Abstract
We consider a family of real-analytic symplectic four-dimensional maps \(F_{\tilde{\nu }}\), \(\tilde{\nu }\in \mathbb{R}^{p}\), p ≥ 1, with respect to the standard symplectic two-form \(\Omega = dx_{1} \wedge dy_{1} + dx_{2} \wedge dy_{2}\), where (x 1, x 2, y 1, y 2) denote the Cartesian coordinates.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
T.J. Bridges and J.E. Furter, “Singularity theory and equivariant symplectic maps”. Lecture Notes in Mathematics 1558. Springer-Verlag, 1993.
H. Broer, R. Roussarie, and C. Simó, “Invariant circles in the Bogdanov–Takens diffeomorphisms”. Ergodic Theory and Dynamical Systems 16 (1996), 1147–1172.
A. Delshams, V. Gelfreich, A. Jorba, and T.M. Seara, “Exponentially small splitting of separatrices under fast quasiperiodic forcing”. Comm. Math. Phys. 189(1) (1997), 35–71.
A. Delshams and P. Gutiérrez, “Exponentially small splitting of separatrices for whiskered tori in Hamiltonian systems”. Journal of Mathematical Sciences 128(2) (2005), 2726–2745.
E. Fontich and C. Simó, “The splitting of separatrices for analytic diffeomorphisms”. Ergod. Th. and Dynam. Sys. 10 (1990), 295–318.
E. Fontich, C. Simó, and A. Vieiro, “Geometrical and analytical aspects of the transition to complex instability within double resonances”. Preprint.
J.P. Gaivão and V. Gelfreich, “Splitting of separatrices for the Hamiltonian–Hopf bifurcation with the Swift–Hohenberg equation as an example”. Nonlinearity 24(3) (2011), 677–698.
V. Gelfreich, C. Simó, and A. Vieiro, “Dynamics of 4D symplectic maps near a double resonance”. Physica D 243(1) (2013), 92–110.
J.D. Hadjidemetriou, “The stability of resonant orbits in planetary systems”. In “Resonances in the Motion of Planets, Satellites and Asteroids”. Ferraz-Melo, Univ. Sao Paulo, 1985.
L.M. Lerman and A.P. Markova, “On stability at the Hamiltonian–Hopf Bifurcation”. Regular and Chaotic Dynamics 14 (2009), 148–162.
P.D. McSwiggen and K.R. Meyer, “The evolution of invariant manifolds in Hamiltonian–Hopf bifurcations”. J. Differential Equations 189(2) (2003), 538–555.
K.M. Meyer, “The evolution of the stable and unstable manifold of an equilibrium point”. Celestial Mech. Dynam. Astronom. 70(3) (1998), 159–165.
K.R. Meyer and G. Hall, “Introduction to Hamiltonian dynamical systems and the N-body problem”. Applied Mathematical Sciences, 90. Springer-Verlag, New York, 1992.
D. Pfenninger, “Numerical study of complex instability”. Atron. Astrophys. 150 (1985), 97–111.
C. Simó, “Averaging under Fast Quasiperiodic Forcing”. In J. Seimenis, editor, “Hamiltonian Mechanics: Integrability and Chaotic Behaviour”, NATO Adv. Sci. Inst. Ser. B Phys. 331, 13–34.
A.G. Sokolskiĭ, “On the stability of an autonomous Hamiltonian system with two degrees of freedom in the case of equal frequencies”. J. Appl. Math. Mech. 38 (1974), 741–749; translated from Prikl. Mat. Meh. 38 (1974), 791–799.
Acknowledgements
This work has been supported by grants MTM2010-16425 (Spain) and 2009 SGR 67 (Catalonia). This manuscript was prepared during a stay of the last two authors at the Centre de Recerca Matemàtica (CRM), Catalunya. They warmly thank the CRM for the facilities and support.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2015 Springer International Publishing Switzerland
About this paper
Cite this paper
Fontich, E., Simó, C., Vieiro, A. (2015). The Discrete Hamiltonian–Hopf Bifurcation for 4D Symplectic Maps. In: Corbera, M., Cors, J., Llibre, J., Korobeinikov, A. (eds) Extended Abstracts Spring 2014. Trends in Mathematics(), vol 4. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-22129-8_14
Download citation
DOI: https://doi.org/10.1007/978-3-319-22129-8_14
Publisher Name: Birkhäuser, Cham
Print ISBN: 978-3-319-22128-1
Online ISBN: 978-3-319-22129-8
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)