), 1283–1298.Google Scholar
Baesens, C., Guckenheimer, J., Kim, S., and MacKay, R. S. 1991. Three coupled oscillators: mode-locking, global bifurcations and toroidal chaos. Phys. D, 49(3), 387–475.Google Scholar
Balibrea, F., Oliveira, H. M., and Valverde, J. C. 2017. Topological equivalences for one-parameter bifurcations of maps. J. Nonlinear Sci., 27(2), 661–685.Google Scholar
Berezovskaya, F. S., and Khibnik, A. I. 1979. 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 (in Russian, with an English summary).Google Scholar
Berezovskaia, F. S., and Khibnik, A. I. 1980. On the bifurcation of separatrices in the problem of stability loss of auto-oscillations near 1:4 resonance. J. Appl. Math. Mech., 44(5), 938–943 (in Russian).Google Scholar
Beyn, W.-J., and Hüls, T. 2004. Error estimates for approximating non-hyperbolic heteroclinic orbits of maps. Numer. Math., 99(2), 289–323.Google Scholar
Beyn, W.-J., and Kleinkauf, J.-M. 1997. The numerical computation of homoclinic orbits for maps. SIAM J. Numer. Anal., 34(3), 1207–1236.Google Scholar
Beyn, W.-J., and Kless, W. 1998. Numerical Taylor expansions of invariant manifolds in large dynamical systems. Numer. Math., 80(1), 1–38.Google Scholar
Beyn, W.-J., Champneys, A., Doedel, E., Govaerts, W., Kuznetsov, Yu.A., and Sandstede, B. 2002. Numerical continuation, and computation of normal forms. Pages 149–219 of: Handbook of Dynamical Systems, Vol. 2. North-Holland, Amsterdam.Google Scholar
Beyn, W.-J., Hüls, T., Kleinkauf, J.-M., and Zou, Y. 2004. Numerical analysis of degenerate connecting orbits for maps. Internat. J. Bifur. Chaos Appl. Sci. Engrg., 14(10), 3385–3407.Google Scholar
Biragov, V. S. 1987. Bifurcations in a two-parameter family of conservative mappings that are close to the Hénon mapping. Pages 10–24 of: Methods of the Qualitative Theory of Differential Equations (in Russian). Gorky State University, Gorky. Translated in Selecta Math. Soviet, 9 (1990), 273–282.Google Scholar
Bischi, G. I., and Kopel, M. 2001. Equilibrium selection in a nonlinear duopoly game with adaptive expectations. J. Econ. Behav. Organ., 46(1), 73–100.Google Scholar
Bischi, G. I. and Tramontana, F. 2010. Three-dimensional discrete-time Lotka–Volterra models with an application to industrial clusters. Commun. Nonlinear. Sci. Numer. Simul., 15(10), 3000–3014Google Scholar
Bogdanov, R. I. 1975. Versal deformation of a singular point of a vector field on the plane in the case of zero eigenvalues. Funkcional Anal. i Prilozen., 9(2), 63.Google Scholar
Bogdanov, R. I. 1976a. Bifurcations of a limit cycle of a certain family of vector fields on the plane. Trudy Sem. Petrovsk., 2, 23–35 (in Russian). Translated in Selecta Math. Soviet. 1 (1981), 373–388.Google Scholar
Bogdanov, R. I. 1976b. The versal deformation of a singular point of a vector field on the plane in the case of zero eigenvalues. Trudy Sem. Petrovsk., 2, 37–65 (in Russian). Translated in Selecta Math. Soviet. 1 (1981), 389–421.Google Scholar
Borland, M., Emery, L., Shang, H., and Soliday, R. 2005. User’s Guide for SDDS Toolkit Version 1.30. www.aps.anl.gov/asd/oag/software.shtmlGoogle Scholar
Braaksma, B. L. J., and Broer, H. W. 1987. On a quasiperiodic Hopf bifurcation. Ann. Inst. H. Poincaré Anal. Non Linéaire, 4(2), 115–168.CrossRefGoogle Scholar
Brauer, F., and Castillo-Chávez, C. 2001. Mathematical Models in Population Biology and Epidemiology. Springer-Verlag, New York.Google Scholar
Broer, H. W., Huitema, G. B., Takens, F., and Braaksma, B. L. J. 1990. Unfoldings and bifurcations of quasi-periodic tori. Mem. Amer. Math. Soc., 83(421).Google Scholar
Broer, H. W., Roussarie, R., and Simó, C. 1996a. Invariant circles in the Bogdanov– Takens bifurcation for diffeomorphisms. Ergodic Theory Dynam. Systems, 16(6), 1147–1172.Google Scholar
Broer, H. W., Huitema, G. B., and Sevryuk, M. B. 1996b. Quasi-Periodic Motions in Families of Dynamical Systems Order Amidst Chaos. Springer-Verlag, Berlin.Google Scholar
