Víctor Lanchares

CHAOS IN THE REORIENTATION PROCESS OF A DUAL-SPIN SPACECRAFT WITH TIME-DEPENDENT MOMENTS OF INERTIA

We study the spin-up dynamics of a dual-spin spacecraft containing one axisymmetric rotor which is parallel to one of the principal axes of the spacecraft. It will be supposed that one of the moments of inertia of the platform is a periodic function of time and that the center of mass of the spacecraft is not modified. Under these assumptions, it is shown that in the absence of external torques and spinning rotors the system possesses chaotic behavior in the sense that it exhibits Smales horseshoes. We prove this statement by means of the Melnikov method. The presence of chaotic behavior results in a random spin-up operation. This randomness is visualized by means of maps of the initial conditions with final nutation angle close to zero. This phenomenon is well described by a suitable parameter that measures the amount of randomness of the process. Finally, we relate this parameter with the Melnikov function in the absence of the spinning rotor and with the presence of subharmonic resonances.

Spin Rotor Stabilization of a Dual-Spin Spacecraft with Time Dependent Moments of Inertia

We consider a dual-spin deformable spacecraft, in the sense that one of the moments of inertia is a periodic function of time such that the center of mass is not altered. In the absence of external torques and spin rotors, by means of the Melnikovs method we prove that the body motion is chaotic. Stabilization is obtained by means of a spinning rotor about one of the principal axes of inertia.

Nonlinear Stability in Resonant Cases: A Geometrical Approach

Summary. In systems with two degrees of freedom, Arnolds theorem is used for studying nonlinear stability of the origin when the quadratic part of the Hamiltonian is a nondefinite form. In that case, a previous normalization of the higher orders is needed, which reduces the Hamiltonian to homogeneous polynomials in the actions. However, in the case of resonances, it could not be possible to bring the Hamiltonian to the normal form required by Arnolds theorem. In these cases, we determine the stability from analysis of the normalized phase flow. Normalization up to an arbitrary order by Lie-Deprit transformation is carried out using a generalization of the Lissajous variables.

Chaotic Rotations of an Asymmetric Body with Time-Dependent Moments of Inertia and Viscous Drag

We study the dynamics of a rotating asymmetric body under the influence of an aerodynamic drag. We assume that the drag torque is proportional to the angular velocity of the body. Also we suppose that one of the moments of inertia of the body is a periodic function of time and that the center of mass of the body is not modified. Under these assumptions, we show that the system exhibits a transient chaotic behavior by means of a higher dimensional generalization of the Melnikovs method. This method give us an analytical criterion for heteroclinic chaos in terms of the system parameters. These analytical results are confirmed by computer numerical simulations of the system rotations.

Bifurcations in biparametric quadratic potentials

Numerous dynamical systems are represented by quadratic Hamiltonians with the phase space on the S (2) sphere. For a class of these Hamiltonians depending on two parameters, we analyze the occurrence of bifurcations and we obtain the bifurcation lines in the parameter plane. As the parameters evolve, the appearance-disappearance of homoclinic orbits in the phase portrait is governed by three types of bifurcations, the pitchfork, the teardrop and the oyster bifurcations. We find that the teardrop bifurcation is associated with a non-elementary fixed point whose Poincare index is zero. (c) 1995 American Institute of Physics.

On the Stability of Equilibria in Two-Degrees-of- Freedom Hamiltonian Systems Under Resonances

AbstractWe consider the problem of stability of equilibrium points in Hamiltonian systems of two degrees of freedom under resonances. Determining the stability or instability is based on a geometrical criterion based on how two surfaces, related with the normal form, intersect one another. The equivalence of this criterion with a result of Cabral and Meyer is proved. With this geometrical procedure, the hypothesis may be extended to more general cases.

Two pitchfork bifurcations in the polar quadratic Zeeman-Stark effect

Abstract In the framework of classical mechanics, a study of the hydrogen atom in the presence of parallel electric and magnetic fields is presented when the magnetic quantum number m is zero. By means of perturbation methods and Poincare surfaces of section, the existence of the three states experimentally detected by Cacciani et al. (the so-called I, II, and III Caccianis states), their energy extensions, their evolution and their disappearance are explained as a result of two pitchfork bifurcations.

Perturbed ion traps: A generalization of the three-dimensional Hénon–Heiles problem

This paper presents an analytical study of an axially symmetric perturbation of the Penning trap. This system is modeled as a generalization of the three-dimensional (3D) Henon-Heiles potential. Thus, the same techniques which succeeded in the study of the 3D Henon-Heiles system apply here. The departure Hamiltonian is three dimensional, although it possesses an axial symmetry. This property, together with an averaging process, is used to reduce the original system to an integrable one. We study the flow of the reduced Hamiltonian: equilibria, bifurcations, and stability, extracting thereafter the relevant information about the dynamics of the original problem. (c) 2002 American Institute of Physics.

Stability of the permanent rotations of an asymmetric gyrostat in a uniform Newtonian field

The stability of the permanent rotations of a heavy gyrostat is analyzed by means of the Energy-Casimir method. Sufficient and necessary conditions are established for some of the permanent rotations. The geometry of the gyrostat and the value of the gyrostatic moment are relevant in order to get stable permanent rotations. Moreover, the necessary conditions are also sufficient, for some configurations of the gyrostat.

On the stability of periodic Hamiltonian systems with one degree of freedom in the case of degeneracy

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.