Geometry of a chaotic layer

An analysis is made of the dependence of the geometric shape of the chaotic layer near the separatrix of a nonlinear resonance of a Hamiltonian system on the parameters of this system. A separatrix algorithmic mapping, which describes the motion near the separatrix in the presence of an asymmetric perturbation having an arbitrary degree of asymmetry. The separatrix algorithmic mapping is an algorithm containing conditional transfer instructions, is considered. An analytic procedure is derived to reduce the separatrix algorithmic mapping to the unified surface of the cross section of the initial Hamiltonian system (mapping synchronization procedure). It is observed that in the case of the high-frequency perturbation λ → +∞ (where λ is the ratio of the perturbation frequency to the frequency of small phase oscillations at resonance), the chaotic layer is subjected to strong bending in the sense that during motion near the separatrix theamplitude of the energy deviations relative to the unperturbed separatrix value is much larger than the layer width. However, the synchronized separatrix algorithmic mapping ensures an accurate representation of the phase portrait of the layer for both low and high values of the parameter λ provided that the amplitude of the perturbation is fairly small. This is demonstrated by comparing the phase portraits obtained using the synchronized separatrix algorithmic mapping with the results of direct numerical integrations of the initial Hamiltonian system.     

Ratchet effect and the transporting islands in the chaotic sea

We study directed transport in a classical deterministic dissipative system. We consider the generic case of mixed phase space and show that large ratchet currents can be generated thanks to the presence, in the Hamiltonian limit, of transporting stability islands embedded in the chaotic sea. Because of the simultaneous presence of chaos and dissipation the stationary value of the current is independent of initial conditions, except for initial states with very small measure.     

Basin topology in dissipative chaotic scattering

Chaotic scattering in open Hamiltonian systems under weak dissipation is not only of fundamental interest but also important for problems of current concern such as the advection and transport of inertial particles in fluid flows. Previous work using discrete maps demonstrated that nonhyperbolic chaotic scattering is structurally unstable in the sense that the algebraic decay of scattering particles immediately becomes exponential in the presence of weak dissipation. Here we extend the result to continuous-time Hamiltonian systems by using the Henon-Heiles system as a prototype model. More importantly, we go beyond to investigate the basin structure of scattering dynamics. A surprising finding is that, in the common case where multiple destinations exist for scattering trajectories, Wada basin boundaries are common and they appear to be structurally stable under weak dissipation, even when other characteristics of the nonhyperbolic scattering dynamics are not. We provide numerical evidence and a geometric theory for the structural stability of the complex basin topology.     

The structure of chaotic magnetic field lines in a tokamak withexternal nonsymmetric magnetic perturbations

We consider the effects of external nonsymmetric magnetostatic perturbations caused by resonant helical windings and a chaotic magnetic limiter on the plasma confined in a tokamak. The main purpose of both types of perturbation is to create a region in which field lines are chaotic in the Lagrangian sense: two initially nearby field lines diverge exponentially through many turns around the tokamak. The equilibrium field is obtained from the equations of magneto-hydrodynamic equilibrium written down in a polar toroidal coordinate system. The magnetic fields generated by the resonant helical windings and the chaotic magnetic limiter are obtained through an analytical solution of Laplace equation. The magnetic field line equations are integrated to give a Hamiltonian mapping of field lines that we use to characterize the structure of chaotic field lines. In the case of resonant windings, we obtained the map by both numerical integration and a Hamiltonian formulation. For a chaotic limiter, we analytically derived a symplectic map by using a Hamiltonian formulation     

Probing microscopic chaotic dynamics by observing macroscopic transport processes

We derive the Kramers equation, namely, the Fokker-Planck equation for an oscillator, from a completely deterministic picture. The oscillator is coupled to a “booster”, i.e., a deterministic system in a fully chaotic state, wherein diffusion is derived from the sensitive dependence of chaos on initial conditions and friction is a consequence of the linear response of the booster to the action exerted on it by the oscillator. To deal with the Hamiltonian nature of the system of interest and of its coupling to the booster, we extend the earlier theoretical derivation of macroscopic transport coefficients from deterministic dynamics. We show that the frequency of the oscillator can be tuned to the microscopic frequencies of the booster without affecting the canonical nature of the “macroscopic” statistics. The theoretical predictions are supported by numerical simulations.     

Semiclassical mechanics of bound chaotic potentials

Semiclassical methods for determining quantum eigenvalues in chaotic systems are discussed. A recent calculation for an open scattering system with Axiom-A properties serves as a starting point for the discussion. How deviation from Axiom-A properties, such as intermittency and occurrence of small stability islands, normally arise in bound Hamiltonian systems, and how these deviations complicate the calculation of semiclassical eigenvalues are demonstrated. It is also stressed that since such deviations are typical of bound Hamiltonian systems, they might be of crucial importance for the statistical properties of the energy levels.     

