Fractional standard map

Properties of the phase space of the standard map with memory are investigated. This map was obtained from a kicked fractional differential equation. Depending on the value of the map parameter and the fractional order of the derivative in the original differential equation, this nonlinear dynamical system demonstrates attractors (fixed points, stable periodic trajectories, slow converging and slow diverging trajectories, ballistic trajectories, and fractal-like structures) and/or chaotic trajectories. At least one type of fractal-like sticky attractors in the chaotic sea was observed.     

Boundary crisis and transient in a dissipative relativistic standard map

Some dynamical properties for a problem concerning the acceleration of particles in a wave packet are studied. The model is described in terms of a two-dimensional nonlinear map obtained from a Hamiltonian which describes the motion of a relativistic standard map. The phase space is mixed in the sense that there are regular and chaotic regions coexisting. When dissipation is introduced, the property of area preservation is broken and attractors emerge. We have shown that a tiny increase of the dissipation causes a change in the phase space. A chaotic attractor as well as its basin of attraction are destroyed thereby leading the system to experience a boundary crisis. We have characterized such a boundary crisis via a collision of the chaotic attractor with the stable manifold of a saddle fixed point. Once the chaotic attractor is destroyed, a chaotic transient described by a power law with exponent −1 is observed.     

Elliptic behaviour in the sawtooth standard map

This paper examines the standard map with sawtooth nonlinearity when the eigenvalues of the Jacobian lie on the unit circle. This is an area-preserving map of the torus to itself that is linear except on a line on which it is discontinuous. We discuss the closure of the set of images of the discontinuity and present numerical evidence that its Lebesgue measure is positive. Moreover, we present evidence that the measure of the closure of images of the discontinuity changes continuously with the parameter k. This means that the sawtooth standard map may exhibit coexistence of two positive measure subsets on which the dynamics is respectively regular and irregular in a certain sense. In the appendix we show that this map is equivalent to a map studied by electronics engineers as a model for the quiescent behaviour of a linear lossless digital filter with “two's complement” overflow.     

Near threshold anomalous transport in the standard map

Anomalous transport is investigated near threshold in the standard map. Very long time flights, and a large anomaly in the transport, are shown to be associated with a new form of multi-island structures causing orbit sticking. The phase space structure of these traps, and the exponents of the characteristic long time tails associated with them are determined. In general these structures are very complex, but some cases, consisting of layers of islands, allow simple modeling. (c) 1998 American Institute of Physics.     

Diffusion and transport phenomena in a collisional magnetoplasma having both streaming and temperature anisotropy: Collisional kinetics

Boltzmann-transport equation is analytically solved for two-component magnetoplasma using Chapman-Enskog analysis to include collisional diffusion transport having anisotropies in both streaming velocity and temperature components. The modified collisional integrals are analytically solved with flux integrals and perturbed kinetic equation to arrive at drift diffusion velocity and resulting transport coefficients which are markedly affected by both streaming and temperature anisotropy. The early isotropic results are recovered in the limit V ₀ = 0 and T _‖ = T _⊥ which reduce to eqs (11.30) and (11.31) of [1] and eqs (2.7) and (2.13) of [2]. The electrical resistivity (η_⊥) diminishes sharply in fusion temperature limit kT _⊥ = 1 keV. The shape of the curves for both electrical resistivity and thermal conductivity is rectangular hyperbolic. However, for low thermal ratio (T _‖/T _⊥ < 1), the curves are raised up and for high thermal ratio (T _‖/T _⊥ > 1), they are lowered down the isotropic case (T _‖/T _⊥ > 1), showing comparatively diminished magnitudes of the quantities.      

The dynamical temperature and the standard map

Numerical experiments with the standard map at high values of the stochasticity parameter reveal the existence of simple analytical relations connecting the volume and the dynamical temperature of the chaotic component of the phase space.     

Ulam method for the Chirikov standard map

We introduce a generalized Ulam method and apply it to symplectic dynamical maps with a divided phase space. Our extensive numerical studies based on the Arnoldi method show that the Ulam approximant of the Perron-Frobenius operator on a chaotic component converges to a continuous limit. Typically, in this regime the spectrum of relaxation modes is characterized by a power law decay for small relaxation rates. Our numerical data show that the exponent of this decay is approximately equal to the exponent of Poincaré recurrences in such systems. The eigenmodes show links with trajectories sticking around stability islands.     

Diffusion theory of the positive column with two-stage ionization and collisional deexcitation

The diffusion theory of the low pressure positive column is extended to account for collisional deexcitation as well as for two-stage ionization by electron impact. This extension results in the equationn″(ξ)+n′(ξ)/ξ+n(ξ)+vn ²(ξ)[1+μn(ξ)]^-1=0, which is solved numerically for the value ofξ ₁ which satisfies the boundary conditionn(ξ ₁)=0 for different values of the parametersν andμ.     

