# American Institute of Mathematical Sciences

October  2001, 7(4): 855-871. doi: 10.3934/dcds.2001.7.855

## Unstable equilibria of Hamiltonian systems

 1 Università di Milano Bicocca, Dipartimento di Matematica e Applicazioni, Via degli Arcimboldi 8, 20126 - Milano, Italy

Received  November 2000 Revised  January 2001 Published  July 2001

It is shown that a Hamiltonian system in the neighbourhood of an equilibrium may be given a special normal form in case four of the eigenvalues of the linearized system are of the form $\lambda_1, -\lambda_1, \lambda_2, -\lambda_2,$ with $\lambda_1$ and $\lambda_2$ independent over the reals, i.e., $\lambda_1/\lambda_2 \notin \mathbf R$. That is, for a real Hamiltonian system and concerning the variables $x_1, y_1, x_2, y_2$ the equilibrium is of either type center–saddle or complex–saddle. The normal form exhibits the existence of a four–parameter family of solutions which has been previously investigated by Moser. This paper completes Moser's result in that the convergence of the transformation of the Hamiltonian to a normal form is proven.
Citation: Antonio Giorgilli. Unstable equilibria of Hamiltonian systems. Discrete & Continuous Dynamical Systems - A, 2001, 7 (4) : 855-871. doi: 10.3934/dcds.2001.7.855
 [1] Ricardo Miranda Martins. Formal equivalence between normal forms of reversible and hamiltonian dynamical systems. Communications on Pure & Applied Analysis, 2014, 13 (2) : 703-713. doi: 10.3934/cpaa.2014.13.703 [2] James Montaldi. Bifurcations of relative equilibria near zero momentum in Hamiltonian systems with spherical symmetry. Journal of Geometric Mechanics, 2014, 6 (2) : 237-260. doi: 10.3934/jgm.2014.6.237 [3] Vivi Rottschäfer. Multi-bump patterns by a normal form approach. Discrete & Continuous Dynamical Systems - B, 2001, 1 (3) : 363-386. doi: 10.3934/dcdsb.2001.1.363 [4] Todor Mitev, Georgi Popov. Gevrey normal form and effective stability of Lagrangian tori. Discrete & Continuous Dynamical Systems - S, 2010, 3 (4) : 643-666. doi: 10.3934/dcdss.2010.3.643 [5] Dario Bambusi, A. Carati, A. Ponno. The nonlinear Schrödinger equation as a resonant normal form. Discrete & Continuous Dynamical Systems - B, 2002, 2 (1) : 109-128. doi: 10.3934/dcdsb.2002.2.109 [6] Jaume Llibre, Y. Paulina Martínez, Claudio Vidal. Phase portraits of linear type centers of polynomial Hamiltonian systems with Hamiltonian function of degree 5 of the form $H = H_1(x)+H_2(y)$. Discrete & Continuous Dynamical Systems - A, 2019, 39 (1) : 75-113. doi: 10.3934/dcds.2019004 [7] Virginie De Witte, Willy Govaerts. Numerical computation of normal form coefficients of bifurcations of odes in MATLAB. Conference Publications, 2011, 2011 (Special) : 362-372. doi: 10.3934/proc.2011.2011.362 [8] Letizia Stefanelli, Ugo Locatelli. Kolmogorov's normal form for equations of motion with dissipative effects. Discrete & Continuous Dynamical Systems - B, 2012, 17 (7) : 2561-2593. doi: 10.3934/dcdsb.2012.17.2561 [9] John Burke, Edgar Knobloch. Normal form for spatial dynamics in the Swift-Hohenberg equation. Conference Publications, 2007, 2007 (Special) : 170-180. doi: 10.3934/proc.2007.2007.170 [10] P. Adda, J. L. Dimi, A. Iggidir, J. C. Kamgang, G. Sallet, J. J. Tewa. General models of host-parasite systems. Global analysis. Discrete & Continuous Dynamical Systems - B, 2007, 8 (1) : 1-17. doi: 10.3934/dcdsb.2007.8.1 [11] Xingwu Chen, Weinian Zhang. Normal forms of planar switching systems. Discrete & Continuous Dynamical Systems - A, 2016, 36 (12) : 6715-6736. doi: 10.3934/dcds.2016092 [12] Lyudmila Grigoryeva, Juan-Pablo Ortega, Stanislav S. Zub. Stability of Hamiltonian relative equilibria in symmetric magnetically confined rigid bodies. Journal of Geometric Mechanics, 2014, 6 (3) : 373-415. doi: 10.3934/jgm.2014.6.373 [13] Stefan Siegmund. Normal form of Duffing-van der Pol oscillator under nonautonomous parametric perturbations. Conference Publications, 2001, 2001 (Special) : 357-361. doi: 10.3934/proc.2001.2001.357 [14] Rong Dong, Dongsheng Li, Lihe Wang. Regularity of elliptic systems in divergence form with directional homogenization. Discrete & Continuous Dynamical Systems - A, 2018, 38 (1) : 75-90. doi: 10.3934/dcds.2018004 [15] Emmanuel Hebey, Jérôme Vétois. Multiple solutions for critical elliptic systems in potential form. Communications on Pure & Applied Analysis, 2008, 7 (3) : 715-741. doi: 10.3934/cpaa.2008.7.715 [16] Denis de Carvalho Braga, Luis Fernando Mello, Carmen Rocşoreanu, Mihaela Sterpu. Lyapunov coefficients for non-symmetrically coupled identical dynamical systems. Application to coupled advertising models. Discrete & Continuous Dynamical Systems - B, 2009, 11 (3) : 785-803. doi: 10.3934/dcdsb.2009.11.785 [17] Susanna Terracini, Juncheng Wei. DCDS-A Special Volume Qualitative properties of solutions of nonlinear elliptic equations and systems. Preface. Discrete & Continuous Dynamical Systems - A, 2014, 34 (6) : i-ii. doi: 10.3934/dcds.2014.34.6i [18] Anna Kostianko, Sergey Zelik. Inertial manifolds for 1D reaction-diffusion-advection systems. Part Ⅰ: Dirichlet and Neumann boundary conditions. Communications on Pure & Applied Analysis, 2017, 16 (6) : 2357-2376. doi: 10.3934/cpaa.2017116 [19] Anna Kostianko, Sergey Zelik. Inertial manifolds for 1D reaction-diffusion-advection systems. Part Ⅱ: periodic boundary conditions. Communications on Pure & Applied Analysis, 2018, 17 (1) : 285-317. doi: 10.3934/cpaa.2018017 [20] Kenneth R. Meyer, Jesús F. Palacián, Patricia Yanguas. Normally stable hamiltonian systems. Discrete & Continuous Dynamical Systems - A, 2013, 33 (3) : 1201-1214. doi: 10.3934/dcds.2013.33.1201

2018 Impact Factor: 1.143