×

zbMATH — the first resource for mathematics

Heteroclinic tangles and homoclinic snaking in the unfolding of a degenerate reversible Hamiltonian-Hopf bifurcation. (English) Zbl 0952.37009
The authors study a two-parameter unfolding of a reversible four-dimensional differential system which undergoes a so-called reversible Hopf bifurcation (the authors call it the reversible Hamiltonian-Hopf bifurcation), i.e., when the system possesses an equilibrium with double nonsemisimple imaginary eigenvalues. This requires a special type of involution action with a two-dimensional fixed plane. Degeneracy here means that a coefficient, which is responsible for the type of bifurcation, in the third order normal form at the equilibrium vanishes. The theoretical analysis is given for the third order normal from for the unfolding. This normal form is integrable (as well as the normal form of any order) and its dynamics is determined by the roots and their types to some polynomial potential (in fact, being a Hamiltonian of the reduced system).
This dynamics is completely the same as in the case of a similar bifurcation for a Hamiltonian system, which was studied and related bifurcation diagrams were plotted by L. Yu. Glebsky and L. M. Lerman [Chaos 5, 424-431 (1995; Zbl 0952.37021)] (it’s funny that the authors mention this paper in their reference list but did not point out that their results for the Hamiltonian case were obtained there). After this theoretical analysis they simulate the model equation, the generalised Swift-Hohenberg equation (which just gives a Hamiltonian reversible system), to understand how homoclinic solutions to a saddle-focus equilibrium arise in nonintegrable setting through a heteroclinic connection between the saddle-focus and a saddle periodic orbit.

