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Numerical Analysis of Bifurcations

  • Chapter
Elements of Applied Bifurcation Theory

Part of the book series: Applied Mathematical Sciences ((AMS,volume 112))

Abstract

In this chapter we shall describe some of the basic techniques used in the numerical analysis of dynamical systems. We assume that low-level numerical routines like those for solving linear systems, finding eigenvectors and eigenvalues, and performing numerical integration of ODEs are known to the reader. Instead we focus on algorithms that are more specific to bifurcation analysis, specifically those for the location of equilibria (fixed points) and their continuation with respect to parameters, and for the detection, analysis, and continuation of bifurcations. Special attention is given to location and continuation of limit cycles and their associated bifurcations, as well as to continuation of homoclinic orbits. We deal mainly with the continuous-time case and give only brief remarks on discrete-time systems. Appendix A summarizes simple estimates of convergence of Newton-like methods. Appendix B gives some background information on the bialternate matrix product used to detect Hopf and Neimark-Sacker bifurcations. Appendix C presents numerical methods for detection of higher-order homoclinic bifurcations. The bibliographical notes in Appendix D include references to standard noninteractive software packages and interactive programs available for continuation and bifurcation analysis of dynamical systems. Actually, the main goal of this chapter is to provide the reader with an understanding of the methods implemented in widely used software for dynamical systems analysis.

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Bibliographical notes

  • Beyn, W.-J. (1991), Numerical methods for dynamical systems, in W. Light, ed., `Advances in Numerical Analysis, Vol. I, Nonlinear Partial Differential Equations and Dynamical Systems’, Oxford University Press, Oxford, pp. 175–236.

    Google Scholar 

  • Doedel, E., Keller, H. and Kernévez, J.-P. (1991a), `Numerical analysis and control of bifurcation problems: (I) Bifurcation in finite dimensions’, Internat. J. Bifur. Chaos Appl. Sci. Engrg. 1, 493–520.

    Article  MathSciNet  MATH  Google Scholar 

  • Doedel, E., Keller, H. and Kernévez, J.-P. (1991b), `Numerical analysis and control of bifurcation problems: (II) Bifurcation in infinite dimensions’, Internat. J. Bifur. Chaos Appl. Sci. Engrg. 1, 745–772.

    Article  MathSciNet  MATH  Google Scholar 

  • Guckenheimer, J. and Worfolk, P. (1993), Dynamical systems: some computational problems, in D. Schlomiuk, ed., `Bifurcations and Periodic Orbits of Vector Fields (Montreal, 1992)’, Vol. 408 of NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci.,Kluwer Acad. Publ., Dordrecht, pp. 241–278.

    Google Scholar 

  • Guckenheimer, J. (2002), Numerical analysis of dynamical systems, in B. Fiedler, ed., `Handbook of Dynamical Systems, Vol. 2’, Elsevier Science, Amsterdam, pp. 345–390.

    Google Scholar 

  • Govaerts, W. (2000), Numerical Methods for Bifurcations of Dynamical Equilibria, SIAM, Philadelphia, PA.

    Google Scholar 

  • Beyn, W.-J. (1990a), Global bifurcations and their numerical computation, in D. Roose, B. De Dier and A. Spence, eds, `Continuation and Bifurcations: Numerical Techniques and Applications (Leuven, 1989)’, Vol. 313 of NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., Kluwer Acad. Publ., Dordrecht, pp. 169–181.

    Google Scholar 

  • Kaas-Petersen, C. (1989), PATH - User’s Guide, University of Leeds, Leeds. Kantorovich, L.V. and Akilov, G.P. (1964), Functional Analysis in Normed Spaces, Pergamon Press, Oxford.

    Google Scholar 

  • Broyden, C. (1965), `A class of methods for solving nonlinear simultaneous equa-tions’, Math. Comp. 19, 577–593.

    Article  MathSciNet  MATH  Google Scholar 

  • Dennis, J. and Schnabel, R. (1983), Numerical Methods for Unconstrained Optimiza-tion and Nonlinear Equations,Prentice-Hall, Englewood Cliffs, NJ.

