Escape dynamics through a continuously growing leak.

We formulate a model that describes the escape dynamics in a leaky chaotic system in which the size of the leak depends on the number of the in-falling particles. The basic motivation of this work is the astrophysical process, which describes the planetary accretion. In order to study the dynamics generally, the standard map is investigated in two cases when the dynamics is fully hyperbolic and in the presence of Kolmogorov-Arnold-Moser islands. In addition to the numerical calculations, an analytic solution to the temporal behavior of the model is also derived. We show that in the early phase of the leak expansion, as long as there are enough particles in the system, the number of survivors deviates from the well-known exponential decay. Furthermore, the analytic solution returns the classical result in the limiting case when the number of particles does not affect the leak size.

Chaotic motion of light particles in an unsteady three-dimensional vortex: experiments and simulation.

We study the chaotic motion of a small rigid sphere, lighter than the fluid in a three-dimensional vortex of finite height. Based on the results of Eulerian and Lagrangian measurements, a sequence of models is set up. The time-independent model is a generalization of the Burgers vortex. In this case, there are two types of attractors for the particle: a fixed point on the vortex axis and a limit cycle around the vortex axis. Time dependence might combine these regular attractors into a single chaotic attractor, however its robustness is much weaker than what the experiments suggest. To construct an aperiodically time-dependent advection dynamics in a simple way, Gaussian noise is added to the particle velocity in the numerical simulation. With an appropriate choice of the noise properties, mimicking the effect of local turbulence, a reasonable agreement with the experimentally observed particle statistics is found.

Dynamics of tidal synchronization and orbit circularization of celestial bodies.

We take a dynamical-systems approach to study the qualitative dynamical aspects of the tidal locking of the rotation of secondary celestial bodies with their orbital motion around the primary. We introduce a minimal model including the essential features of gravitationally induced elastic deformation and tidal dissipation that demonstrates the details of the energy transfer between the orbital and rotovibrational degrees of freedom. Despite its simplicity, our model can account for both synchronization into the 1:1 spin-orbit resonance and the circularization of the orbit as the only true asymptotic attractors, together with the existence of relatively long-lived metastable orbits with the secondary in p:q synchronous rotation.

Dynamics of chaotic driving: rotation in the restricted three-body problem.

We investigate the rotation of a small nonspherical body in the planar restricted three-body problem along periodic, quasi-periodic, and chaotic orbits of the small body's center of mass. The rotation dynamics is chaotic in all three cases, but a systematic overview of it via stroboscopic mappings is possible only in the periodic case. We propose to explore the structured phase space patterns by following an ensemble of trajectories, a droplet, in the phase space. The temporal evolution of the pattern can be characterized by a time-dependent fractal dimension. It is shown to converge exponentially to a time-independent value for long times. In the presence of dissipation, the droplet typically converges to a so-called snapshot chaotic attractor whose shape might change chaotically in time, but whose asymptotic fractal dimension is constant.