Broer, H. W., Simó, C., and Tatjer, J. C. 1998. Towards global models near homoclinic tangencies of dissipative diffeomorphisms. Nonlinearity, 11(3), 667–770.Google Scholar
Broer, H. W., Takens, F., and Wagener, F. O. O. 1999. Integrable and non-integrable deformations of the skew Hopf bifurcation. Regul. Chaotic Dyn., 4(2), 16–43.Google Scholar
Broer, H. W., Simó, C., and Vitolo, R. 2002. Bifurcations and strange attractors in the Lorenz-84 climate model with seasonal forcing. Nonlinearity, 15(4), 1205–1267.Google Scholar
Broer, H. W., Golubitsky, M., and Vegter, G. 2003. The geometry of resonance tongues: a singularity theory approach. Nonlinearity, 16(4), 1511–1538.Google Scholar
Broer, H. W., Simó, C., and Vitolo, R. 2008a. Hopf saddle-node bifurcation for fixed points of 3D-diffeomorphisms: analysis of a resonance ‘bubble’. Phys. D, 237(13), 1773–1799.Google Scholar
Broer, H. W., Simó, C., and Vitolo, R. 2008b. The Hopf–saddle-node bifurcation for fixed points of 3D-diffeomorphisms: the Arnol’d resonance web. B. Belg. Math. Soc.-Sim., 15(5), 769–787.Google Scholar
Broer, H. W., and Holtman, S. J., Vegter, G., and Vitolo, R. 2011. Dynamics and geometry near resonant bifurcations Regul. Chaotic Dyn., 16(1–2), 39–50.Google Scholar
Bruschi, C. 2010. Growing 1D stable and unstable manifolds of n-dimensional maps. Master’s project, Utrecht University.Google Scholar
Carcassès, J. P., and Kawakami, H. 1999. Existence of a cusp point on a fold bifurcation curve and stability of the associated fixed point: case of an n-dimensional map. Internat. J. Bifur. Chaos Appl. Sci. Engrg., 9(5), 875–894.Google Scholar
Champneys, A. R., Härterich, J., and Sandstede, B. 1996. A non-transverse homoclinic orbit to a saddle-node equilibrium. Ergodic Theory Dynam. Systems, 16(3), 431– 450.Google Scholar
Chenciner, A. 1985a. Bifurcations de points fixes elliptiques: I. Courbes invariantes. Inst. Hautes Études Sci. Publ. Math., 61, 67–127.CrossRefGoogle Scholar
Chenciner, A. 1985b. Bifurcations de points fixes elliptiques: II. Orbites periodiques et ensembles de Cantor invariants. Invent. Math., 80(1), 81–106.Google Scholar
Chenciner, A. 1988. Bifurcations de points fixes elliptiques: III. Orbites périodiques de “petites” périodes et élimination résonnante des couples de courbes invariantes. Inst. Hautes Études Sci. Publ. Math., 66, 5–91.CrossRefGoogle Scholar
Chenciner, A., Gasull, A., and Llibre, J. 1987. Une description complète du portrait de phase d’un modèle d’élimination résonante. C. R. Acad. Sci. Paris Sér. I Math., 305(13), 623–626.Google Scholar
Cheng, C. Q. 1990. Hopf bifurcations in nonautonomous systems at points of resonance. Sci. China Ser. A, 33(2), 206–219.Google Scholar
Cheng, C. Q., and Sun, Y. S. 1992. Metamorphoses of phase portraits of vector field in the case of symmetry of order 4. J. Differential Equations, 95(1), 130–139.Google Scholar
Chow, S.-N., Li, C. Z., and Wang, D. 1994. Normal Forms and Bifurcation of Planar Vector Fields. Cambridge University Press, Cambridge.CrossRefGoogle Scholar
Cincotta, P. M., Giordano, C. M., and Simó, C. 2003. Phase space structure of multidimensional systems by means of the mean exponential growth factor of nearby orbits. Phys. D, 182(3–4), 151–178.Google Scholar
Coullet, P. H., and Spiegel, E. A. 1983. Amplitude equations for systems with competing instabilities. SIAM J. Appl. Math., 43(4), 776–821.Google Scholar
Cournot, A. 1838. Recherches sur les Principes Mathematiques de la Theorie des Richesses. Irwin, Homewood, IL, 1963 (English translation).Google Scholar
Cushing, J. M., Henson, S. M., and Roeger, L.-I. 2007. Coexistence of competing juvenile–adult structured populations. J. Biol. Dyn., 1(2), 201–231.Google Scholar
Davydova, N. V. 2004. Old and young: can they coexist? PhD thesis, University of Utrecht.Google Scholar
