Shrinking shrimp-shaped domains and multistability in the dissipative asymmetric kicked rotor map.

An interesting feature in dissipative nonlinear systems is the emergence of characteristic domains in parameter space that exhibit periodic temporal evolution, known as shrimp-shaped domains. We investigate the parameter space of the dissipative asymmetric kicked rotor map and show that, in the regime of strong dissipation, the shrimp-shaped domains repeat themselves as the nonlinearity parameter increases while maintaining the same period. We analyze the dependence of the length of each periodic domain with the nonlinearity parameter, revealing that it follows a power law with the same exponent regardless of the dissipation parameter.

Isochronous bifurcations in a two-parameter twist map.

Isochronous islands in phase space emerge in twist Hamiltonian systems as a response to multiple resonant perturbations. According to the Poincaré-Birkhoff theorem, the number of islands depends on the system characteristics and the perturbation. We analyze, for the two-parameter standard map, also called two-harmonic standard map, how the island chains are modified as the perturbation amplitude increases.

Nontwist field line mapping in a tokamak with ergodic magnetic limiter.

For tokamaks with uniform magnetic shear, Martin and Taylor have proposed a symplectic map which has been used to describe the magnetic field lines at the plasma edge perturbed by an ergodic magnetic limiter. We propose an analytical magnetic field line map, based on the Martin-Taylor map, for a tokamak with arbitrary safety factor profile. With the inclusion of a nonmonotonic profile, we obtain a nontwist map which presents the characteristic properties of degenerate systems, such as the twin islands scenario, shearless curve, and separatrix reconnection.

Stickiness and recurrence plots: An entropy-based approach.

The stickiness effect is a fundamental feature of quasi-integrable Hamiltonian systems. We propose the use of an entropy-based measure of the recurrence plots (RPs), namely, the entropy of the distribution of the recurrence times (estimated from the RP), to characterize the dynamics of a typical quasi-integrable Hamiltonian system with coexisting regular and chaotic regions. We show that the recurrence time entropy (RTE) is positively correlated to the largest Lyapunov exponent, with a high correlation coefficient.

Dynamics, multistability, and crisis analysis of a sine-circle nontwist map.

We propose a one-dimensional dynamical system, the sine-circle nontwist map, that can be considered a local approximation of the standard nontwist map and an extension of the paradigmatic sine-circle map. The map depends on three parameters, exhibiting a simple mathematical form but with a rich dynamical behavior. We identify periodic, quasiperiodic, and chaotic solutions for different parameter sets with the Lyapunov exponent and Slater's theorem.

Curry-Yorke route to shearless attractors and coexistence of attractors in dissipative nontwist systems.

The routes to chaos play an important role in predictions about the transitions from regular to irregular behavior in nonlinear dynamical systems, such as electrical oscillators, chemical reactions, biomedical rhythms, and nonlinear wave coupling. Of special interest are dissipative systems obtained by adding a dissipation term in a given Hamiltonian system. If the latter satisfies the so-called twist property, the corresponding dissipative version can be called a "dissipative twist system.

Ratchet current in nontwist Hamiltonian systems.

Non-monotonic area-preserving maps violate the twist condition locally in phase space, giving rise to shearless invariant barriers surrounded by twin island chains in these regions of phase space. For the extended standard nontwist map, with two resonant perturbations with distinct wave numbers, we investigate the presence of such barriers and their associated island chains and compare our results with those that have been reported for the standard nontwist map with only one perturbation. Furthermore, we determine in the control parameter space the existence of the shearless barrier and the influence of the additional wave number on this condition.

Using rotation number to detect sticky orbits in Hamiltonian systems.

In Hamiltonian systems, depending on the control parameter, orbits can stay for very long times around islands, the so-called stickiness effect caused by a temporary trapping mechanism. Different methods have been used to identify sticky orbits, such as recurrence analysis, recurrence time statistics, and finite-time Lyapunov exponent. However, these methods require a large number of map iterations and to know the island positions in the phase space.

Recurrence-based analysis of barrier breakup in the standard nontwist map.

We study the standard nontwist map that describes the dynamic behaviour of magnetic field lines near a local minimum or maximum of frequency. The standard nontwist map has a shearless invariant curve that acts like a barrier in phase space. Critical parameters for the breakup of the shearless curve have been determined by procedures based on the indicator points and bifurcations of periodical orbits, a methodology that demands high computational cost.