    Google Scholar 

  • Hassard, B. (1980), Computation of invariant manifolds, in P. Holmes, ed., `New Ap-proaches to Nonlinear Problems in Dynamics’, SIAM, Philadelphia, PA, pp. 27–42.

    Google Scholar 

  • Kuznetsov, Yu.A. (1983), One-dimensional invariant manifolds of saddles in ordinary differential equations depending upon parameters, FORTRAN Software Series, Vol. 8, Research Computing Centre, USSR Academy of Sciences, Pushchino, Moscow Region. In Russian.

    Google Scholar 

  • Osinga, H. (2003), `Nonorientable manifolds in threee-dimensional vector fields’, Internat. J. Bifur. Chaos Appl. Sci. Engrg. 13, 553–570.

    Article  MathSciNet  MATH  Google Scholar 

  • Dieci, L. and Lorenz, J. (1995), `Computation of invariant tori by the method of characteristics’, SIAM J. Numer. Anal. 32, 1436–1474.

    Article  MathSciNet  MATH  Google Scholar 

  • Dieci, L. and Lorenz, J. (1997), `Lyapunov-type numbers and torus breakdown: nu-merical aspects and a case study’, Numer. Algorithms 14, 79–102.

    Article  MathSciNet  MATH  Google Scholar 

  • Krauskopf, B. and Osinga, H. (1999), `Two-dimensional global manifolds of vector fields’, Chaos 9, 768–774.

    Article  MathSciNet  MATH  Google Scholar 

  • Friedman, M. and Doedel, E. (1993), `Computational methods for global analysis of homoclinic and heteroclinic orbits: A case study’, J. Dynamics Differential Equations 5, 37–57.

    Article  MathSciNet  MATH  Google Scholar 

  • Berezovskaya, F.S. and Khibnik, A.I. (1981), `On the bifurcations of separatrices in the problem of stability loss of auto-oscillations near 1:4 resonance’, J. Appl. Math. Mech. 44, 938–942.

    Google Scholar 

  • Holodniok, M. and Kubicek, M. (1984a), `DERPER–An algorithm for the continuation of periodic solutions in ordinary differential equations’, J. Comput. Phys. 55, 254–267.

    Article  MathSciNet  MATH  Google Scholar 

  • Guckenheimer, J. and Meloon, B. (2000), `Computing periodic orbits and their bifur- cations with automatic differentiation’, SIAM J. Sci. Comput. 22, 951–985.

    Article  MathSciNet  MATH  Google Scholar 

  • Doedel, E., Keller, H. and Kernévez, J.-P. (1991b), `Numerical analysis and control of bifurcation problems: (II) Bifurcation in infinite dimensions’, Internat. J. Bifur. Chaos Appl. Sci. Engrg. 1, 745–772.

    Article  MathSciNet  MATH  Google Scholar 

  • de Boor, C. and Swartz, B. (1973), `Collocation at Gaussian points’, SIAM J. Numer. Anal. 10, 582–606.

    Google Scholar 

  • Doedel, E. (1981), `AUTO, a program for the automatic bifurcation analysis of autonomous systems’, Congr. Numer. 30, 265–384.

    MathSciNet  Google Scholar 

  • Balabaev, N.K. and Lunevskaya, L.V. (1978), Continuation of a curve in the n-dimensional space, FORTRAN Software Series, Vol. 1, Research Computing

    Google Scholar 

  • Kaas-Petersen, C. (1989), PATH - User’s Guide, University of Leeds, Leeds. Kantorovich, L.V. and Akilov, G.P. (1964), Functional Analysis in Normed Spaces, Pergamon Press, Oxford.

    Google Scholar 

  • Deuflhard, P., Fiedler, B. and Kunkel, P. (1987), `Efficient numerical pathfollowing beyond critical points’, SIAM J. Numer. Anal. 24, 912–927.