Demmel, J. W., Dieci, L., and Friedman, M. J. 2000. Computing connecting orbits via an improved algorithm for continuing invariant subspaces. SIAM J. Sci. Comput., 22(1), 81–94.Google Scholar
Devaney, R. L. 2003. An Introduction to Chaotic Dynamical Systems, 2nd ed. Westview Press, Boulder, CO.Google Scholar
Dhooge, A., Govaerts, W., Kuznetsov, Yu. A., Mestrom, W., and Riet, A. M. 2003a. CL MATCONT: a continuation toolbox in MATLAB. Proceedings of the 2003 ACM symposium on applied computing. ACM Press, pp. 161–166.Google Scholar
Dhooge, A., Govaerts, W., and Kuznetsov, Yu. A. 2003b. MATCONT: a MATLAB package for numerical bifurcation analysis of ODEs. ACM Trans. Math. Software, 29(2), 141–164.Google Scholar
Dhooge, A., Govaerts, W., Kuznetsov, Yu. A., Meijer, H. G. E., and Sautois, B. 2008. New features of the software MatCont for bifurcation analysis of dynamical systems. Math. Comput. Model. Dyn. Syst., 14(2), 147–175.Google Scholar
Dieci, L., and Eirola, T. 1999. On smooth decompositions of matrices. SIAM J. Matrix Anal. Appl., 20(3), 800–819.Google Scholar
Diekmann, O., and Heesterbeek, J. A. P. 2000. Mathematical Epidemiology of Infectious Diseases: Model Building, Analysis and Interpretation. Wiley, Chichester.Google Scholar
Ding, W.-C., Xie, J. H., and Sun, Q. G. 2004. Interaction of Hopf and period doubling bifurcations of a vibro-impact system. Journal of Sound and Vibration, 275(1), 27–45.CrossRefGoogle Scholar
Doedel, E. J., Champneys, A. R., Fairgrieve, T. F., Kuznetsov, Yu. A., Sandstede, B., and Wang, X.-J. 1997–2000. Continuation and Bifurcation Software for Ordinary Differential Equations (with HomCont), User’s Guide. http://indy.cs.concordia.caGoogle Scholar
Edmunds, J., Cushing, J. M., Costantino, R.F., Henson, S. M., Dennis, B., and Desharnais, R. A. 2003. Park’s Tribolium competition experiments: a non-equilibrium species coexistence hypothesis. J. Anim. Ecol., 72(5), 703–712.Google Scholar
Elphick, C., Tirapegui, E., Brachet, M. E., Coullet, P., and Iooss, G. 1987. A simple global characterization for normal forms of singular vector fields. Phys. D, 29(1– 2), 95–127.Google Scholar
England, J. P., Krauskopf, B., and Osinga, H. M. 2004. Computing one-dimensional stable manifolds and stable sets of planar maps without the inverse. SIAM J. Appl. Dyn. Syst., 3(2), 161–190.CrossRefGoogle Scholar
Frouzakis, C. E., Adomaitis, R. A., and Kevrekidis, I. G. 1991. Resonance phenomena in an adaptively-controlled system. Internat. J. Bifur. Chaos Appl. Sci. Engrg., 1(1), 83–106.CrossRefGoogle Scholar
Gaunersdorfer, A., Hommes, C. H. and Wagener, F. O. O. 2008. Bifurcation routes to volatility clustering under evolutionary learning. J. Econ. Behav. Organ., 67(1), 27–47, Working paper available at https://ideas.repec.org/p/ams/ ndfwpp/00-04.html.Google Scholar
Gavrilov, N. K., and Shilnikov, L. P. 1972. Three-dimensional dynamical systems that are close to systems with a structurally unstable homoclinic curve: I. Mat. Sb. (N.S.), 88(130), 475–492.Google Scholar
Gavrilov, N. K., and Shilnikov, L. P. 1973. Three-dimensional dynamical systems that are close to systems with a structurally unstable homoclinic curve: II. Mat. Sb. (N.S.), 90(132), 139–156.Google Scholar
Gheiner, J. 1994. Codimension-two reflection and nonhyperbolic invariant lines. Nonlinearity, 7(1), 109–184.Google Scholar
Gheiner, J. 1998. Codimension n flips. Ergodic Theory Dynam. Systems, 18(5), 1115– 1137.Google Scholar
Ginelli, F., Poggi, P., Turchi, A., Chaté, H., Livi, R., and Politi, A. 2007. Characterizing dynamics with covariant Lyapunov vectors. Phys. Rev. Lett., 99(Sep), 130601.Google Scholar
Golden, M. P., and Ydstie, B. E. 1988. Bifurcation in model reference adaptive control systems. Systems Control Lett., 11(5), 413–430.Google Scholar
Gonchenko, S. V., and Gonchenko, V. S. 2000. On Andronov–Hopf bifurcations of two-dimensional diffeomorphisms with homoclinic tangencies. Technical report 556. WIAS, Berlin.Google Scholar
