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Investigating the consequences of global bifurcations for two-dimensional invariant manifolds of vector fields

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  • We consider a homoclinic bifurcation of a vector field in $\R^3$, where a one-dimensional unstable manifold of an equilibrium is contained in the two-dimensional stable manifold of this same equilibrium. How such one-dimensional connecting orbits arise is well understood, and software packages exist to detect and follow them in parameters.
       In this paper we address an issue that it is far less well understood: how does the associated two-dimensional stable manifold change geometrically during the given homoclinic bifurcation? This question can be answered with the help of advanced numerical techniques. More specifically, we compute two-dimensional manifolds, and their one-dimensional intersection curves with a suitable cross-section, via the numerical continuation of orbit segments as solutions of a boundary value problem. In this way, we are able to explain how homoclinic bifurcations may lead to quite dramatic changes of the overall dynamics. This is demonstrated with two examples. We first consider a Shilnikov bifurcation in a semiconductor laser model, and show how the associated change of the two-dimensional stable manifold results in the creation of a new basin of attraction. We then investigate how the basins of the two symmetrically related attracting equilibria change to give rise to preturbulence in the first homoclinic explosion of the Lorenz system.
    Mathematics Subject Classification: Primary: 34C45, 34C37; Secondary: 65L10.


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  • [1]

    R. H. Abraham and C. D. Shaw, "Dynamics -- The Geometry Of Behavior, Part Three: Global Behavior," Aerial Press, Santa Cruz, 1985.


    U. M. Ascher, J. Christiansen and R. D. Russell, Colsys -- A collocation code for boundary-value problems, Lecture Notes in Computer Science, 76 (1979), 164-185.


    U. M. Ascher and R. J. Spiteri, Collocation software for boundary value differential-algebraic equations, SIAM J. Sci. Comput., 15 (1994), 938-952.doi: 10.1137/0915056.


    M. R. Bassett and J. L. Hudson, Shil'nikov chaos during copper electrodissolution, J. Phys. Chem., 92 (1988), 6963-6966.doi: 10.1021/j100335a025.


    W.-J. Beyn, On well-posed problems for connecting orbits in dynamical systems, in "Chaotic Numerics (Geelong, 1993)," Contemp. Math., 172, Amer. Math. Soc., (1994), 131-168.


    C. J. Budd and J. P. Wilson, Bogdanov-Takens bifurcation points and Shilnikov homoclinicity in a simple power system model of voltage collapse, IEEE Trans. Circuits Systems I, 43 (2002), 575-590.doi: 10.1109/TCSI.2002.1001947.


    A. R. Champneys, V. Kirk, E. Knobloch, B. E. Oldeman and J. Sneyd, When Shil'nikov meets Hopf in excitable systems, SIAM J. Appl. Dyn. Syst., 6 (2007), 663-693.doi: 10.1137/070682654.


    A. R. Champneys, Y. Kuznetsov and B. Sandstede, A numerical toolbox for homoclinic bifurcation analysis, Int. J. Bifurc. Chaos, 6 (1996), 867-887.doi: 10.1142/S0218127496000485.


    B. Deng and G. Hines, Food chain chaos due to Shilnikov's orbit, Chaos, 12 (2002), 533-538.doi: 10.1063/1.1482255.


    A. Dhooge, W. Govaerts and Yu. A. Kuznetsov, MATCONT: A Matlab package for numerical bifurcation analysis of ODEs, ACM Trans. Math. Software, 29 (2003), 141-164.doi: 10.1145/779359.779362.


    E. J. Doedel, AUTO: A program for the automatic bifurcation analysis of autonomous systems, Congr. Numer., 30 (1981), 265-284.


    E. J. Doedel and B. E. Oldeman, with major contributions from A. R. Champneys, F. Dercole, T. F. Fairgrieve, Yu. A. Kuznetsov, R. C. Paffenroth, B. Sandstede, X. J. Wang, and C. H. Zhang, AUTO-07p Version 0.7: Continuation and bifurcation software for ordinary differential equations, Department of Computer Science, Concordia University, Montreal, Canada, 2010. Available from http://cmvl.cs.concordia.ca/auto/


    E. J. Doedel, Lecture notes on numerical analysis of nonlinear equations, in "Numerical Continuation Methods for Dynamical Systems," (eds. B. Krauskopf, H. M. Osinga and J. Galán-Vioque), Underst. Complex Syst., Springer-Verlag, New York, (2007), 1-49.doi: 10.1007/978-1-4020-6356-5_1.