    Article  MathSciNet  MATH  Google Scholar 

  • Allgower, E. and Georg, K. (1993), Continuation and path following, in ‘Acta Numerica’, Cambridge University Press, Cambridge, pp. 1–64.

    Google Scholar 

  • Keller, H. (1977), Numerical solution of bifurcation and nonlinear eigenvalue problems, in P. Rabinowitz, ed., `Applications of Bifurcation Theory’, Academic Press, New York, pp. 359–384.

    Google Scholar 

  • Rheinboldt, W. (1986), Numerical Analysis of Parametrized Nonlinear Equations, Wiley, New York.

    MATH  Google Scholar 

  • Allgower, E. and Georg, K. (1993), Continuation and path following, in ‘Acta Numerica’, Cambridge University Press, Cambridge, pp. 1–64.

    Google Scholar 

  • Khibnik, A.I. (1990), LINLBF: A program for continuation and bifurcation analysis of equilibria up to codimension three, in D. Roose, B. De Dier and A. Spence, eds, `Continuation and Bifurcations: Numerical Techniques and Applications (Leuven, 1989)’, Vol. 313 of NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., Kluwer Acad. Publ., Dordrecht, pp. 283–296.

    Google Scholar 

  • Fuller, A. (1968), `Condition for a matrix to have only characteristic roots with negative real parts’, J. Math. Anal. Appl. 23, 71–98.

    Article  MathSciNet  MATH  Google Scholar 

  • Jury, E. and Gutman, S. (1975), `On the stability of the A matrix inside the unit circle’, IEEE Trans. Automatic Control AC - 20,533–535.

    Google Scholar 

  • Guckenheimer, J. and Myers, M. (1996), `Computing Hopf bifurcations. II. Three examples from neurophysiology’, SIAM J. Sci. Comput. 17, 1275–1301.

    Article  MathSciNet  MATH  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 

  • Govaerts, W. (2000), Numerical Methods for Bifurcations of Dynamical Equilibria, SIAM, Philadelphia, PA.

    Google Scholar 

  • Guckenheimer, J., Myers, M. and Sturmfels, B. (1997), `Computing Hopf bifurcations. I’, SIAM J. Numer. Anal. 34, 1–21.

    Google Scholar 

  • Moore, G., Garret, T. and Spence, A. (1990), The numerical detection of Hopf bifurcation points, in D. Roose, B. De Dier and A. Spence, eds, `Continuation and Bifurcations: Numerical Techniques and Applications (Leuven, 1989)’, Vol. 313 of NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., Kluwer Acad. Publ., Dordrecht, pp. 227–246.

    Google Scholar 

  • Friedman, M. (2001), `Improved detection of bifurcations in large nonlinear systems via the continuation of invariant subspaces algorithm’, Internat. J. Bifur. Chaos Appl. Sci. Engrg. 11, 2277–2285.

    Article  MathSciNet  MATH  Google Scholar 

  • Dieci, L. and Friedman, M. (2001), `Continuation of invariant subspaces’, Numer. Linear Algebra Appl. 8, 317–327.

    Article  MathSciNet  MATH  Google Scholar 

  • Griewank, A. and Reddien, G. (1984), `Characterization and computation of general-ized turning points’, SIAM J. Numer. Anal. 21, 176–184.

    Article  MathSciNet  MATH  Google Scholar 

  • Govaerts, W. and Pryce, J. (1993), `Mixed block elimination for linear systems with wider borders’, IMA J. Numer. Anal. 13, 161–180.

    Google Scholar 

  • Govaerts, W., Guckenheimer, J. and Khibnik, A.I. (1997), `Defining functions for multiple Hopf bifurcations’, SIAM J. Numer. Anal. 34, 1269–1288.

    Google Scholar 

  • Roose, D. and Hlavacek, V. (1985), `A direct method for the computation of Hopf bifurcation points’, SIAM J. Appl. Math. 45, 897–894.

    MathSciNet  Google Scholar 

  • Allgower, E. and Georg, K. (1993), Continuation and path following, in ‘Acta Numerica’, Cambridge University Press, Cambridge, pp. 1–64.

