We explain in detail the definition, construction and generalisation of the Galois group of Chebyshev polynomials of high degree to the Galois group of chaotic chains. The calculations in this paper are performed for Chebyshev polynomials and chaotic chains of degree $N=2$. Insides into possible further steps are given.
We present here a new approach of the partial control method, which is a useful control technique applied to transient chaotic dynamics affected by a bounded noise. Usually we want to avoid the escape of these chaotic transients outside a certain region $Q$ of the phase space. For that purpose, there exists a control bound such that for controls smaller than this bound trajectories are kept in a special subset of $Q$ called the safe set. The aim of this new approach is to go further, and to compute for every point of $Q$ the minimal control bound that would keep it in $Q$. This defines a special function that we call the safety function, which can provide the necessary information to compute the safe set once we choose a particular value of the control bound. This offers a generalized method where previous known cases are included, and its use encompasses more diverse scenarios.
We study the quantum probability to survive in an open chaotic system in the framework of the van Vleck-Gutzwiller propagator and present the first such calculation that accounts for quantum interference effects. Specifically we calculate quantum deviations from the classical decay after the break time for both broken and preserved time-reversal symmetry. The source of these corrections is identified in interfering pairs of correlated classical trajectories. In our approach the quantized chaotic system is modelled by a quatum graph.
Many-site Bose-Hubbard lattices display complex semiclassical dynamics, with both chaotic and regular features. We have characterised chaos in the semiclassical dynamics of short Bose-Hubbard chains using both stroboscopic phase space projections and finite-time Lyapunov exponents. We found that chaos was present for intermediate collisional nonlinearity in the open trimer and quatramer systems, with soft chaos and Kolmogoroff-Arnold-Moser islands evident. We have found that the finite-time Lyapunov exponents are consistent with stroboscopic maps for the prediction of chaos in these small systems. This gives us confidence that the finite-time Lyapunov exponents will be a useful tool for the characterisation of chaos in larger systems, where meaningful phase-space projections are not possible and the dimensionality of the problem can make the standard methods intractable.
We shall use symmetry breaking as a tool to attack the problem of identifying the topology of chaotic scatteruing with more then two degrees of freedom. specifically we discuss the structure of the homoclinic/heteroclinic tangle and the connection between the chaotic invariant set, the scattering functions and the singularities in the cross section for a class of scattering systems with one open and two closed degrees of freedom.
The behaviors of coupled chaotic oscillators before complete synchronization were investigated. We report three phenomena: (1) The emergence of long-time residence of trajectories besides one of the saddle foci; (2) The tendency that orbits of the two oscillators get close becomes faster with increasing the coupling strength; (3) The diffusion of two oscillators phase difference is first enhanced and then suppressed. There are exact correspondences among these phenomena. The mechanism of these correspondences is explored. These phenomena uncover the route to synchronization of coupled chaotic oscillators.