Gonchenko, S. V., and Gonchenko, V. S. 2004. On bifurcations of the birth of closed invariant curves in the case of two-dimensional diffeomorphisms with homoclinic tangencies. Tr. Mat. Inst. Steklova, 244, 87–114 (in Russian).Google Scholar
Gonchenko, S. V., and Shilnikov, L. P. 2003. On two-dimensional area-preserving maps with homoclinic tangencies that have infinitely many generic elliptic periodic points. Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI), 300(8), 155–166, 288–289 (in Russian).Google Scholar
Gonchenko, S. V., Shilnikov, L. P., and Turaev, D. V. 1996. Dynamical phenomena in systems with structurally unstable Poincaré homoclinic orbits. Chaos, 6(1), 15–31.Google Scholar
Gonchenko, S. V., Shilnikov, L. P. and Turaev, D.V. 2002a. On Dynamical Properties of Diffeomorphisms with Homoclinic Tangencies. Technical report 795. WIAS, Berlin.Google Scholar
Gonchenko, S. V., Shilnikov, L. P., and Stenkin, O. V. 2002b. On Newhouse regions with infinitely many stable and unstable invariant tori. Pages 80–102 of: Progress in Nonlinear Science, Vol. 1 (Nizhny Novgorod, 2001). RAS, Institute of Applied Physics, Nizhniy Novgorod.Google Scholar
Gonchenko, S. V., Gonchenko, V. S., and Tatjer, J. C. 2002c. Three-dimensional dissipative diffeomorphisms with codimension two homoclinic tangencies and generalized Hénon maps. Pages 63–79 of: Progress in Nonlinear Science, Vol. 1 (Nizhny Novgorod, 2001). RAS, Institute of Applied Physics, Nizhniy Novgorod.Google Scholar
Gonchenko, V. S. 2002. On bifurcations of Two-Dimensional Diffeomorphisms with a Homoclinic Tangency of Manifolds of a “Neutral” Saddle. Trudy Mat. Inst. Steklova, 236, 86–93 (in Russian).Google Scholar
Gonchenko, V. S., and Ovsyannikov, I. I. 2005. On bifurcations of three-dimensional diffeomorphisms with a homoclinic tangency to a “neutral” saddle fixed point. Journal of Mathematical Sciences, 128(2), 2774–2777.Google Scholar
Gonchenko, V. S., Kuznetsov, Yu. A., and Meijer, H. G. E. 2005. Generalized Hénon map and bifurcations of homoclinic tangencies. SIAM J. Appl. Dyn. Syst., 4(2), 407–436.Google Scholar
Govaerts, W. J. F. 2000. Numerical Methods for Bifurcations of Dynamical Equilibria. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA.Google Scholar
Govaerts, W., and Khoshsiar Ghaziani, R. 2006. Numerical bifurcation analysis of a nonlinear stage structured cannibalism population model. J. Difference Equ. Appl., 12(10), 1069–1085.Google Scholar
Govaerts, W., and Khoshsiar Ghaziani, R. 2008. Stable cycles in a Cournot duopoly model of Kopel. J. Comput. Appl. Math., 218(2), 247–258.Google Scholar
Govaerts, W., Kuznetsov, Yu. A., and Sijnave, B. 1999. Bifurcations of maps in the software package CONTENT. Pages 191–206 of: Computer Algebra in Scientific Computing – CASC’99 (Munich). Springer, Berlin.Google Scholar
Govaerts, W., Khoshsiar Ghaziani, R., Kuznetsov, Yu. A., and Meijer, H. G. E. 2007. Numerical methods for two-parameter local bifurcation analysis of maps. SIAM J. Sci. Comput., 29(6), 2644–2667.Google Scholar
Gramchev, T. and Walcher, S. 2005, Normal forms of maps: formal and algebraic aspects. Acta Applicandae Mathematica, 87(1), 123–146Google Scholar
Griewank, A. and Walther, A. 2008. Evaluating Derivatives: Principles and Techniques of Algorithmic Differentiation 2nd edition. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA.Google Scholar
Griewank, A., Juedes, D., and Utke, J. 1996. ADOL-C: a package for the automatic differentiation of algorithms written in C/C++. ACM Trans. Math. Softw., 22(2), 131–167.Google Scholar
Guckenheimer, J., and Holmes, P. 1990. Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields. Springer-Verlag, New York. Revised and corrected reprint of the 1983 original.Google Scholar
Guckenheimer, J., and Meloon, B. 2000. Computing periodic orbits and their bifurcations with automatic differentiation. SIAM J. Sci. Comput., 22(3), 951–985.Google Scholar
Hassard, B. D., Kazarinoff, N. D., and Wan, Y. H. 1981. Theory and Applications of Hopf Bifurcation. Cambridge University Press, Cambridge.Google Scholar