    Google Scholar 

  • Moore, G. (1980), `The numerical treatment of non-trivial bifurcation points’, Nu-mer. Funct. Anal. Optimiz. 2, 441–472.

    Article  MATH  Google Scholar 

  • Borisyuk, R.M. (1981), Stationary solutions of a system of ordinary differential equations depending upon a parameter, FORTRAN Software Series, Vol. 6, Research Computing Centre, USSR Academy of Sciences, Pushchino, Moscow Region. In Russian.

    Google Scholar 

  • Hassard, B., Kazarinoff, N. and Wan, Y.-H. (1981), Theory and Applications of Hopf Bifurcation, Cambridge University Press, London.

    MATH  Google Scholar 

  • Khibnik, A.I. (1990), LINLBF: A program for continuation and bifurcation analysis of equilibria up to codimension three, in D. Roose, B. De Dier and A. Spence, eds, `Continuation and Bifurcations: Numerical Techniques and Applications (Leuven, 1989)’, Vol. 313 of NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., Kluwer Acad. Publ., Dordrecht, pp. 283–296.

    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’, Physica D 62, 360–371.

    Article  MathSciNet  MATH  Google Scholar 

  • Feudel, U. and Jansen, W. (1992), `CANDYS/QA–A software system for qualitative analysis of nonlinear dynamical systems’, Internat. J. Bifur. Chaos Appl. Sci. Engrg. 2, 773–794.

    Article  MathSciNet  MATH  Google Scholar 

  • Dhooge, A., Govaerts, W. and Kuznetsov, Yu.A. (2003), `MATCONT:A MATLAB package for numerical bifurcation analysis of ODEs’, ACM Trans. Math. Software 29, 141–164.

    Article  MathSciNet  MATH  Google Scholar 

  • Bazykin, A.D., Kuznetsov, Yu.A. and Khibnik, A.I. (1989), Portraits of Bifurcations: Bifurcation Diagrams of Dynamical Systems on the Plane,Znanie, Moscow. In Russian.

    Google Scholar 

  • Doedel, E. and Friedman, M. (1989), `Numerical computation of heteroclinic orbits’, J. Comput. Appl. Math. 26, 159–170.

    Article  MathSciNet  Google Scholar 

  • Doedel, E. and Kernévez, J.-P. (1986), AUTO: Software for continuation problems in ordinary differential equations with applications, Applied Mathematics, California Institute of Technology, Pasadena, CA.

    Google Scholar 

  • Kuznetsov, Yu.A. (1990), Computation of invariant manifold bifurcations, in D. Roose, B. De Dier and A. Spence, eds, `Continuation and Bifurcations: Numerical Techniques and Applications (Leuven, 1989)’, Vol. 313 of NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., Kluwer Acad. Publ., Dordrecht, pp. 183–195.

    Google Scholar 

  • Kuznetsov, Yu.A. (1991), Numerical analysis of the orientability of homoclinic trajectories, in R. Seydel, F. Schneider, T. Küpper and H. Troger, eds, `Bifurcation and Chaos: Analysis, Algorithms, Applications (Würzburg, 1990)’, Vol. 97 of Internat. Ser. Numer. Math., Birkhäuser, Basel, pp. 237–242.

    Google Scholar 

  • Rodriguez-Luis, A., Freire, E. and Ponce, E. (1990), A method for homoclinic and heteroclinic continuation in two and three dimensions, in D. Roose, B. De Dier and A. Spence, eds, `Continuation and Bifurcations: Numerical Techniques and Applications (Leuven, 1989)’, Vol. 313 of NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., Kluwer Acad. Publ., Dordrecht, pp. 197–210.

    Google Scholar 

  • Hassard, B. (1980), Computation of invariant manifolds, in P. Holmes, ed., `New Ap-proaches to Nonlinear Problems in Dynamics’, SIAM, Philadelphia, PA, pp. 27–42.

    Google Scholar 

  • Miura, R. (1982), `Accurate computation of the stable solitary wave for the FitzHughNagumo equations’, J. Math. Biol. 13, 247–269.

