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)     

The Physical Nature of Mesoscopic Inhomogeneities in Highly Diluted Aqueous Suspensions of Protein Particles

The dispersed composition of water samples prepared by the multiple-dilution technique has been investigated opto-physically using two laser diagnostics methods: dynamic light scattering and laser phase microscopy. We studied suspensions of antibodies (Ab) to the human interferon-gamma (IFNγ) in water with an initial concentration of 0.125 mg ml^?1 (5×10¹⁴ mol cm^?3), which were subjected to 12 successive 100-multiple volume dilutions (C12) in water. Individual antibodies and particles of about 300 nm in size, which can be considered as aggregates of individual antibodies, have been found in the initial suspensions. It turned out that some mesoparticles with a concentration on the order of 10³ cm^?3 exist also in suspensions subjected to C12 dilution. These particles have a refractive index close to typical values of protein refractive indices. A possible explanation of the origin of particles with sizes on the order of several hundreds of nanometers, revealed in highly diluted suspensions, is proposed. On the one hand, some of these particles may be aggregates of antibodies from the initial suspension, which remained (due to the flotation effect) in the bulk of the liquid after dilutions. On the other hand, the appearance of solid contaminant particles at the same mesoscopic scale (i.e., on the order of several hundreds of nanometers) cannot be excluded in the dilution process.

     

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.     

Possibility of the occurrence of dissociation reactions of multiatomic molecules during rapid cooling in an expanding supersonic stream

We investigate the possibility of the occurrence of the reaction involving the dissociation of N₂O and the exchange reaction N₂O+CON₂+CO₂ during rapid cooling of a gas mixture due to supersonic expansion in a nozzle. A numerical solution of the equations for the balance of vibrational energy and the kinetics of the chemical reactions under nonequilibrium conditions is given. We find the values of the parameters that are necessary for the experimental observation of the phenomena considered.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 2, pp. 48–53, March–April, 1973.