Hénon, M. 1976. A two-dimensional mapping with a strange attractor. Comm. Math. Phys., 50(1), 69–77.Google Scholar
Hirsch, M. W., Pugh, C. C., and Shub, M. 1977. Invariant Manifolds. Springer-Verlag, Berlin.Google Scholar
Holmes, P.J., and Rand, D. A. 1977/1978. Bifurcations of the forced van der Pol oscillator. Quart. Appl. Math., 35(4), 495–509.Google Scholar
Holmes, P., and Whitley, D. 1984a. Bifurcations of one- and two-dimensional maps. Philos. Trans. Roy. Soc. London Ser. A, 311(1515), 43–102.Google Scholar
Holmes, P., and Whitley, D. 1984b. Erratum: “Bifurcations of one- and two-dimensional maps.” Philos. Trans. Roy. Soc. London Ser. A, 312(1523), 601.Google Scholar
Horozov, E. I. 1979. Versal deformations of equivariant vector fields for cases of symmetry of order 2 and 3. Trudy Sem. Petrovsk., 5, 163–192 (in Russian).Google Scholar
Iooss, G. 1979. Bifurcation of Maps and Applications. North-Holland Publishing Co., Amsterdam.Google Scholar
Iooss, G. 1988. Global characterization of the normal form for a vector field near a closed orbit. J. Differential Equations, 76(1), 47–76.Google Scholar
Iooss, G., and Adelmeyer, M. 1998 Topics in Bifurcation Theory and Applications. 2nd edition. World Scientific Publishing, River Edge, NJ.Google Scholar
Iooss, G., and Los, J. E. 1988. Quasi-genericity of bifurcations to high-dimensional invariant tori for maps. Comm. Math. Phys., 119(3), 453–500.Google Scholar
Ipsen, M., Hynne, F., and Sörensen, P. G. 1998. Systematic derivation of amplitude equations and normal forms for dynamical systems. Chaos, 8(4), 834–852.Google Scholar
Jorba, A. 2001. Numerical computation of the normal behaviour of invariant curves of n-dimensional maps. Nonlinearity, 14(5), 943–976.Google Scholar
Kamiyama, K., Komuro, M., Endo, T., and Aihara, K. 2014. Classification of bifurcations of quasi-periodic solutions using Lyapunov bundles. Internat. J. Bifur. Chaos Appl. Sci. Engrg., 24(12), 1430034, 30.Google Scholar
Katok, A., and Hasselblatt, B. 1995 Introduction to the Modern Theory of Dynamical Systems. Cambridge University Press, Cambridge.Google Scholar
Kawakami, H. 1981. Algorithme numérique définissant la bifurcation d’un point homocline. C. R. Acad. Sci. Paris Sér. I Math., 293(8), 401–403.Google Scholar
Keller, H. B. 1977. Numerical solution of bifurcation and nonlinear eigenvalue problems. Pages 359–384 of: Applications of Bifurcation Theory. Academic Press, New York.Google Scholar
Khibnik, A. I., Kuznetsov, Yu. A., Levitin, V. V., and Nikolaev, E. V. 1993. Continuation techniques and interactive software for bifurcation analysis of ODEs and iterated maps. Phys. D, 62(1–4), 360–371.Google Scholar
Khoshsiar Ghaziani, R., Govaerts, W., Kuznetsov, Yu. A., and Meijer, H. G. E. 2009. Numerical continuation of connecting orbits of maps in MATLAB. J. Difference Equ. Appl., 15(8–9), 849–875.Google Scholar
Khoshsiar Ghaziani, R., Govaerts, W., and Sonck, C. 2012. Resonance and bifurcation in a discrete-time predator–prey system with Holling functional response. Nonlinear Anal. Real World Appl., 13(3), 1451–1465.Google Scholar
Kleinkauf, J.-M. 1998. Numerische Analyse Tangentialer Homokliner Orbits. PhD thesis, Universität Bielefeld.Google Scholar
Komuro, M., Kamiyama, K., Endo, T., and Aihara, K. 2016. Quasi-periodic bifurcations of higher-dimensional tori. Internat. J. Bifur. Chaos Appl. Sci. Engrg., 26(7), 1630016, 40.Google Scholar
Kopel, M. 1996. Simple and complex adjustment dynamics in Cournot duopoly models. Chaos, Solitons & Fractals, 7(12), 2031–2048.Google Scholar
Krauskopf, B. 1994. The bifurcation set for the 1:4 resonance problem. Experiment. Math., 3(2), 107–128.Google Scholar
Krauskopf, B. 1995. The extended parameter space of a model for 1:4 resonance. CWI Quarterly, 8(3), 237–255.Google Scholar
Krauskopf, B. 1997. Bifurcations at ∞ in a model for 1:4 resonance. Ergodic Theory Dynam. Systems, 17(4), 899–931.Google Scholar