Exponential decay and scaling laws in noisy chaotic scattering

In this Letter we present a numerical study of the effect of noise on a chaotic scattering problem in open Hamiltonian systems. We use the second order Heun method for stochastic differential equations in order to integrate the equations of motion of a two-dimensional flow with additive white Gaussian noise. We use as a prototype model the paradigmatic Hénon-Heiles Hamiltonian with weak dissipation which is a well-known example of a system with escapes. We study the behavior of the scattering particles in the scattering region, finding an abrupt change of the decay law from algebraic to exponential due to the effects of noise. Moreover, we find a linear scaling law between the coefficient of the exponential law and the intensity of noise. These results are of a general nature in the sense that the same behavior appears when we choose as a model a two-dimensional discrete map with uniform noise (bounded in a particular interval and zero otherwise), showing the validity of the algorithm used. We believe the results of this work be useful for a better understanding of chaotic scattering in more realistic situations, where noise is presented.     

Chaotic and Arnold stripes in weakly chaotic Hamiltonian systems

The dynamics in weakly chaotic Hamiltonian systems strongly depends on initial conditions (ICs) and little can be affirmed about generic behaviors. Using two distinct Hamiltonian systems, namely one particle in an open rectangular billiard and four particles globally coupled on a discrete lattice, we show that in these models, the transition from integrable motion to weak chaos emerges via chaotic stripes as the nonlinear parameter is increased. The stripes represent intervals of initial conditions which generate chaotic trajectories and increase with the nonlinear parameter of the system. In the billiard case, the initial conditions are the injection angles. For higher-dimensional systems and small nonlinearities, the chaotic stripes are the initial condition inside which Arnold diffusion occurs.     

Quantization of chaotic systems

Starting from the semiclassical dynamical zeta function for chaotic Hamiltonian systems we use a combination of the cycle expansion method and a functional equation to obtain highly excited semiclassical eigenvalues. The power of this method is demonstrated for the anisotropic Kepler problem, a strongly chaotic system with good symbolic dynamics. An application of the transfer matrix approach of Bogomolny is presented leading to a significant reduction of the classical input and to comparable accuracy for the calculated eigenvalues.     

Measurements on quantum chaotic systems

The problems of the feedback of a measurement on the dynamics of quantum mechanical systems, which are chaotic in some way are studied. The system can be Hamiltonian or dissipative. For the latter case it is shown that measurements can be devised which do not affect the evolution of the system. Hamiltonian systems are discussed in terms of two models, one being the kicked quantum rotator and the other a two-state system driven by a field with two incommensurate frequencies. Both destructive and continuous measurements are discussed. For the quantum kicked rotator, in the absence of measurement, there is Anderson localisation due to quantum interference. Surprisingly the act of measurement, which might be expected to destroy the delicate interference, does not lead to delocalisation. Measurements however destroy the time-reversal invariance of the evolution of the Hamiltonian systems. In most circumstances it is shown that quantum chaotic systems can be effectively measured.     

Classical and quantum Hamiltonian ratchets

We explain the mechanism leading to directed chaotic transport in Hamiltonian systems with spatial and temporal periodicity. We show that a mixed phase space comprising both regular and chaotic motion is required and we derive a classical sum rule which allows one to predict the chaotic transport velocity from properties of regular phase-space components. Transport in quantum Hamiltonian ratchets arises by the same mechanism as long as uncertainty allows one to resolve the classical phase-space structure. We derive a quantum sum rule analogous to the classical one, based on the relation between quantum transport and band structure.     

Escaping from nonhyperbolic chaotic attractors

The noise-induced escape process from a nonhyperbolic chaotic attractor is of physical and fundamental importance. We address this problem by uncovering the general mechanism of escape in the relevant low noise limit using the Hamiltonian theory of large fluctuations and by establishing the crucial role of the primary homoclinic tangency closest to the basin boundary in the dynamical process. In order to demonstrate that, we provide an unambiguous solution of the variational equations from the Hamiltonian theory. Our results are substantiated with the help of physical and dynamical paradigms, such as the Hénon and the Ikeda maps. It is further pointed out that our findings should be valid for driven flow systems and for experimental data.     

Non-linear diffusion in Hamiltonian systems exhibiting chaotic motion

The exact evolution equation for the angle averaged phase space density in action-angle space is derived from the Liouville equation using projection operator techniques. This equation involves a correlation function of the initial value of the phase space density with the angle dependent part of the Hamiltonian and a correlation function of the angle dependent part of the Hamiltonian and a correlation function of the angle dependent part of the Hamiltonian with itself. Each of these correlation functions develops in time with angle projected dynamics. We show their relation to the correlation functions which develop in time with usual Hamiltonian dynamics. These correlation functions are then studied in the standard model of Chirikov, and we conclude that they behave as $\text{e}{}^{\mathrm{-\sigma t}}\text{cos}\text{(Ωt + φ)}$ in regions of irregular motion. We conjecture that angle averaged correlation functions behave this way in general, and we give an argument based on the mixing property of the Hamiltonian system. Our argument goes beyond the usual mixing, so we regard it as a quasi-mixing hypothesis. Under this hypothesis the equation for the angle averaged phase space density becomes a diffusion equation which incorporates much of the non-linear dynamics of Hamiltonian systems exhibiting chaotic motion.     

