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Bifurcation and coalescence of a plethora of homoclinic orbits for a Hamiltonian system. (English) Zbl 0854.34047
The authors investgate the set of homoclinic solutions of the reversible Hamiltonian system \(u^{iv}+ Pu''+ u- u^2= 0\). It is shown rigorously that for \(P\leq - 2\) there is a unique homoclinic solution of the system, that solution is given, and on the zero energy surface its orbit coincides with the transverse intersection of the global stable and unstable manifold. At \(P= -2\) infinitely many families of homoclinic solutions are created which exist for \(P\in (- 2, -2+ \varepsilon)\) for some \(\varepsilon> 0\). Second the development of the set of symmetric solutions is monitored numerically as \(P\) increases from \(- 2\). It is observed that two branches extend from \(P= - 2\) to \(P= 2\). All other symmetric branches are in the form of closed loops with a turning point between \(P= -2\) and \(P= 2\).
Numerically, it is observed that close to each turning point a bifurcation of a branch of nonsymmetrical homoclinic orbits occurs, which can be followed back to \(P= - 2\). Finally, the observed phenomena are explained in the language of geometric dynamical systems theory. It is shown how the different orbits are related to each other and how the highly complicated order of turning points can be related to the intersection of stable and unstable manifolds. This paper demonstrates nicely, how geometric dynamical systems theory can contribute to the understanding of very complicated dynamic phenomena.
Reviewer: A.Steindl (Wien)

MSC:
34C37 Homoclinic and heteroclinic solutions to ordinary differential equations
37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems
34C23 Bifurcation theory for ordinary differential equations
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AUTO
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[1] Amick, C. J., and Kirchgässner, K. (1989). A theory of solitary water-waves in the presence of surface tension.Arch. Ration. Mech. Anal. 105, 1–49. · Zbl 0666.76046 · doi:10.1007/BF00251596
[2] Amick, C. J., and Toland, J. F. (1991). Points of egress in problems of Hamiltonian dynamics.Math. Proc. Cambr. Phil. Soc. 109, 405–417. · Zbl 0728.58012 · doi:10.1017/S030500410006984X
[3] Amick, C. J., and Toland, J. F. (1992). Homoclinic orbits in the dynamic phase space analogy of an elastic strut.Eur. J. Appl. Math. 3, 97–114. · Zbl 0755.73023 · doi:10.1017/S0956792500000735
[4] Belyakov, L. A., and Šil’nikov, L. P. (1990). Homoclinic curves and complex solitary waves.Selecta Math. Sovietica 9, 219–228. · Zbl 0739.35086
[5] Beyn, W.-J. (1990). The numerical computation of connecting orbits in dynamical systems.IMA J. Num. Anal. 9, 379–405. · Zbl 0706.65080 · doi:10.1093/imanum/10.3.379
[6] Buffoni, B. (1995). Infinitely many large amplitude homoclinic orbits for a class of autonomous Hamiltonian systems.J. Diff. Eq. 121, 109–120. · Zbl 0832.34032 · doi:10.1006/jdeq.1995.1123
[7] Buffoni, B. (1996). Periodic and homoclinic orbits for Lorentz-Lagrangian systems via variational methods.Nonlinear Analysis, Theory Method Applications 26, 443–467. · Zbl 0838.49004 · doi:10.1016/0362-546X(94)00290-X
[8] Buffoni, B., and Toland, J. F. (1995). Global existence of homoclinic and periodic orbits for a class of autonomous Hamiltonian systems.J. Diff. Eq. 118, 104–120. · Zbl 0828.34032 · doi:10.1006/jdeq.1995.1068
[9] Buffoni, B., Groves, M. D., and Toland, J. F. (1996). A plethora of solitary gravitycapillary water waves with nearly critical Bond and Froude numbers.Phil. Trans. R. Soc. Lond. A 354, 575–607. · Zbl 0861.76012 · doi:10.1098/rsta.1996.0020