Krauskopf, B. 2001. Strong resonances and Takens’s Utrecht preprint. Pages 89–111 of: Global Analysis of Dynamical Systems. Institute of Physics, Bristol.Google Scholar
Krauskopf, B., and Oldeman, B. E. 2004. A planar model system for the saddle-node Hopf bifurcation with global reinjection. Nonlinearity, 17(4), 1119–1151.Google Scholar
Krauskopf, B., and Osinga, H. M. 1998a. Globalizing two-dimensional unstable manifolds of maps. Internat. J. Bifur. Chaos Appl. Sci. Engrg., 8(3), 483–503.Google Scholar
Krauskopf, B., and Osinga, H. M. 1998b. Growing 1D and quasi-2D unstable manifolds of maps. J. Comput. Phys., 146(1), 404–419.Google Scholar
Kurakin, L. G. 1989. 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., 1, 31–35 (in Russian).Google Scholar
Kuznetsov, Yu. A. 1999. Numerical normalization techniques for all codim 2 bifurcations of equilibria in ODE’s. SIAM J. Numer. Anal., 36(4), 1104–1124.Google Scholar
Kuznetsov, Yu A. 2004. Elements of Applied Bifurcation Theory. 3rd edition. Springer-Verlag, New York.Google Scholar
Kuznetsov, Yu. A. 2005. Practical computation of normal forms on center manifolds at degenerate Bogdanov–Takens bifurcations. Internat. J. Bifur. Chaos Appl. Sci. Engrg., 15(11), 3535–3546.Google Scholar
Kuznetsov, Yu. A., and Levitin, V. V. 1995–1997. CONTENT: A Multiplatform Environment for Analyzing Dynamical Systems. www.staff.science.uu.nl/∼kouzn101/CONTENT.Google Scholar
Kuznetsov, Yu. A., and Meijer, H. G. E. 2005. Numerical normal forms for codim 2 bifurcations of fixed points with at most two critical eigenvalues. SIAM J. Sci. Comput., 26(6), 1932–1954.Google Scholar
Kuznetsov, Yu. A., and Meijer, H. G. E. 2006. Remarks on interacting Neimark–Sacker bifurcations. J. Difference Equ. Appl., 12(10), 1009–1035.Google Scholar
Kuznetsov, Yu. A., and Piccardi, C. 1994. Bifurcation analysis of periodic SEIR and SIR epidemic models. J. Math. Biol., 32(2), 109–121.Google Scholar
Kuznetsov, Yu. A., Muratori, S., and Rinaldi, S. 1992. Bifurcations and chaos in a periodic predator–prey model. Internat. J. Bifur. Chaos Appl. Sci. Engrg., 2(1), 117–128.Google Scholar
Kuznetsov, Yu. A., Meijer, H. G. E., and van Veen, L. 2004. The fold–flip bifurcation. Internat. J. Bifur. Chaos Appl. Sci. Engrg., 14(7), 2253–2282.Google Scholar
Laskar, J., Froeschlé, C., and Celletti, A. 1992. The measure of chaos by the numerical analysis of the fundamental frequencies: application to the standard mapping. Phys. D, 56(2–3), 253–269.Google Scholar
Leslie, P. H., and Gower, J. C. 1958. The properties of a stochastic model for two competing species. Biometrika, 45, 316–330.Google Scholar
Leslie, P. H., Park, T., and Mertz, D. B. 1968. The effect of varying the initial numbers on the outcome of competition between two Tribolium species. J. Animal Ecology, 37, 9–23.Google Scholar
Lindtner, E., Steindl, A., and Troger, H. 1989. Generic one-parameter bifurcations in the motion of a simple robot. J. Comput. Appl. Math., 26(1– 2), 199–218.Google Scholar
Lorenz, E. N. 1984. Irregularity: a fundamental property of the atmosphere. Tellus Series A, 36, 98–110.Google Scholar
Los, J. E. 1988. Small divisor phenomena in an invariant curve doubling bifurcation on a cylinder. Ann. Inst. H. Poincaré Anal. Non Linéaire, 5(1), 37–95.Google Scholar
Los, J. E. 1989. Non-normally hyperbolic invariant curves for maps in R3 and doubling bifurcation. Nonlinearity, 2(1), 149–174.Google Scholar
Maynard Smith, J. 1968. Mathematical Ideas in Biology. Cambridge University Press, LondonGoogle Scholar
Melnikov, V. K. 1963. Qualitative description of strong resonance in a nonlinear system. Dokl. Akad. Nauk SSSR, 148, 1257–1260.Google Scholar
Mira, C. 1997. Some historical aspects of nonlinear dynamics: possible trends for the future. J. Franklin Inst. B, 334(5–6), 1075–1113.Google Scholar
Murdock, J. 2003. Normal Forms and Unfoldings for Local Dynamical Systems. Springer-Verlag, New York.Google Scholar