    E. J. Doedel and M. J. Friedman, Numerical computation of heteroclinic orbits, J. Comput. Appl. Math., 26 (1989), 155-170.doi: 10.1016/0377-0427(89)90153-2.


    E. J. Doedel, B. Krauskopf and H. M. Osinga, Global bifurcations of the Lorenz manifold, Nonlinearity, 19 (2006), 2947-2972.doi: 10.1088/0951-7715/19/12/013.


    E. J. Doedel, R. C. Paffenroth, H. B. Keller, D. Dichmann, J. Galán-Vioque and A. Vanderbauwhede, Computation of periodic solutions of conservative systems with application to the 3-body problem, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 13 (2003), 1353-1381.doi: 10.1142/S0218127403007291.


    E. J. Doedel, V. Romanov, R. C. Paffenroth, H. B. Keller, D. Dichmann, J. Galán-Vioque and A. Vanderbauwhede, Elemental periodic orbits associated with the libration points in the circular restricted 3-body problem, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 17 (2007), 2625-2677.doi: 10.1142/S0218127407018671.


    J. P. England, B. Krauskopf and H. M. Osinga, Computing one-dimensional global manifolds of Poincaré maps by continuation, SIAM J. Appl. Dyn. Sys., 4 (2005), 1008-1041.doi: 10.1137/05062408X.


    J. P. England, B. Krauskopf and H. M. Osinga, Computing two-dimensional global invariant manifolds in slow-fast systems, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 17 (2007), 805-822.doi: 10.1142/S0218127407017562.


    J. A. Feroe, Homoclinic orbits in a parametrized saddle-focus system, Physica D, 62 (1993), 254-262.doi: 10.1016/0167-2789(93)90285-9.


    M. Friedman and E. J. Doedel, Numerical computation and continuation of invariant manifolds connecting fixed points, SIAM J. Numer. Anal., 28 (1991), 789-808.doi: 10.1137/0728042.


    P. Glendinning and C. Sparrow, Local and global behavior near homoclinic orbits, J. Statist. Phys., 35 (1984), 645-696.


    D. Goulding, S. P. Hegarty, O. Rasskazov, S. Melnik, M. Hartnett, G. Greene, J. G. McInerney, D. Rachinskii and G. Huyet, Excitability in a quantum dot semiconductor laser with optical injection, Phys. Rev. Lett., 98 (2007), 153903.


    G. Gómez, W. S. Koon, M. W. Lo, J. E. Marsden, J. Masdemont and S. D. Ross, "Invariant Manifolds, the Spatial Three-Body Problem and Space Mission Design," Astrodynamics Specialist Meeting, Quebec City, Canada, August 2001, AAS 01-31.


    J. Guckenheimer and P. Holmes, "Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields," 2nd edition, Springer-Verlag, New York/Berlin, 1986.


    M. E. Henderson, Multiple parameter continuation: Computing implicitly defined $k$-manifolds, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 12 (2002), 451-476.


    M. E. Henderson, Computing invariant manifolds by integrating fat trajectories, SIAM J. Appl. Dyn. Sys., 4 (2005), 832-882.


    M. W. Hirsch, C. C. Pugh and M. Shub, "Invariant Manifolds," Lecture Notes in Mathematics, Vol. 583, Springer-Verlag, Berlin, 1977.


    A. J. Homburg and B. Krauskopf, Resonant homoclinic flip bifurcations, J. Dynam. Diff. Eqs, 12 (2000), 807-850.


    A. J. Homburg and B. SandstedeHomoclinic and heteroclinic bifurcations in vector fields, in B. Fiedler (Ed.) "Handbook of Dynamical Systems III" North Holland, Amsterdam (to appear); available from http://www.dam.brown.edu/people/sandsted/publications.php.