Influence of phase-space localization on the energy diffusion in a quantum chaotic billiard

The quantum dynamics of a chaotic billiard with moving boundary is considered in this paper. We found a shape parameter Hamiltonian expansion, which enables us to obtain the spectrum of the deformed billiard for deformations so large as the characteristic wavelength. Then, for a specified time-dependent shape variation, the quantum dynamics of a particle inside the billiard is integrated directly. In particular, the dispersion of the energy is studied in the Bunimovich stadium billiard with oscillating boundary. The results showed that the distribution of energy spreads diffusively for the first oscillations of the boundary (=2Dt). We studied the diffusion constant D as a function of the boundary velocity and found differences with theoretical predictions based on random matrix theory. By extracting highly phase-space localized structures from the spectrum, previous differences were reduced significantly. This fact provides numerical evidence of the influence of phase-space localization on the quantum diffusion of a chaotic system.     

A novel image block cryptosystem based ona spatiotemporal chaotic system and a chaotic neural network

In this paper, we propose a novel block cryptographic scheme based on a spatiotemporal chaotic system and a chaotic neural network (CNN). The employed CNN comprises a 4-neuron layer called a chaotic neuron layer (CNL), where the spatiotemporal chaotic system participates in generating its weight matrix and other parameters. The spatiotemporal chaotic system used in our scheme is the typical coupled map lattice (CML), which can be easily implemented in parallel by hardware. A 160-bit-long binary sequence is used to generate the initial conditions of the CML. The decryption process is symmetric relative to the encryption process. Theoretical analysis and experimental results prove that the block cryptosystem is secure and practical, and suitable for image encryption.     

Invariants of chaotic Hamiltonian systems

The invariants of chaotic bounded Hamiltonian systems and their relation to the solutions of the first variational equations of the equations of motion are studied. We show that these invariants are characterized by the fact that they either lose the property of differentiability as functions on phase space or that a certain formal power series defined in terms of the derivatives of the invariants has zero radius of convergence. For a specific example, we show that the former possibility appears to apply.     

Quasi-stationary chaotic states in multi-dimensional Hamiltonian systems

We study numerically statistical distributions of sums of chaotic orbit coordinates, viewed as independent random variables, in weakly chaotic regimes of three multi-dimensional Hamiltonian systems: Two Fermi-Pasta-Ulam (FPU-β) oscillator chains with different boundary conditions and numbers of particles and a microplasma of identical ions confined in a Penning trap and repelled by mutual Coulomb interactions. For the FPU systems we show that, when chaos is limited within “small size” phase space regions, statistical distributions of sums of chaotic variables are well approximated for surprisingly long times (typically up to t≈10⁶) by a q-Gaussian (1<q<3) distribution and tend to a Gaussian (q=1) for longer times, as the orbits eventually enter into “large size” chaotic domains. However, in agreement with other studies, we find in certain cases that the q-Gaussian is not the only possible distribution that can fit the data, as our sums may be better approximated by a different so-called “crossover” function attributed to finite-size effects. In the case of the microplasma Hamiltonian, we make use of these q-Gaussian distributions to identify two energy regimes of “weak chaos”—one where the system melts and one where it transforms from liquid to a gas state-by observing where the q-index of the distribution increases significantly above the q=1 value of strong chaos.     

Targeting in Hamiltonian systems that have mixed regular/chaotic phase spaces

The problem of directing a trajectory of a chaotic dynamical system to a target has been previously considered, and it has been shown that chaos allows targeting using only small controls. In this paper we consider targeting in a Hamiltonian system, whose phase space contains a mixture of regular quasi-periodic and chaotic regions. A multistep forward-backward method targeting strategic intermediate points is found to be efficient and robust. It takes full advantage of the phase space structure and is believed to yield optimal transport times. It is robust under the influence of small noise and small modeling errors and recovers from temporary loss of control. Two illustrative examples, the standard map and the restricted circular three body problem, are presented. (The latter corresponds to motion of a space probe in the presence of the earth and the moon.) Comparisons are made of our method to other targeting strategies. (c) 1997 American Institute of Physics.     

Stages of chaotic synchronization

In an experimental investigation of the response of a chaotic system to a chaotic driving force, we have observed synchronization of chaos of the response system in the forms of generalized synchronization, phase synchronization, and lag synchronization to the driving signal. In this paper we compare the features of these forms of synchronized chaos and study their relations and physical origins. We found that different forms of chaotic synchronization could be interpreted as different stages of nonlinear interaction between the coupled chaotic systems. (c) 1998 American Institute of Physics.