Neimark, Ju. I. 1959. Some cases of the dependence of periodic motions on parameters. Dokl. Akad. Nauk SSSR, 129, 736–739.Google Scholar
Neimark, Ju. I. 1972. The Method of Point Transformations in the Theory of Nonlinear Oscillations. Nauka, Moscow (in Russian).Google Scholar
Neirynck, N., Govaerts, W., Kuznetsov, Yu. A., and Meijer, H. G. E. 2018. Numerical bifurcation analysis of homoclinic orbits embedded in one-dimensional manifolds of maps. ACM Trans. Math. Softw. 44 (3), article no. 25.Google Scholar
Neishtadt, A. I. 1978. Bifurcations of the phase pattern of an equation system arising in the problem of stability loss of self-oscillations close to 1:4 resonance. J. Appl. Math. Mech., 830–840.Google Scholar
Newhouse, S., Palis, J., and Takens, F. 1983. Bifurcations and stability of families of diffeomorphisms. Inst. Hautes Études Sci. Publ. Math., 57, 5–71.Google Scholar
Nusse, H. E., and Yorke, J. A. 1998. Dynamics: Numerical Explorations. 2nd edition. Springer-Verlag, New York.Google Scholar
Paez Chavez, J. 2010 Starting homoclinic tangencies near 1:1 resonances. Internat. J. Bifur. Chaos Appl. Sci. Engrg, 20(10), 3157–3172.Google Scholar
Palis, J., and de Melo, W. 1982. Geometric Theory of Dynamical Systems. Springer-Verlag, New York.Google Scholar
Palis, J., and Takens, F. 1993. Hyperbolicity and Sensitive Chaotic Dynamics at Homoclinic Bifurcations. Cambridge University Press, Cambridge.Google Scholar
Palmer, T. N. 1995. The influence of north-west Atlantic sea surface temperature: an unplanned experiment. Weather, 50(12), 413–419.Google Scholar
Peckham, B. B., and Kevrekidis, I. G. 1991. Period doubling with higher-order degeneracies. SIAM J. Math. Anal., 22(6), 1552–1574.Google Scholar
Peckham, B. B., and Kevrekidis, I. G. 2002. Lighting Arnold flames: resonance in doubly forced periodic oscillators. Nonlinearity, 15(2), 405–428.Google Scholar
Pecora, N. 2018. Analysis of 1:4 resonance in a monopoly model with memory. Chaos, Solitons & Fractals, 110, 95–104.Google Scholar
Pryce, J. D., Khoshiar Ghaziani, R., de Witte, V., and Govaerts, W. 2010, Computation of normal form coefficients of cycle bifurcations of maps by algorithmic differentiation. Math. Comput. Sim., 81(1), 109–119.Google Scholar
Roose, D., and Hlavacek, V. 1985. A direct method for the computation of Hopf bifurcation points. SIAM J. Appl. Math., 45(6), 879–894.Google Scholar
Rousseau, C. 1990. Codimension 1 and 2 bifurcations of fixed points of diffeomorphisms and periodic solutions of vector fields. Ann. Sci. Math. Québec, 13(2), 55–91.Google Scholar
Sacker, R. J. 1965. A new approach to the perturbation theory of invariant surfaces. Comm. Pure Appl. Math., 18, 717–732.Google Scholar
Saleh, K. 2005. Organising centres in the semi-global analysis of dynamical systems. PhD thesis, University of Groningen.Google Scholar
Saleh, K., and Wagener, F. O. O. 2010. Semi-global analysis of periodic and quasi-periodic normal-internal k:1 and k:2 resonances. Nonlinearity, 23(9), 2219–2252.Google Scholar
Sanders, J. A., and Verhulst, F. 1985. Averaging Methods in Nonlinear Dynamical Systems. Springer-Verlag, New York.Google Scholar
Shilnikov, A., Nicolis, G., and Nicolis, C. 1995. Bifurcation and predictability analysis of a low-order atmospheric circulation model. Internat. J. Bifur. Chaos Appl. Sci. Engrg., 5(6), 1701–1711.Google Scholar
Shoshitaishvili, A. N. 1975. The bifurcation of the topological type of the singular points of vector fields that depend on parameters. Trudy Sem. Petrovsk., 1, 279– 309 (in Russian).Google Scholar
Simó, C. 1990. On the analytical and numerical approximation of invariant manifolds. Pages 285–329 of: Les Méthodes Modernes de la Mécanique Céleste [Modern Methods in Celestial Mechanics]. Available at www.maia.ub.es/dsg/2004/index.html.Google Scholar