    J. L. Kaplan and J. A. Yorke, Preturbulence: A regime observed in a fluid flow model of Lorenz, Commun. Math. Phys., 67 (1979), 93-108.


    B. Krauskopf and H. M. Osinga, Growing 1D and quasi-2D unstable manifolds of maps, J. Comput. Phys., 146 (1998), 406-419.


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


    B. Krauskopf and H. M. Osinga, Computing geodesic level sets on global (un)stable manifolds of vector fields, SIAM J. Appl. Dyn. Sys., 2 (2003), 546-569.


    B. Krauskopf and H. M. Osinga, Computing invariant manifolds via the continuation of orbit segments, in "Numerical Continuation Methods for Dynamical Systems," (eds. B. Krauskopf, H. M. Osinga and J. Galán-Vioque), Underst. Complex Syst., Springer-Verlag, New York, (2007), 117-154.


    B. Krauskopf, H. M. Osinga and E. J. Doedel, Visualizing global manifolds during the transition to chaos in the Lorenz system, in "Topology-Based Methods in Visualization II" (eds. H.-C. Hege, K. Polthier and G. Scheuermann), Mathematics and Visualization, Springer-Verlag, Berlin, (2009), 115-126.doi: 10.1007/978-3-540-88606-8_9.


    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, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 15 (2005), 763-791.doi: 10.1142/S0218127405012533.


    B. Krauskopf, K. Schneider, J. Sieber, S. M. Wieczorek and M. Wolfrum, Excitability and self-pulsations near homoclinic bifurcations in semiconductor lasers, Optics Communications, 215 (2003), 230-249.doi: 10.1016/S0030-4018(02)02239-3.


    B. Krauskopf and T. Riess, A Lin's method approach to finding and continuing heteroclinic orbits connections involving periodic orbits, Nonlinearity, 21 (2008), 1655-1690.doi: 10.1088/0951-7715/21/8/001.


    Yu. A. Kuznetsov, "CONTENT - Integrated Environment for Analysis of Dynamical Systems. Tutorial," École Normale Supérieure de Lyon, Rapport de Recherche UPMA-98-224, 1998.


    Yu. A. Kuznetsov, "Elements of Applied Bifurcation Theory," 3nd edition, Springer-Verlag, New York/Berlin, 2004.


    C. M. Lee, P. J. Collins, B. Krauskopf and H. M. Osinga, Tangency bifurcations of global Poincaré maps, SIAM J. Appl. Dyn. Syst., 7 (2008), 712-754.doi: 10.1137/07069972X.


    X.-B. Lin, Using Melnikov's method to solve Shilnikov's problems, Proc. R. Soc. Edinb. A, 116 (1990), 295-325.


    E. N. Lorenz, Deterministic nonperiodic flows, J. Atmosph. Sci., 20 (1963), 130-141.doi: 10.1175/1520-0469(1963)020<0130:DNF>2.0.CO;2.


    T. Noh, Shilnikov's chaos in the oxidation of formic acid with bismuth ion on Pt ring electrode, Electrochimica Acta, 54 (2009), 3657-3661.doi: 10.1016/j.electacta.2009.01.043.


    B. E. Oldeman, A. R. Champneys and B. Krauskopf, Homoclinic branch switching: A numerical implementation of Lin's method, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 13 (2003), 2977-2999.doi: 10.1142/S0218127403008326.


    H. M. Osinga and B. Krauskopf, Visualizing the structure of chaos in the Lorenz system, Computers and Graphics, 25 (2002), 815-823.doi: 10.1016/S0097-8493(02)00136-X.


    H. M. Osinga and B. Krauskopf, Crocheting the Lorenz manifold, The Mathematical Intelligencer, 26 (2004), 25-37.doi: 10.1007/BF02985416.


    H. M. Osinga and B. Krauskopf, Visualizing curvature on the Lorenz manifold, Journal of Mathematics and the Arts, 1 (2007), 113-123.doi: 10.1080/17513470701503632.


