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.     

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. Zh. éksp. Teor. Fiz. 112, 1132–1146 (September 1997) Published in English in the original Russian journal. Reproduced here with stylistic changes by the Translation Editor.     

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=K ²/3 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.     

High-precision measurement of separatrix splitting in a nonlinear resonance

We present the results of numerical modeling and a theoretical analysis of the splitting of a nonlinear-resonance separatrix in the intermediate asymptotic region for the standard-map model. Direct measurements of the splitting angle α(K), where K is the small parameter of the system, have been carried out over a huge range, 0.1≳α≳10⁻²⁰⁸ (1⩾K⩾0.0004), with a relative accuracy greater than one part in 10⁻²⁵ and an average accuracy of roughly one part in 10⁻³⁰. This made it possible to compare in detail our results with those of the existing asymptotic theory and to detect a number of new effects. We find a relatively simple empirical expression for the α vs. K dependence in the intermediate asymptotic region, and this region proves to be surprisingly broad: K≲10⁻². We also study the effect of noise, in particular, errors in measuring the angle, which proved to be much more significant and complicated than expected. Finally, we point out unresolved questions and possible directions of research involving this problem. Zh. éksp. Teor. Fiz. 114, 1516–1531 (October 1998)     

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.     

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.     

Formation of a chaotic layer of nonlinear resonance by a two-frequency perturbation

A new effect [V. V. Vecheslavov, Zh. éksp. Teor. Fiz. 109, 2208 (1996) (JETP 82, 1190 (1996)]—the appearance of low-frequency secondary harmonics in the separatrix mapping of a system—is discussed in detail for the example of a pendulum with a two-frequency perturbation. It is shown that there exist regions of values of the perturbation parameters where these harmonics make the main contribution to the formation of the chaotic layer of the fundamental resonance. The results of analytical and numerical determinations of the amplitudes of the secondary harmonics are compared. Pis’ma Zh. éksp. Teor. Fiz. 63, No. 12, 989–994 (25 June 1996)     

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.     

Suppression of dynamic chaos in Hamiltonian systems

The paper describes the results of a recent numerical study on the canonical mapping with a sawtooth force. The dynamic effects of the formation of invariant resonance structures of various orders, whose presence prevents the development of global chaos and restricts momentum diffusion in the phase space, are discussed. The dynamic situation near an integer resonance separatrix in the neighborhood of the critical state is studied, and the conditions responsible for the stability of this separatrix in the critical state are determined. Along with the mapping, the related continuous Hamiltonian system is considered. For this system, the separatrix mapping and the Mel’nikov-Arnold integral are introduced, whose analysis facilitates understanding the reasons responsible for the unusual dynamics. This dynamics is shown to be preserved under substantial saw shape changes. Relevant new problems and open questions are formulated.     

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.