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2013
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# Víctor Lanchares

Mathematics and Computer Science Department. Madre de Dios 53, 26006, Logrono, La Rioja, Spain
Universidad de La Rioja, Spain
Professor of Applied Mathematics at University of La Rioja, Spain.

## Publications:

 Bardin B. S., Lanchares V. Stability of a One-degree-of-freedom Canonical System in the Case of Zero Quadratic and Cubic Part of a Hamiltonian 2020, vol. 25, no. 3, pp.  237-249 Abstract We consider the stability of the equilibrium position of a periodic Hamiltonian system with one degree of freedom. It is supposed that the series expansion of the Hamiltonian function, in a small neighborhood of the equilibrium position, does not include terms of second and third degree. Moreover, we focus on a degenerate case, when fourth-degree terms in the Hamiltonian function are not enough to obtain rigorous conclusions on stability or instability. A complete study of the equilibrium stability in the above degenerate case is performed, giving sufficient conditions for instability and stability in the sense of Lyapunov. The above conditions are expressed in the form of inequalities with respect to the coefficients of the Hamiltonian function, normalized up to sixth-degree terms inclusive. Keywords: Hamiltonian systems, Lyapunov stability, normal forms, KAM theory, case of degeneracy Citation: Bardin B. S., Lanchares V.,  Stability of a One-degree-of-freedom Canonical System in the Case of Zero Quadratic and Cubic Part of a Hamiltonian, Regular and Chaotic Dynamics, 2020, vol. 25, no. 3, pp. 237-249 DOI:10.1134/S1560354720030016
 Iñarrea M., Lanchares V., Pascual A. I., Elipe A. On the Stability of a Class of Permanent Rotations of a Heavy Asymmetric Gyrostat 2017, vol. 22, no. 7, pp.  824-839 Abstract We consider the motion of an asymmetric gyrostat under the attraction of a uniform Newtonian field. It is supposed that the center of mass lies along one of the principal axes of inertia, while a rotor spins around a different axis of inertia. For this problem, we obtain the possible permanent rotations, that is, the equilibria of the system. The Lyapunov stability of these permanent rotations is analyzed by means of the Energy–Casimir method and necessary and sufficient conditions are derived, proving that there exist permanent stable rotations when the gyrostat is oriented in any direction of the space. The geometry of the gyrostat and the value of the gyrostatic momentum are relevant in order to get stable permanent rotations. Moreover, it seems that the necessary conditions are also sufficient, but this fact can only be proved partially. Keywords: gyrostat rotation, stability, Energy –Casimir method Citation: Iñarrea M., Lanchares V., Pascual A. I., Elipe A.,  On the Stability of a Class of Permanent Rotations of a Heavy Asymmetric Gyrostat, Regular and Chaotic Dynamics, 2017, vol. 22, no. 7, pp. 824-839 DOI:10.1134/S156035471707005X
 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 Abstract 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. Keywords: Hamiltonian systems, Lyapunov stability, stability theory, normal forms, KAM theory, Chetaev’s function, resonance Citation: Bardin B. S., Lanchares V.,  On the Stability of Periodic Hamiltonian Systems with One Degree of Freedom in the Case of Degeneracy, Regular and Chaotic Dynamics, 2015, vol. 20, no. 6, pp. 627-648 DOI:10.1134/S1560354715060015
 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 Abstract 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. Keywords: nonlinear stability, resonances, normal forms Citation: Lanchares V., Pascual A. I., Elipe A.,  Determination of Nonlinear Stability for Low Order Resonances by a Geometric Criterion, Regular and Chaotic Dynamics, 2012, vol. 17, no. 3-4, pp. 307-317 DOI:10.1134/S1560354712030070