    Article  MathSciNet  MATH  Google Scholar 

  • Schecter, S. (1993), `Numerical computation of saddle-node homoclinic bifurcation points’, SIAM J. Numer. Anal. 30, 1155–1178.

    Article  MathSciNet  MATH  Google Scholar 

  • Bai, F. and Champneys, A. (1996), `Numerical detection and continuation of saddle-node homoclinic bifurcations of codimension one and two’, Dynam. Stability Systems 11, 325–346.

    Article  MathSciNet  MATH  Google Scholar 

  • Demmel, J., Dieci, L. and Friedman, M. (2000), `Computing connecting orbits via an improved algorithm for continuing invariant subspaces’, SIAM J. Sci. Comput. 22, 81–94

    Article  MathSciNet  MATH  Google Scholar 

  • Deng, B. (1989), `The Sil’nikov problem, exponential expansion, strong.A-lemma, C1-linearization, and homoclinic bifurcation’, J. Differential Equations 79, 189–231.

    Article  MathSciNet  MATH  Google Scholar 

  • Champneys, A., Kuznetsov, Yu.A. and Sandstede, B. (1996), `A numerical toolbox for homoclinic bifurcation analysis’, Internat. J. Bifur. Chaos Appl. Sci. Engrg. 6, 867–887.

    Article  MathSciNet  MATH  Google Scholar 

  • Champneys, A., Härterich, J. and Sandstede, B. (1996), `A non-transverse homoclinic orbit to a saddle-node equilibrium’, Ergodic Theory Dynamical Systems 16, 431450.

    Google Scholar 

  • Champneys, A., Kuznetsov, Yu.A. and Sandstede, B. (1995), HomCoNT: An AUTO86 driver for homoclinic bifurcation analysis. Version 2.0, Report AM-R9516, Centrum voor Wiskunde en Informatica, Amsterdam.

    Google Scholar 

  • Carr, J. (1981), Applications of Center Manifold Theory, Springer-Verlag, New York. Carr, J., Chow, S.-N. and Hale, J. (1985), `Abelian integrals and bifurcation theory’, J. Differential Equations 59, 413–436.

    MathSciNet  Google Scholar 

  • Champneys, A. and Kuznetsov, Yu.A. (1994), `Numerical detection and continuation of codimension-two homoclinic bifurcations’, Internat. J. Bifur. Chaos Appl. Sci. Engrg. 4, 795–822.

    MathSciNet  Google Scholar 

  • Carr, J. (1981), Applications of Center Manifold Theory, Springer-Verlag, New York. Carr, J., Chow, S.-N. and Hale, J. (1985), `Abelian integrals and bifurcation theory’, J. Differential Equations 59, 413–436.

    MathSciNet  Google Scholar 

  • Champneys, A. and Kuznetsov, Yu.A. (1994), `Numerical detection and continuation of codimension-two homoclinic bifurcations’, Internat. J. Bifur. Chaos Appl. Sci. Engrg. 4, 795–822.

    MathSciNet  Google Scholar 

  • Doedel, E., Friedman, M. and Kunin, B (1997), `Successive continuation for locating connecting orbits’, Numerical Algorithms 14, 103–124.

    Article  MathSciNet  MATH  Google Scholar 

  • Lou, Z., Kostelich, E. and Yorke, J. (1992), `Erratum: “Calculating stable and unstable manifolds”’, Internat. J. Bifur. Chaos Appl. Sci. Engrg. 2, 215.

    Article  MathSciNet  Google Scholar 

  • Krauskopf, B. and Osinga, H. (1999), `Two-dimensional global manifolds of vector fields’, Chaos 9, 768–774.

    Article  MathSciNet  MATH  Google Scholar 

  • Nusse, H. and Yorke, J. (1998), Dynamics: Numerical Explorations, 2nd ed.,Springer-Verlag, New York.

    Google Scholar 

  • Edoh, K. and Lorenz, J. (2001), `Computation of Lyapunov-type numbers for invariant curves of planar maps’, SIAM J. Sci. Comput. 23, 1113–1134.

