997 research outputs found
Kolmogorov : la "k" de KAM
La teoria de Kolmogorov-Arnold-Moser (o kam) va ser desenvolupada per sistemes dinàmics conservatius que estan prop d’integrables. Típicament els sistemes integrables contenen molts tors invariants en el seu espai de fases. La teoria kam estableix resultats de persistència d’aquests tors, en els quals el moviment és quasiperiòdic. Fem un esbós d’aquesta teoria al voltant de la figura de Kolmogorov
A Global Kam-Theorem: Monodromy in Near-Integrable Perturbations of Spherical Pendulum
The KAM Theory for the persistence of Lagrangean invariant tori in nearly integrable Hamiltonian systems is lobalized to bundles of invariant tori. This leads to globally well-defined conjugations between near-integrable systems and their integrable approximations, defined on nowhere dense sets of positive measure associated to Diophantine frequency vectors. These conjugations are Whitney smooth diffeomorphisms between the corresponding torus bundles. Thus the geometry of the integrable torus bundle is inherited by the near-integrable perturbation. This is of intereet in cases where these bundles are nontrivial. The paper deals with the spherical pendulum as a leading example
Quasi-periodic stability of normally resonant tori
We study quasi-periodic tori under a normal-internal resonance, possibly with
multiple eigenvalues. Two non-degeneracy conditions play a role. The first of
these generalizes invertibility of the Floquet matrix and prevents drift of the
lower dimensional torus. The second condition involves a Kolmogorov-like
variation of the internal frequencies and simultaneously versality of the
Floquet matrix unfolding. We focus on the reversible setting, but our results
carry over to the Hamiltonian and dissipative contexts
Resonances in a spring-pendulum: algorithms for equivariant singularity theory
A spring-pendulum in resonance is a time-independent Hamiltonian model system for formal reduction to one degree of freedom, where some symmetry (reversibility) is maintained. The reduction is handled by equivariant singularity theory with a distinguished parameter, yielding an integrable approximation of the Poincaré map. This makes a concise description of certain bifurcations possible. The computation of reparametrizations from normal form to the actual system is performed by Gröbner basis techniques.
Pacer cell response to periodic Zeitgebers
Almost all organisms show some kind of time periodicity in their behavior.
Especially in mammals the neurons of the suprachiasmatic nucleus form a
biological clock regulating the activity-inactivity cycle of the animal. This
clock is stimulated by the natural 24-hour light-dark cycle. In our model of
this system we consider each neuron as a so called phase oscillator, coupled to
other neurons for which the light-dark cycle is a Zeitgeber. To simplify the
model we first take an externally stimulated single phase oscillator. The first
part of the phase interval is called the active state and the remaining part is
the inactive state. Without external stimulus the oscillator oscillates with
its intrinsic period. An external stimulus, be it from activity of neighboring
cells or the periodic daylight cycle, acts twofold, it may delay the change
form active to inactive and it may advance the return to the active state. The
amount of delay and advance depends on the strength of the stimulus. We use a
circle map as a mathematical model for this system. This map depends on several
parameters, among which the intrinsic period and phase delay and advance. In
parameter space we find Arnol'd tongues where the system is in resonance with
the Zeitgeber. Thus already in this simplified system we find entrainment and
synchronization. Also some other phenomena from biological experiments and
observations can be related to the dynamical behavior of the circle map
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