Meanders and reconnection-collision sequences in the standard nontwist map

New global periodic orbit collision and separatrix reconnection scenarios exhibited by the standard nontwist map are described in detail, including exact methods for determining reconnection thresholds, methods that are implemented numerically. Results are compared to a parameter space breakup diagram for shearless invariant curves. The existence of meanders, invariant tori that are not graphs, is demonstrated numerically for both odd and even period reconnection for certain regions in parameter space. Implications for transport are discussed.     

Critical behaviour of invariant curves in the standard map: a perturbative approach

The classical perturbation series is used to follow an invariant curve, of fixed winding number, up to its break-up point. The result obtained for the critical value of the perturbation and the corresponding behaviour of the invariant curve are in complete agreement with the results of Kadanoff and Shenker.     

Badly ordered periodic orbits of the standard map

The criterion of Boyland and Hall for non-existence of invariant circles for twist maps in terms of existence of badly ordered periodic orbits is tested on the standard map. We conclude that it is not as efficient as some other criteria.     

Invariant circles for the piecewise linear standard map

We investigate invariant circles for a one-parameter family of piecewise linear twist homeomorphisms of the annulus. We show that invariant circles of all types and rotation numbers occur and we classify them into families. We compute parameter ranges in which there are no invariant circles.     

Visual explorations of dynamics: The standard map

The Macintosh application StdMap allows easy exploration of many of the phenomena of area-preserving mappings. This tutorial explains some of these phenomena and presents a number of simple experiments centered on the use of this program. Dedicated to the memory of John Greene. His ideas are as timeless as his generosity is legendary.     

Heteroclinic bifurcations and chaotic transport in the two-harmonic standard map

We study a two-parameter family of standard maps: the so-called two-harmonic family. In particular, we study the areas of lobes formed by the stable and unstable manifolds. Variational methods are used to find heteroclinic orbits and their action. A specific pair of heteroclinic orbits is used to define a difference in action function and to study bifurcations in the stable and unstable manifolds. Using this idea, two phenomena are studied: the change of orientation of lobes and tangential intersections of stable and unstable manifolds.     

Long time behavior in the quantized standard map with dissipation

A dissipative version of the quantized standard map is constructed by analytical means and iterated numerically to study the long time behavior in various regions of the damping rate. For weak dissipation, stochastic transitions induced by the heat bath disrupt the localization in the action variable, which suppresses chaotic motion in the conservative quantized standard map, and tend to restore diffusion of action. A steady state is reached on the time scale of classical relaxation. For strong dissipation, observable deviations from classical behavior both in the transients and in the statey state are due to quantum noise. They are reproduced by a classical stochastic map which is approached by the dissipative quantum map as its semi-classical limit.     

Ion wave instability in a collisional magnetoplasma

Temperature relaxation mechanism is found to have a destabilizing effect on ion waves propagating nearly across the magnetic field in the presence of streaming electrons.     

Ion-acoustic instability in a two-temperature collisional plasma

It is shown that, if one takes account of the non-stationary nature of the background equilibrium, then ion acoustic waves in a two-temperature, collisional, fully ionized plasma are stable.     

Dynamics of a collisional ion sheath

Experiments on collisional ion sheaths are carried out by applying a pulsed negative bias on a disc electrode immersed in a collisional plasma. The pulse is characterized by a linear rise, followed by a constant voltage phase and then exponential decay. The measured currents to the electrode are compared to predictions from a dynamic collisional ion sheath model which is developed from the basic two fluid equations. The parameter determining the degree of collisionality is also defined. The agreement between the two in the rising and the flat top phases of the pulse is found to be good. Some residual discrepancies as well as the disagreement in the decay phase are discussed.     

A fast image encryption scheme based on chaotic standard map

In recent years, a variety of effective chaos-based image cryptosystems have been proposed. One of the architectures of this kind of cryptosystems is composed of multiple rounds of substitution and diffusion. As the confusion and diffusion effects are solely contributed by the substitution and the diffusion stages, respectively, the required overall rounds of operations in achieving a certain level of security is found more than necessary. In this Letter, we suggest to introduce a certain diffusion effect in the substitution stage by simple sequential add-and-shift operations. Although this leads to a longer processing time in a single round, the overall encryption time is reduced as fewer rounds are required. Simulation results show that at a similar performance level, the proposed cryptosystem needs less than one-third the encryption time of an existing fast cryptosystem. The effective acceleration of chaos-based image cryptosystems is thus achieved.