Abstract:
For a 2 d.o.f. Hamiltonian system we prove the Lyapunov stability of its equilibrium with two double pure imaginary eigenvalues and non-semisimple Jordan form for the linearization matrix, when some coefficient in the 4th order normal form is positive (the equilibrium is known to be unstable, if this coefficient is negative). Such the degenerate equilibrium is met generically in one-parameter unfoldings, the related bifurcation is called to be the Hamiltonian Hopf Bifurcation. Though the stability is known since 1977, proofs that were published are either incorrect or not complete. Our proof is based on the KAM theory and a work with the Weierstrass elliptic functions, estimates of power series and scaling.
Keywords:
Hamiltonian Hopf Bifurcation, KAM theory, Lyapunov stability, normal form, action-angle variables, elliptic functions, scaling.
\Bibitem{LerMar09}
\by L. M. Lerman, A. P. Markova
\paper On Stability at the Hamiltonian Hopf Bifurcation
\jour Regul. Chaotic Dyn.
\yr 2009
\vol 14
\issue 1
\pages 148--162
\mathnet{http://mi.mathnet.ru/rcd544}
\crossref{https://doi.org/10.1134/S1560354709010109}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=2480956}
\zmath{https://zbmath.org/?q=an:1229.37056}
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https://www.mathnet.ru/eng/rcd544
https://www.mathnet.ru/eng/rcd/v14/i1/p148
This publication is cited in the following 19 articles:
B. S. Bardin, E. V. Volkov, “The Lyapunov Stability of Central Configurations of the Planar Circular Restricted Four-Body Problem”, Cosmic Res, 62:5 (2024), 388
Tatiana Titova, Lecture Notes in Networks and Systems, 574, XV International Scientific Conference “INTERAGROMASH 2022”, 2023, 1282
Kenneth R. Meyer, Dieter S. Schmidt, “Normal Forms for Hamiltonian Systems
in Some Nilpotent Cases”, Regul. Chaotic Dyn., 27:5 (2022), 538–560
B. S. Bardin, A. N. Avdyushkin, “On Stability of the Collinear Libration Point $L_1$
in the Planar Restricted Circular Photogravitational
Three-Body Problem”, Rus. J. Nonlin. Dyn., 18:4 (2022), 543–562
O. V. Kholostova, “On Nonlinear Oscillations of a Near-Autonomous
Hamiltonian System in the Case of Two Identical
Integer or Half-Integer Frequencies”, Rus. J. Nonlin. Dyn., 17:1 (2021), 77–102
A. I. Neishtadt, D. V. Treschev, “Dynamical phenomena connected with stability loss of equilibria and periodic trajectories”, Russian Math. Surveys, 76:5 (2021), 883–926
B S Bardin, A N Avdyushkin, “On stability of a collinear libration point in the planar circular restricted photogravitational three-body problem in the cases of first and second order resonances”, J. Phys.: Conf. Ser., 1959:1 (2021), 012004
Daniela Cárcamo-Díaz, Jesús F. Palacián, Claudio Vidal, Patricia Yanguas, “On the Nonlinear Stability of the Triangular Points in the Circular Spatial Restricted Three-body Problem”, Regul. Chaotic Dyn., 25:2 (2020), 131–148
B S Bardin, E V Volkov, “Stability Study of a Relative Equilibrium in the Planar Circular Restricted Four-Body Problem”, IOP Conf. Ser.: Mater. Sci. Eng., 927:1 (2020), 012012
Tatiana Titova, “Properties of canonical transformations of linear Hamiltonian systems”, IOP Conf. Ser.: Mater. Sci. Eng., 365 (2018), 042017
Kenneth R. Meyer, Daniel C. Offin, Applied Mathematical Sciences, 90, Introduction to Hamiltonian Dynamical Systems and the N-Body Problem, 2017, 305
Leonid Kurakin, Andrey Melekhov, Irina Ostrovskaya, “A survey of the stability criteria of Thomson's vortex polygons outside a circular domain”, Bol. Soc. Mat. Mex., 22:2 (2016), 733
Claudio Vidal, Jhon Vidarte, “Stability of the equilibrium solutions in a charged restricted circular three-body problem”, Journal of Differential Equations, 260:6 (2016), 5128
Ernest Fontich, Carles Simó, Arturo Vieiro, Trends in Mathematics, 4, Extended Abstracts Spring 2014, 2015, 77
Kenneth R. Meyer, Jesús F. Palacián, Patricia Yanguas, “Stability of a Hamiltonian System in a Limiting Case”, Regul. Chaotic Dyn., 17:1 (2012), 24–35
Leonid G. Kurakin, “On the Stability of Thomson’s Vortex Pentagon Inside a Circular Domain”, Regul. Chaotic Dyn., 17:2 (2012), 150–169
Víctor Lanchares, Ana I. Pascual, Antonio Elipe, “Determination of Nonlinear Stability for Low Order Resonances by a Geometric Criterion”, Regul. Chaotic Dyn., 17:3 (2012), 307–317
L. G. Kurakin, “Ob ustoichivosti tomsonovskogo vikhrevogo pyatiugolnika vnutri kruga”, Nelineinaya dinam., 7:3 (2011), 465–488
José Pedro Gaivão, Vassili Gelfreich, “Splitting of separatrices for the Hamiltonian-Hopf bifurcation with the Swift–Hohenberg equation as an example”, Nonlinearity, 24:3 (2011), 677
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