MSC:
37G20 Hyperbolic singular points with homoclinic trajectories in dynamical systems
34C23 Bifurcation theory for ordinary differential equations
37G05 Normal forms for dynamical systems
37J20 Bifurcation problems for finite-dimensional Hamiltonian and Lagrangian systems
70H05 Hamilton’s equations
Software:
AUTO
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Devaney, R, Reversible diffeomorphisms and flows, Trans. am. math. soc., 218, 89-113, (1976) · Zbl 0363.58003
[2] Lamb, J; Roberts, J, Time-reversal symmetry in dynamical systems: a survey, Physica D, 112, 1-39, (1998) · Zbl 1194.34072
[3] J. van den Meer, The Hamiltonian-Hopf bifurcation, Lecture Notes in Mathematics 1160, Springer, Berlin, 1985. · Zbl 0585.58019
[4] Champneys, A.R, Homoclinic orbits in reversible systems and their applications in mechanics, fluids and optics, Physica D, 112, 158-186, (1998) · Zbl 1194.37154
[5] Iooss, G; Pérouème, M, Perturbed homoclinic solutions in reversible 1:1 resonance vector fields, J. diff. eqns., 102, 62-88, (1993) · Zbl 0792.34044
[6] Härterich, J, Cascades of reversible homoclinic orbits to a saddle-focus equilibrium, Phusica D, 112, 187-200, (1998) · Zbl 1194.34083
[7] Dias, F; Iooss, G, Capillary-gravity interfacial waves in infinite depth, Eur. J. mech. B/fluids, 15, 3, 367-393, (1996) · Zbl 0887.76014
[8] Iooss, G, Existence d’orbites homoclines á unéquilibre elliptique pour un systéme réversible, C. R. acad. sci. Paris, ser. 1, 324, 1, 993-997, (1997) · Zbl 0874.34044
[9] Hunt, G; Bolt, H; Thompson, J, Structural localization phenomena and the dynamical phase-space analogy, Proc. R. soc. London A, 425, 245-267, (1989) · Zbl 0697.73043
[10] Hunt, G; Wadee, M, Comparative Lagrangian formulations for localised buckling, Proc. R. soc. London A, 434, 485-502, (1991) · Zbl 0753.73037
[11] G. Hunt, P. Woods, A. Champneys, M. Peletier, M. Wadee, C. Budd, G. Lord, Cellular buckling in long structures, Nonlinear Dynamics, to appear. · Zbl 0974.74024
[12] Glebsky, L; Lerman, L, On small stationary localized solutions for the generalized 1-d swift – hohenburg equation, Chaos, 5, 2, 424-431, (1995) · Zbl 0952.37021
[13] Hilali, M; Métens, S; Brockmans, P; Dewel, G, Pattern selection in the generalized swift – hohenburg model, Physcial rev. E, 51, 3, 2046-2052, (1995)
[14] Iooss, G; Kirchgässner, K, Bifurcation d’ondes solitaires en présence d’une fiable tension superficielle, C. R. acad. sci. Paris, ser. 1, 311, 1, 265-268, (1990) · Zbl 0705.76020
[15] Buffoni, B; Groves, M; Toland, J, A plethora of solitary gravity-capillary water waves with nearly critical bond and Froude numbers, Phil. trans. roy. soc. London A, 354, 575-607, (1996) · Zbl 0861.76012
[16] Buffoni, B; Champneys, A; Toland, J, Bifurcation and coalescence of a plethora of homoclinic orbits for a Hamiltonian system, J. dyn. diff. eqns., 8, 221-281, (1996) · Zbl 0854.34047
[17] Buffoni, B; Séré, E, A global condition for quasi-random behaviour in a class of conservative systems, Commun. pure appl. math., 49, 285-305, (1996) · Zbl 0860.58027
[18] Elphick, C; Trapegui, E; Brachet, M; Coullet, P; Iooss, G, A simple global characterization for normal forms of singular vector fields, Physica D, 29, 95-127, (1987) · Zbl 0633.58020
[19] Devaney, R, Homoclinic orbits in Hamiltonian systems, J. diff. eqns., 21, 431-438, (1976) · Zbl 0343.58005
[20] Yang, T.-S; Akylas, T, On asymmetric gravity-capillary solitary waves, J. fluid mech., 330, 215-232, (1997) · Zbl 0913.76011
[21] G. Hunt, G. Lord, A. Champneys, Homoclinic and heteroclinic orbits underlying the post-buckling of axially-compressed cylindrical shells, Computer Methods Applied Mechanics Engineering theme issue on Computational Methods and Bifurcation Theory with Applications, 1999, in press. · Zbl 0958.74021
[22] Y. Kuznetsov, Elements of Applied Bifurcation Theory, Springer, 1995. · Zbl 0829.58029
[23] P. Woods, Localized buckling and fourth-order equations, Ph.D. thesis, University of Bristol, Bristol, UK, 1999.
[24] E. Doedel, A. Champneys, T. Fairgrieve, Y. Kuznetsov, B. Sandstede, X. Wang, \scauto97 continuation and bifurcation software for ordinary differential equations, Technical Report, Concordia University, Software available by anonymous ftp from \scftp.cs.concordia.ca, directory \scpub/doedel/auto, 1997.
[25] Kichenassamy, S; Olver, P, Existence and non-existence of solitary wave solutions to higher-order model evolution equations, SIAM J. math. anal., 23, 1141-1166, (1996) · Zbl 0755.76023
[26] Gottwald, G; Grimshaw, R; Malomed, B, Parametric envelope solitons in coupled korteweg – de Vries equations, Phys. lett. A, 227, 47-54, (1997) · Zbl 0962.35506
[27] Dias, F; Menasce, D; Vanden-Broek, J.-M, Numerical study of capillary-gravity solitary waves, Eur. J. mech. B, 15, 17-36, (1996) · Zbl 0863.76011
[28] Amick, C; Toland, J, Homoclinic orbits in the dynamic phase-space analogy of an elastic strut, Eur. J. appl. mech., 3, 97-114, (1992) · Zbl 0755.73023
[29] van den Berg, J, Uniqueness of solutions for the extended fisher – kolmogorov equation, Comp. R. acad. sci. I, 326, 447-452, (1998) · Zbl 0913.34052
[30] Kalies, W; Vandervorst, R, Multitransition homoclinic and heteroclinic solutions of the extended fisher – kolmogorov equation, J. diff. eqns., 131, 209-228, (1996) · Zbl 0872.34033
[31] Peletier, L; Troy, W, A topological shooting method and the existence of kinks of the extended fisher – kolmogorov equation, Topological methods in nonlinear analysis, 6, 331-355, (1996) · Zbl 0862.34030
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.