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Stability of Equilibrium Solutions in Planar Hamiltonian Difference Systems

  Published:2015-01-28
 Printed: Dec 2015
  • Cristian Carcamo,
    Depto. de Mat., Fac. de Ciencias, Universidad del Bío-Bío, Casilla 5--C, Concepción, VIII--región, Chile
  • Claudio Vidal,
    Depto. de Mat., Fac. de Ciencias, Universidad del Bío-Bío, Casilla 5--C, Concepción, VIII--región, Chile
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Abstract

In this paper, we study the stability in the Lyapunov sense of the equilibrium solutions of discrete or difference Hamiltonian systems in the plane. First, we perform a detailed study of linear Hamiltonian systems as a function of the parameters, in particular we analyze the regular and the degenerate cases. Next, we give a detailed study of the normal form associated with the linear Hamiltonian system. At the same time we obtain the conditions under which we can get stability (in linear approximation) of the equilibrium solution, classifying all the possible phase diagrams as a function of the parameters. After that, we study the stability of the equilibrium solutions of the first order difference system in the plane associated to mechanical Hamiltonian system and Hamiltonian system defined by cubic polynomials. Finally, important differences with the continuous case are pointed out.
Keywords: difference equations, Hamiltonian systems, stability in the Lyapunov sense difference equations, Hamiltonian systems, stability in the Lyapunov sense
MSC Classifications: 34D20, 34E10 show english descriptions Stability
Perturbations, asymptotics
34D20 - Stability
34E10 - Perturbations, asymptotics
 

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