Simó, C. 2005. On the use of Lyapunov exponents to detect global properties of the dynamics. Pages 631–636 of: EQUADIFF 2003. World Scientific Publishing, Hackensack, NJ.Google Scholar
Takens, F. 1974. Forced oscillations and bifurcations. Pages 1–59 of: Applications of global analysis: I. Mathematical Institute, Utrecht. Reprinted in Global Analysis of Dynamical Systems, edited by Broer, H.W., Krauskopf, B. and Vegter, G.. Institute of Physics, Bristol, 2001.Google Scholar
Takens, F., and Wagener, F. O. O. 2000. Resonances in skew and reducible quasi-periodic Hopf bifurcations. Nonlinearity, 13(2), 377–396.Google Scholar
Turaev, D. 1996. On dimension of non-local bifurcational problems. Internat. J. Bifur. Chaos Appl. Sci. Engrg., 6(5), 919–948.Google Scholar
van Gils, S. A. 1984. On a Formula for the Direction of Hopf Bifurcation. Technical Report TW/225. CWI, Amsterdam.Google Scholar
Vanderbauwhede, A. 1982. Local Bifurcation and Symmetry. Pitman (Advanced Publishing Program), Boston, MA.Google Scholar
Vanderbauwhede, A. 1988. Center manifolds and their basic properties: an introduction. Delft Progr. Rep., 12(1), 57–78.Google Scholar
Vanderbauwhede, A., and van Gils, S. A. 1987. Center manifolds and contractions on a scale of Banach spaces. J. Funct. Anal., 72(2), 209–224.Google Scholar
Vanhulle, L. 2017. Computational detection of homoclinic and heteroclinic connections for iterated maps (in Dutch). MSc thesis, Ghent University. Available at https://lib.ugent.be/en/catalog/rug01:002376270?i=0&q=Laura+Vanhulle.Google Scholar
Veen, L. van. 2003. Baroclinic flow and the Lorenz-84 model. Internat. J. Bifur. Chaos Appl. Sci. Engrg., 13(8), 2117–2139.Google Scholar
Vitolo, R. 2003. Bifurcations in 3D diffeomorphisms: a study in experimental mathematics. PhD thesis, University of Groningen.Google Scholar
Vitolo, R., Broer, H. W., and Simó, C. 2010. Routes to chaos in the Hopf-saddle-node bifurcation for fixed points of 3D-diffeomorphisms. Nonlinearity, 23(8), 1919.Google Scholar
Vitolo, R., Glendinning, P., and Gallas, J. A. C. 2011a. Global structure of periodicity hubs in Lyapunov phase diagrams of dissipative flows. Phys. Rev. E, 84, 016216.Google Scholar
Vitolo, R., Broer, H. W., and Simó, C. 2011b. Quasi-periodic bifurcations of invariant circles in low-dimensional dissipative dynamical systems. Regul. Chaotic Dyn., 16(1–2), 154–184.Google Scholar
Wan, Y. H. 1978. Bifurcation into invariant tori at points of resonance. Arch. Rational Mech. Anal., 68(4), 343–357.Google Scholar
Wen, G.-L., and Xu, D. 2004a. Feedback control of Hopf–Hopf interaction bifurcation with development of torus solutions in high-dimensional maps. Phys. Lett. A, 321(1), 24–33.Google Scholar
Wen, G.-L., and Xu, D. 2004b. Implicit criteria of eigenvalue assignment and transversality for bifurcation control in four-dimensional maps. Internat. J. Bifur. Chaos Appl. Sci. Engrg., 14(10), 3489–3503.Google Scholar
Wen, G.-L., Wang, Q.-G., and Chiu, M.-S. 2006. Delay feedback control for interaction of Hopf and period doubling bifurcations in discrete-time systems. Internat. J. Bifur. Chaos Appl. Sci. Engrg., 16(1), 101–112.Google Scholar
Wiggins, S. 2003. Introduction to Applied Nonlinear Dynamical Systems and Chaos. 2nd edition. Springer-Verlag, New York.Google Scholar
Research, Wolfram, Inc. 2017. Mathematica, Version 11.1. Wolfram Research, Champaign, IL.Google Scholar
Xie, J., and Ding, W. 2005. Hopf–Hopf bifurcation and invariant torus T2 of a vibro-impact system. Internat. J. Non-Linear Mech., 40(4), 531–543.Google Scholar
Yagasaki, K., 1998. Numerical detection and continuation of homoclinic points and their bifurcations for maps and periodically forced systems, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 8(7), 1617–1627Google Scholar
Zegeling, A. 1993. Equivariant unfoldings in the case of symmetry of order 4. Serdica, 19(1), 71–79.Google Scholar
Zholondek, K. 1983. Versality of a family of symmetric vector fields on the plane. Mat. Sb. (N.S.), 120(162)(4), 473–499.Google Scholar