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Constructing dynamical systems having homoclinic bifurcation points of codimension two. (English) Zbl 0879.34051
Summary: A procedure is derived which allows for a systematic construction of three-dimensional ordinary differential equations having homoclinic solutions. The equations are proved to exhibit codimension-two homoclinic bifurcation points. Examples include the non-orientable resonant bifurcation, the inclination-flip, and the orbit-flip. In addition, an equation is constructed which has a homoclinic orbit converging to a saddle-focus satisfying Shilnikov’s condition. The vector fields are polynomial and non-stiff in that the eigenvalues are of moderate size.
34C37Homoclinic and heteroclinic solutions of ODE
37G99Local and nonlocal bifurcation theory
[1]L. A. Belyakov. The bifurcation set in a system with a homoclinic saddle curve.Mat. Zam. 28 (1980), 910–916.
[2]W.-J. Beyn. The numerical computation of connecting orbits in dynamical systems.IMA J. Numer. Anal. 9 (1990), 379–405. · Zbl 0706.65080 · doi:10.1093/imanum/10.3.379
[3]S.-N. Chow, B. Deng, and B. Fiedler. Homoclinic bifurcation at resonant eigenvalues.J. Dyn. Diff. Eq. 2 (1990), 177–244. · Zbl 0703.34050 · doi:10.1007/BF01057418
[4]A. R. Champneys, J. HÄrterich, and B. Sandstede. A non-transverse homoclinic orbit to a saddle-node equilibrium.Ergod. Theory Dyn. Syst. 16 (1996), 431–450. · Zbl 0853.58079 · doi:10.1017/S0143385700008919
[5]A. R. Champneys and Yu. A. Kuznetsov. Numerical detection and continuation of codimension-two homoclinic bifurcations.Int. J. Bifurc. Chaos 4 (1994), 795–822.
[6]A. R. Champneys, Yu. A. Kuznetsov, and B. Sandstede.HomCont: An AUTO86 Driver for Homoclinic Bifurcation Analysis, Version 2.0, Technical report, CWI, Amsterdam, 1995.
[7]A. R. Champneys, Yu. A. Kuznetsov, and B. Sandstede. A numerical toolbox for homoclinic bifurcation analysis.Int. J. Bifurc. Chaos 6 (1996), 867–887. · Zbl 0877.65058 · doi:10.1142/S0218127496000485
[8]B. Deng. Constructing homoclinic orbits and chaotic attractors.Int. J. Bifurc. Chaos 4 (1994), 823–841. · Zbl 0873.34036 · doi:10.1142/S0218127494000599
[9]F. Dumortier, H. Kokubu, and H. Oka. A degenerate singularity generating geometric Lorenz attractors.Ergod. Theory Dyn. Syst. 15 (1995), 833–856. · Zbl 0836.58030 · doi:10.1017/S0143385700009664
[10]M. J. Friedman and E. J. Doedel. Numerical computation and continuation of invariant manifolds connecting fixed points.SIAM J. Numer. Anal. 28 (1991), 789–808. · Zbl 0735.65054 · doi:10.1137/0728042
[11]A. J. Homburg, H. Kokubu, and M. Krupa. The cusp horseshoe and its bifurcations from inclination-flip homoclinic orbits.Ergod. Theory Dyn. Syst. 14 (1994), 667–693. · Zbl 0864.58044 · doi:10.1017/S0143385700008117
[12]M. Kisaka, H. Kokubu, and H. Oka. Bifurcation toN-homoclinic orbits andN-periodic orbits in vector fields.J. Dyn. Diff. Eq. 5 (1993), 305–358. · Zbl 0784.34038 · doi:10.1007/BF01053164
[13]K. J. Palmer. Exponential dichotomies and transversal homoclinic points.J. Diff. Eq. 55 (1984), 225–256. · Zbl 0539.58028 · doi:10.1016/0022-0396(84)90082-2
[14]B. Sandstede.Verzweigungstheorie homokliner Verdopplungen, Doctoral thesis, University of Stuttgart, Stuttgart, 1993.
[15]B. Sandstede. Convergence estimates for the numerical approximation of homoclinic solutions.IMA J. Numer. Anal. (1997), to appear.
[16]B. Sandstede. A unified approach to homoclinic bifurcations with codimension two. II. Applications, in preparation (1996).
[17]L. P. Shilnikov. A contribution to the problem of the structure of an extended neighborhood of a rough equilibrium state of saddle-focus type.Mat. USSR Sb. 10 (1970), 91–102. · Zbl 0216.11201 · doi:10.1070/SM1970v010n01ABEH001588
[18]B. Sandstede and A. Scheel. Forced symmetry breaking of heteroclinic cycles.Nonlinearity 8 (1995), 333–365. · Zbl 0841.58048 · doi:10.1088/0951-7715/8/3/003
[19]D. Terman. The transition from bursting to continuous spiking in excitable membrane models.J. Nonl. Sci. 2 (1992), 135–182. · Zbl 0900.92059 · doi:10.1007/BF02429854
[20]E. Yanagida. Branching of double pulse solutions from single solutions in nerve axon equations.J. Diff. Eq. 66 (1987), 243–262. · Zbl 0661.35003 · doi:10.1016/0022-0396(87)90034-9