Bifurcation of a reversible Hamiltonian system from a fixed point with fourfold eigenvalue zero

References

• Bridges , T. J. and Furter , J. E. 1993 . Singularity Theory and Equivariant Symplectic Maps, Lecture Notes in Mathematics 1558 , Berlin : Springer-Verlag .
   Google Scholar
• Bröcker , Th. and Lander , L. 1975 . Differentiable Germs and Catastrophes , Cambridge : Cambridge University Press .
   Google Scholar
• Broer , H. W. , Chow , S. N. , Kirm , Y. and Vegter , G. 1993 . A normally elliptic Hamiltonian bifurcation . Zeitschrift für angewandte Mathematik und Physik , 44 : 389 – 432 .
   Web of Science ®Google Scholar
• Buffoni , B. 1999 . Shooting methods and topological tranversality . Proceedings of the Royal Society of Edinburgh , 129A : 1137 – 1155 .
   Google Scholar
• Buryak , A. V. 1996 . Solitons due to quadratic nonlinearities , Australian National University . PhD thesis
   Google Scholar
• Champneys , A. R. 1997 . Homoclinic orbits in reversible systems and their application in mechanics, fluids and optics . Physica D , 112 : 158 – 186 .
   Web of Science ®Google Scholar
• Champneys , A. R. and Toland , J P. 1993 . Bifurcation of a plethora of large amplitude homoclinic orbits for autonomous Hamiltonian systems . Nonlinearity , 6 : 665 – 721 .
   Web of Science ®Google Scholar
• Golubitsky , M. and Stewart , I. 1987 . Generic bifurcation of Hamiltonian systems with symmetry . Physica D , 24 : 391 – 405 .
   Web of Science ®Google Scholar
• Hanßmann , H. 1997 . The reversible umbilic bifurcation . Physica D , 112 : 81 – 94 .
   Web of Science ®Google Scholar
• Hirsch , M. W. 1976 . Differential Topology , Berlin : Springer-Verlag .
   Google Scholar
• Hofer , H. and Toland , J F. 1984 . Homoclinic, heteroclinic, and periodic orbits for a class of indefinite Hamiltonian systems . Mathematische Annalen , 269 : 387 – 403 .
   Google Scholar
• Hoveijn , I. 1996 . Versal deformations and normal forms for reversible and Hamiltonian linear systems . Journal of Differential Equations , 126 (2) : 408 – 442 .
   Web of Science ®Google Scholar
• Iooss , G. 1995 . A codimension 2 bifurcation for reversible vector fields . Fields Institute Communications , 4 : 201 – 217 .
   Google Scholar
• Iooss , G. and Pérouéme , M. C. 1993 . Perturbed homoclinic solutions in reversible 1:1 resonance vector fields . Journal of Defferential Equations , 102 (1) : 62 – 88 .
   Web of Science ®Google Scholar
• Kalies , W. D. , Kwapisz , J. and van der Vorst , R. C. A. M. 1998 . Homotopy classes for stable connections between Hamiltonian saddle-focus equilibria . Communication in Mathematical Physics , 193 : 337 – 371 .
   Web of Science ®Google Scholar
• Knobloch , J. , Rieß , T. and Wagenknecht , T. 2001 . “ A numerical investigation of connecting orbits in a system of second order ODEs ” . in preparation
   Google Scholar
• Poènaru , V. 1976 . Singularités C? en présence de symétrie, Lecture Notes in Mathematical 510 , Berlin : Springer-Verlag .
   Google Scholar
• Teixeira , M. A. 1997 . Singularities of reversible vector fields . Physica D , 100 : 101 – 118 .
   Google Scholar
• Vanderbauwhede , A. 1990 . Hopf bifurcation for equivariant conservative and time-reversible systems . Proceedings of the Royal Society of Edinburgh , 116A : 103 – 128 .
   Google Scholar
• Vanderbauwhede , A. and Fiedler , B. 1992 . Homoclinic period blow-up in reversible and conservative systems . Zeitschrift für angewandte Mathematik und Physik , 43 : 292 – 318 .
   Web of Science ®Google Scholar
• Wagenknecht , T. 1999 . An analytical study of a two degrees of freedom Hamiltonian system associated to the reversible hyperbolic umbilic , TU Ilmenau . Diploma thesis, Available online at http:// imath.mathematik.tu-ilmenau.de/~murmel
   Google Scholar
• Wagenknecht , T. 2001a . Continuation of heteroclinic cycles via topological transversality , Preprint MO2/ 01 TU Ilmenau .
   Google Scholar
• Wagenknecht , T. 2001b . “ Bifurcation of heteroclinic cycles in reversible systems ” . in preparation
   Google Scholar
• Wiggins , S. 1988 . Global Bifurcations and Chaos , Berlin : Springer-Verlag .
   Google Scholar
• Yew , A. C. 1998 . An analytical study of solitary-wave s in quadratic media , Providence , RI : Brown University . PhD thesis
   Google Scholar