    Article  MathSciNet  MATH  Google Scholar 

  • Taylor, M. and Kevrekidis, I. (1990), Interactive AUTO: A graphical interface for AUTO86, Department of Chemical Engineering, Princeton University.

    Google Scholar 

  • Ermentrout, B. (2002), Simulating, Analyzing, and Animating Dynamical Systems: A Guide to XPPAUT for Researchers and Students, Vol. 14 of Software, Environments, and Tools, SIAM, Philadelphia, PA.

    Google Scholar 

  • Doedel, E., Champneys, A., Fairgrieve, T., Kuznetsov, Yu.A., Sandstede, B. and Wang, X.-J. (1997), AUTO97: Continuation and bifurcation software for ordinary differential equations (with HomCont), Computer Science, Concordia University, Montreal.

    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’, Physica D 62, 360–371.

    Article  MathSciNet  MATH  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 

  • Kuznetsov, Yu.A., Levitin, V.V. and Skovoroda, A.R. (1996), Continuation of stationary solutions to evolution problems in CONTENT, Report AM-R9611, Centrum voor Wiskunde en Informatica, Amsterdam.

    Google Scholar 

  • Dhooge, A., Govaerts, W. and Kuznetsov, Yu.A. (2003), `MATCONT:A MATLAB package for numerical bifurcation analysis of ODEs’, ACM Trans. Math. Software 29, 141–164.

    Article  MathSciNet  MATH  Google Scholar 

  • Char, B., Geddes, K., Gonnet, G., Leong, B., Monagan, M. and Watt, S. (1991b), Maple V Library Reference Manual, Springer-Verlag, New York.

    Book  MATH  Google Scholar 

  • Char, B., Geddes, K., Gonnet, G., Leong, B., Monagan, M. and Watt, S. (1991a), Maple V Language Reference Manual, Springer-Verlag, New York.

    Book  MATH  Google Scholar 

  • Wolfram, S. (1991), Mathematica: A System for Doing Mathematics by Computer, Addison-Wesley, Redwood City, CA.

    Google Scholar 

  • Hearn, A. (1993), REDUCE User’s Manual, Version 3.5, The RAND Corporation, Santa Monica.

    Google Scholar 

  • Cox, D., Little, J. and O’Shea, D. (1992), Ideals, Varieties, and Algorithms,Springer-Verlag, New York.

    Google Scholar 

  • Chow, S.-N., Drachman, B. and Wang, D. (1990), `Computation of normal forms’, J. Comput. Appl. Math. 29, 129–143.

    Article  MathSciNet  MATH  Google Scholar 

  • Sanders, J. (1994), Versa1 normal form computation and representation theory, in E. Tournier, ed., `Computer Algebra and Differential Equations’, Cambridge University Press, Cambridge, pp. 185–210.

    Google Scholar 

  • Murdock, J. (2003), Normal Forms and Unfoldings for Local Dynamical Systems, Springer-Verlag, New York.

    MATH  Google Scholar 

  • Gatermann, K. and Hohmann, A. (1991), `Symbolic exploitation of symmetry in nu-merical pathfollowing’, Impact Comput. Sci. Engrg. 3, 330–365.

    Article  MathSciNet  MATH  Google Scholar 

  • Levitin, V.V. (1995), Computation of functions and their derivatives in CONTENT, Report AM-R9512, Centrum voor Wiskunde en Informatica, Amsterdam.

    Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Utrecht University, Budapestlaan 6, 3584 CD, Utrecht, The Netherlands

    Yuri A. Kuznetsov

  2. Institute of Mathematical Problems of Biology, Russian Academy of Sciences, 142290, Pushchino, Moscow Region, Russia

    Yuri A. Kuznetsov

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Kuznetsov, Y.A. (2004). Numerical Analysis of Bifurcations. In: Elements of Applied Bifurcation Theory. Applied Mathematical Sciences, vol 112. Springer, New York, NY. https://doi.org/10.1007/978-1-4757-3978-7_10

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