Univ. de La Rioja, 26004 Logroño, Spain
Departamento Matemáticas y Computación, CIME, Universidad de La Rioja
Bardin B. S., Lanchares V.
On the Stability of Periodic Hamiltonian Systems with One Degree of Freedom in the Case of Degeneracy
2015, vol. 20, no. 6, pp. 627-648
We deal with the stability problem of an equilibrium position of a periodic Hamiltonian system with one degree of freedom. We suppose the Hamiltonian is analytic in a small neighborhood of the equilibrium position, and the characteristic exponents of the linearized system have zero real part, i.e., a nonlinear analysis is necessary to study the stability in the sense of Lyapunov. In general, the stability character of the equilibrium depends on nonzero terms of the lowest order $N (N >2)$ in the Hamiltonian normal form, and the stability problem can be solved by using known criteria.
We study the so-called degenerate cases, when terms of order higher than $N$ must be taken into account to solve the stability problem. For such degenerate cases, we establish general conditions for stability and instability. Besides, we apply these results to obtain new stability criteria for the cases of degeneracy, which appear in the presence of first, second, third and fourth order resonances.
Lanchares V., Pascual A. I., Elipe A.
Determination of Nonlinear Stability for Low Order Resonances by a Geometric Criterion
2012, vol. 17, no. 3-4, pp. 307-317
We consider the problem of stability of equilibrium points in Hamiltonian systems of two degrees of freedom under low order resonances. For resonances of order bigger than two there are several results giving stability conditions, in particular one based on the geometry of the phase flow and a set of invariants. In this paper we show that this geometric criterion is still valid for low order resonances, that is, resonances of order two and resonances of order one. This approach provides necessary stability conditions for both the semisimple and non-semisimple cases, with an appropriate choice of invariants.