V. V. Vecheslavov

Arnold diffusion in large systems

A new regime of Arnold diffusion in which the diffusion rate has a power-law dependence on the perturbation strength is studied theoretically and in numerical experiments. The theory developed predicts this new regime to be universal in the perturbation intermediate asymptotics, the width of the latter increasing with the dimensionality of the perturbation frequency space, particularly in large systems with many degrees of freedom. The results of numerical experiments agree satisfactorily with the theoretical estimates.

Adiabatic invariance and separatrix: Single separatrix crossing

Detailed numerical experiments on the dynamics and statistics of a single crossing of the separatrix of a nonlinear resonance with a time-varying amplitude are described. The results are compared with a simple approximate theory first developed by Timofeev and further improved and generalized by Tennyson and coworkers. The main attention is paid to a new, ballistic, regime of separatrix crossing in which the violation of adiabaticity is maximal. Some unsolved problems and open questions are also discussed.

Multiple Separatrix Crossing: A Chaos Structure ¶

Numerical experiments on the structure of the chaotic component of motion under multiple-crossing of the separatrix of a nonlinear resonance with a time-varying amplitude are described with the emphasis on the ergodicity problem. The results clearly demonstrate nonergodicity of this motion due to the presence of a regular component of a relatively small measure with a very complicated structure. A simple 2D-map per crossing is constructed that qualitatively describes the main properties of both chaotic and regular components of the motion. An empirical relation for the correlation-affected diffusion rate is found including a close vicinity of the chaos border where evidence of the critical structure is observed. Some unsolved problems and open questions are also discussed.

Diffusion in Smooth Hamiltonian Systems

A family of models determined by a smooth canonical 2D-map that depends on two parameters is studied. Preliminary results of numerical experiments are reported; they are evidence of substantial suppression of global diffusion in a wide range of perturbation values. This effect is caused by the little-known phenomenon of the conservation of resonance separatrices and other invariant curves under the conditions of strong local dynamic chaos. Such a total suppression of diffusion occurs although invariant curves are only conserved for a countable zero-measure set of parameter values. Simple refined estimates of diffusion rates in smooth systems without invariant curves were obtained and numerically substantiated. The principal boundary of diffusion suppression in a family of models with invariant curves was described by a semiempirical equation in dimensionless variables. The results were subjected to a statistical analysis, and an integral distribution for diffusion suppression probability was obtained.

Fractal diffusion in smooth dynamical systems with virtual invariant curves

Preliminary results of extensive numerical experiments with a family of simple models specified by the smooth canonical strongly chaotic 2D map with global virtual invariant curves are presented. We focus on the statistics of the diffusion rate D of individual trajectories for various fixed values of the model perturbation parameters K and d. Our previous conjecture on the fractal statistics determined by the critical structure of both the phase space and the motion is confirmed and studied in some detail. In particular, we find additional characteristics of what we earlier termed the virtual invariant curve diffusion suppression, which is related to a new very specific type of critical structure. A surprising example of ergodic motion with a “hidden” critical structure strongly affecting the diffusion rate was also encountered. At a weak perturbation (K ≪ 1), we discovered a very peculiar diffusion regime with the diffusion rate D=K2/3 as in the opposite limit of a strong (K ≫ 1) uncorrelated perturbation, but in contrast to the latter, the new regime involves strong correlations and exists for a very short time only. We have no definite explanation of such a controversial behavior.

Separatrix conservation mechanism for nonlinear resonance in strong chaos

We propose a simple, approximate theory of the fairly general mechanism for the separatrix conservation of nonlinear resonance, which leads to the complete suppression of global diffusion despite the strong local chaos of motion. This theory allows the separatrix splitting angle to be plotted against system parameters and, in particular, yields their values at which the separatrix remains unsplit. We present the results of our numerical experiments confirming theoretical conclusions for a certain class of dynamical Hamiltonian systems. New features of chaos suppression have been found in such systems. In conclusion, we discuss the range of application of the proposed theory.

Dynamics of hamiltonian systems under piecewise linear forcing

A two-parameter family of smooth Hamiltonian systems perturbed by a piecewise linear force is analyzed. The systems are represented both as maps and as dynamical systems. Currently available analytical and numerical results concerning the onset of chaos and global diffusion in such systems are reviewed. Dynamical behavior that has no analogs in the class of systems with analytic Hamiltonians is described. A comparison with the well-studied dynamics of a driven pendulum is presented, and essential differences in dynamics between smooth and analytic systems are highlighted.

Chaos in a driven pendulum under asymmetric forcing

An analysis of the stochastic layer in a pendulum driven by an asymmetric high-frequency perturbation of fairly general form is continued. Analytical expressions are found for the amplitudes of secondary harmonics, and their contributions to the amplitude of the separatrix map responsible for onset of dynamical chaos are evaluated. Additional evidence is presented of the previously established fact that the secondary harmonics completely determine the stochasticl-ayer width when the primary frequencies lie in certain intervals. The mechanism of the onset of chaos in the vicinity of zeros of Melnikov integrals is shown to be substantially different as compared to the previously analyzed case of symmetric perturbation.

Structure of the stochastic layer in a driven pendulum

An analysis of the stochastic layer in a driven pendulum is extended to the case when the separatrix map contains both single-and double-frequency harmonics. Resonance invariants of the first three orders are found for the double-frequency harmonic. Combined with the previously known single-frequency invariants, they can be used to obtain further information about the layer, in particular, to examine the neighborhoods of zeros of Melnikov integrals.

Fine structure of splitting of the separatrix of a nonlinear resonance

This report is a continuation of an analysis, initiated elsewhere V.V. Vecheslavov and B. V. Chirikov, Zh. Éksp. Teor. Fiz. 114, 1516 (1998) [JETP 86, 823 (1998)], of the effect of splitting of the separatrix of a nonlinear resonance for the model of standard mapping, based on results of direct measurements of the splitting angle α(K), where K is the system parameter. Measurements were made in the previously used wide range 0.1≳α≳10−208 (1⩾K⩾0.0004), but with significantly higher relative (better than 1050) and average (∼10−55) accuracy. This procedure made it possible to substantially refine the effects observed in Ref. 1 and construct qualitatively new empirical dependences providing reliable extrapolation of the data obtained for the angle and the invariant in the intermediate asymptotic limit K≲10−2 beyond the limits of the investigated region. The results obtained by us can be useful for further development of the theory of separatrix splitting and formation of the stochastic layer of a nonlinear resonance.