[10] Champneys, A. R. (1993). Subsidiary homoclinic orbits to a saddle-focus for reversible systems.Int. J. Bifurcat. Chaos 4, 1447–1482. · Zbl 0873.34037
[11] Champneys, A. R., and Spence, A. (1993). Hunting for homoclinic orbits in reversible systems: A shooting technique.Adv. Comp. Math. 1, 81–108. · Zbl 0824.65080 · doi:10.1007/BF02070822
[12] Champneys, A. R. and Toland, J. F. (1993). Bifurcation of a plethora of multi-modal homoclinic orbits for autonomous Hamiltonian systems.Nonlinearity 6, 665–721. · Zbl 0789.58035 · doi:10.1088/0951-7715/6/5/002
[13] Devaney, R. L. (1976). Homoclinic orbits in Hamiltonian systems.J. Diff. Eq. 21, 431–438. · Zbl 0343.58005 · doi:10.1016/0022-0396(76)90130-3
[14] Doedel, E. (1981). AUTO, a program for the automatic bifurcation analysis of autonomous systems.Cong. Numer. 30, 265–284. · Zbl 0511.65064
[15] Doedel, E. J., Keller, H. B., and Kernévez, J. P. (1991). Numerical analysis and control of bifurcation problems: (I) Bifurcation in finite dimensions.Int. J. Bifurcat. Chaos 1, 493–520. · Zbl 0876.65032 · doi:10.1142/S0218127491000397
[16] Doedel, E. J., Keller, H. B., and Kernévez, J. P. (1991). Numerical analysis and control of bifurcation problems: (II) Bifurcation in infinite dimensions.Int. J. Bifurcat. Chaos 1, 745–772. · Zbl 0876.65060 · doi:10.1142/S0218127491000555
[17] Friedman, M. J., and Doedel, E. (1991). Numerical computation of invariant manifolds connecting fixed points.SIAM J. Numer. Anal. 28, 789–808. · Zbl 0735.65054 · doi:10.1137/0728042
[18] Grimshaw, R., Malomed, B., and Benilove, E. (1994). Solitary waves with damped oscillatory tails: an analysis of the fifth-order Korteweg-de-Vries equation.Physica D 77, 473–485. · Zbl 0824.35113 · doi:10.1016/0167-2789(94)90302-6
[19] Hofer, H., and Toland, J. F. (1984). Homoclinic, heteroclinic and periodic orbits for a class of indefinite Hamiltonian Systems.Math. Annalen 268, 387–403. · Zbl 0569.70017 · doi:10.1007/BF01457066
[20] Hunt, G. W., Bolt, H. M., and Thompson, J. M. T. (1989). Structural localisation phenomena and the dynamical phase-space analogy analogy.Proc. R. Soc. Lond. A 425, 245–267. · Zbl 0697.73043 · doi:10.1098/rspa.1989.0105
[21] Hunt, G. W., and Wadee, M. K. (1991). Comparative Lagrangian formulations for localised buckling,Proc. R. Soc. Lond. A 434, 485–502. · Zbl 0753.73037 · doi:10.1098/rspa.1991.0109
[22] Iooss, G., and Kirchgässner, K. (1990). Bifurcation d’ondes solitaires en présence d’une faible tension superficielle.C.R. Acad. Sci. Paris Ser. I 311, 265–268. · Zbl 0705.76020
[23] Iooss, G., and Pérouème, M. C. (1993). Perturbed homoclinic solutions in reversible 1 resonance vector fields.J. Diff. Eq. 102, 62–88. · Zbl 0792.34044 · doi:10.1006/jdeq.1993.1022
[24] Knobloch, J. (1994). Bifurcation of degenerate homoclinics in reversible and conservative systems. Preprint M 15/94, Technische Universität Ilmenau.
[25] Kuznetsov, Yu. A. (1990). Computation of invariant manifold bifurcations. In D. Roose, A. Spence, and B. De Dier (eds.),Continuation and Bifurcations: Numerical Techniques and Applications, Kluwer, Dordrecht, Netherlands, pp. 183–195. · Zbl 0705.34013
[26] Lau, Y.-T. (1993). Cocoon bifurcation for systems with two fixed points.Int. J. Bifurcat. Chaos,2, 543.
[27] Peletier, L. A., and Troy, W. C. (1995). A topological shooting method and the existence of kinks of the extended Fischer-Kolmogorov equation. Preprint W95-08, Mathematical Institute, University of Leiden, Leiden. · Zbl 0862.34030
[28] Toland, J. F. (1986). Existence of symmetric homoclinic orbits for systems of Euler-Lagrange equations.Amer. Math. Soc. Proc. Symp. Pure Math. 45(2), 447–459. · Zbl 0589.34029
[29] Wiggins, S. (1988).Global Bifurcations and Chaos: Analytical Methods, Springer-Verlag, New York. · Zbl 0661.58001
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