    J. Palis and W. de Melo, "Geometric Theory of Dynamical Systems," Springer-Verlag, New York/Berlin, 1982.


    J. Palis and F. Takens, "Hyperbolicity & Sensitive Chaotic Dynamics at Homoclinic Bifurcations," Cambridge University Press, Cambridge, 1993.


    T. Peacock and T. Mullin, Homoclinic bifurcations in a liquid crystal flow, J. Fluid Mech., 432 (2001), 369-386.


    C. Perelló, Intertwining invariant manifolds and Lorenz attractor, in "Global Theory of Dynamical Systems (Proc. Internat. Conf., Northwestern Univ., Evanston, Ill., 1979)," Lecture Notes in Math., 819, Springer-Verlag, Berlin, (1979), 375-378.


    M. Phillips, S. Levy and T. Munzner, Geomview: An interactive geometry viewer, Not. Am. Math. Soc., 40 (1993), 985-988. Available from http://www.geomview.org/.


    A. M. Rucklidge, Chaos in a low-order model of magnetoconvection, Physica D, 62 (1993), 323-337.doi: 10.1016/0167-2789(93)90291-8.


    A. L. Shilnikov, On bifurcations of the Lorenz attractor in the Shimizu-Morioka model, Physica D, 62 (1993), 338-346.doi: 10.1016/0167-2789(93)90292-9.


    L. P. Shilnikov, A case of the existence of a countable number of periodic orbits, Sov. Math. Dokl., 6 (1965), 163-166.


    L. P. Shilnikov, A contribution to the problem of the structure of an extended neighborhood of a rough state to a saddle-focus type, Math. USSR-Sb, 10 (1970), 91-102.doi: 10.1070/SM1970v010n01ABEH001588.


    L. P. Shilnikov, A. L. Shilnikov, D. V. Turaev and L. Chua, "Methods of Qualitative Theory in Nonlinear Dynamics, Part II," World Scientific Series on Nonlinear Science, Series A, Vol. 5, 2001.doi: 10.1142/9789812798558.


    C. Sparrow, "The Lorenz Equations: Bifurcations, Chaos and Strange Attractors," Appl. Math. Sci. No. 41, Springer-Verlag, New York, 1982.


    J. V. Stern, H. M. Osinga, A. LeBeau and A. Sherman, Resetting behavior in a model of bursting in secretory pituitary cells: Distinguishing plateaus from pseudo-plateaus, Bulletin Math. Biology, 70 (2008), 68-88.doi: 10.1007/s11538-007-9241-x.


    S. H. Strogatz, "Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry and Engineering," Adison-Wesley, Reading, MA, 1994.


    O. Vaudel, N. Péraud and P. Besnard, Synchronization on excitable pulses in optically injected semiconductor lasers, Proc. SPIE, 6997 (2008), 69970F.doi: 10.1117/12.781568.


    K. Watada, T. Endo and H. Seishi, Shilnikov orbits in an autonomous third-order chaotic phase-locked loop, IEEE Trans. on Circ. and Syst. I, 45 (1998), 979-983.


    S. M. Wieczorek and B. Krauskopf, Bifurcations of $n-$homoclinic orbits in optically injected lasers, Nonlinearity, 18 (2005), 1095-1120.doi: 10.1088/0951-7715/18/3/010.


    S. M. Wieczorek, B. Krauskopf and D. Lenstra, A unifying view of bifurcations in a semiconductor laser subject to optical injection, Optics Communications, 172 (1999), 279-295.doi: 10.1016/S0030-4018(99)00603-3.


    S. M. Wieczorek, B. Krauskopf and D. Lenstra, Multipulse excitability in a semiconductor laser with optical injection, Physical Review Letters, 88 (2002), 1-4.doi: 10.1103/PhysRevLett.88.063901.


    S. M. Wieczorek, B. Krauskopf, T. B. Simpson and D. Lenstra, The dynamical complexity of optically injected semiconductor lasers, Phys. Reports, 416 (2005), 1-128.doi: 10.1016/j.physrep.2005.06.003.


    J. A. Yorke and E. D. Yorke, Metastable chaos: The transition to sustained chaotic behavior in the Lorenz model, J. Stat. Phys., 21 (1979), 263-277.doi: 10.1007/BF01011469.

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