Regular Dynamics Research Articles

Controlling chaos by the system size

Despite the ubiquity of physical systems evolving on time-dependent spatial domains, understanding their regular and chaotic dynamics is still in a rudimentary state. While chaos implies that the system’s behavior can be altered by small perturbations, this sensitivity proves to be useful for control purposes. Here we report on the experimental discovery of a novel mechanism to control chaos by time-variation of the system (spatial domain) size: depending upon the rate of the latter, the chaotic state may be completely prevented. Our experimental observations are disentangled with theoretical insights and numerical modeling, which also reveals the ability to control spatio-temporal chaos, thus making the findings relevant to a wide range of natural phenomena.

Cycles in asymptotically stable and chaotic fractional maps

The presence of the power-law memory is a significant feature of many natural (biological, physical, etc.) and social systems. Continuous and discrete fractional calculus is the instrument to describe the behavior of systems with the power-law memory. Existence of chaotic solutions is an intrinsic property of nonlinear dynamics (regular and fractional). Behavior of fractional systems can be very different from the behavior of the corresponding systems with no memory. Finding periodic points is essential for understanding regular and chaotic dynamics. Fractional systems do not have periodic points except fixed points. Instead, they have asymptotically periodic points (sinks). There have been no reported results (formulae) which would allow calculations of asymptotically periodic points of nonlinear fractional systems so far. In this paper, we derive the equations that allow calculations of the coordinates of the asymptotically periodic sinks.

Chaotic vibrations of size-dependent flexible rectangular plates.

A mathematical model describing nonlinear vibrations of size-dependent rectangular plates is proposed. The plates are treated as the Cosserat continuum with bounded rotations of their particles (pseudo-continuum). The governing partial differential equations (PDEs) and boundary/initial conditions are obtained using the von Kármán geometric relations, and they are yielded by the energetic Hamilton principle. The derived mixed-form PDEs are reduced to ordinary differential equations and algebraic equations (AEs) using (i) the Galerkin-Krylov-Bogoliubov method (GKBM) in higher approximations, and then they are solved with the help of a combination of the Runge-Kutta methods of the second and fourth order, (ii) the finite difference method (FDM), and (iii) the Newmark method. The convergence of FDM vs the interval of the space coordinate grids and of GKBM vs the number of employed terms of the approximating function is investigated. The latter approach allows for achieving reliable results by taking account of almost infinite-degree-of-freedom approximation to the regular and chaotic dynamics of the studied plates. The problem of stability loss of the size-dependent plates under harmonic load is also tackled.

Emergence of Self-Organized Dynamical Domains in a Ring of Coupled Population Oscillators

We show that interactions of inherently chaotic oscillators can lead to coexistence of regular oscillatory regimes and chaotic oscillations in the rings of coupled oscillators provided that the level of interaction between the oscillators exceeds a threshold value. The transformation of the initially chaotic dynamics into the regular dynamics in a number of the coupled oscillators is shown to result from suppression of chaos by separation of certain oscillation periods from the continuous spectra, which are characteristic of chaotic oscillations.

Second-order Saddle Dynamics in Isomerization Reaction

The role of second-order saddle in the isomerization dynamics was investigated by considering the $$E-Z$$ isomerization of guanidine. The potential energy profile for the reaction was mapped using the ab initio wavefunction method. The isomerization path involved a torsional motion about the imine (C-N) bond in a clockwise or an anticlockwise fashion resulting in two degenerate transition states corresponding to a barrier of 23.67 kcal/mol. An alternative energetically favorable path ( $$\sim$$ 1 kcal/mol higher than the transition states) by an in-plane motion of the imine (N-H) bond via a second-order saddle point on the potential energy surface was also obtained. The dynamics of the isomerization was investigated by ab initio classical trajectory simulations. The trajectories reveal that isomerization happens via the transition states as well as the second-order saddle. The dynamics was found to be nonstatistical with trajectories exhibiting recrossing and the higher energy second-order saddle pathway preferred over the traditional transition state pathway. Wavelet based time-frequency analysis of internal coordinates indicate regular dynamics and existence of long-lived quasi-periodic trajectories in the phase space.

Classical and Quantum Dynamical Manifestations of Index-$$2$$ Saddles: Concerted Versus Sequential Reaction Mechanisms

The presence of higher-index saddles on a multidimensional potential energy surface is usually assumed to be of little significance in chemical reaction dynamics. Such a viewpoint requires careful reconsideration, thanks to elegant experiments and novel theoretical approaches that have come about in recent years. In this work, we perform a detailed classical and quantum dynamical study of a model two-degree-of-freedom Hamiltonian, which captures the essence of the debate regarding the dominance of a concerted or a stepwise reaction mechanism. We show that the ultrafast shift of the mechanism from a concerted to a stepwise one is essentially a classical dynamical effect. In addition, due to the classical phase space being a mixture of regular and chaotic dynamics, it is possible to have a rich variety of dynamical behavior, including a Murrell – Laidler type mechanism, even at energies sufficiently above that of the index-2 saddle. We rationalize the dynamical results using an explicit construction of the classical invariant manifolds in the phase space.Supplementary InformationThe online version contains supplementary material with details on computation of the invariant manifolds which available at 10.1134/S1560354721020052.

Construction of Explicit Symplectic Integrators in General Relativity. II. Reissner–Nordström Black Holes

Abstract In a previous paper, second- and fourth-order explicit symplectic integrators were designed for a Hamiltonian of the Schwarzschild black hole. Following this work, we continue to trace the possibility of construction of explicit symplectic integrators for a Hamiltonian of charged particles moving around a Reissner–Nordström black hole with an external magnetic field. Such explicit symplectic methods are still available when the Hamiltonian is separated into five independently integrable parts with analytical solutions as explicit functions of proper time. Numerical tests show that the proposed algorithms share desirable properties in their long-term stability, precision, and efficiency for appropriate choices of step size. For the applicability of one of the new algorithms, the effects of black hole’s charge, the Coulomb part of the electromagnetic potential and the magnetic parameter on the dynamical behavior are surveyed. Under some circumstances, the extent of chaos becomes strong with an increase of the magnetic parameter from a global phase-space structure. No variation of the black hole’s charge other than the Coulomb part affects the regular and chaotic dynamics of the particles’ orbits. A positive Coulomb part more easily induces chaos than a negative one.

Dynamical Systems and Topology of Magnetic Fields in Conducting Medium

We discuss application of contemporary methods of the theory of dynamical systems with regular and chaotic hyperbolic dynamics to investigation of topological structure of magnetic fields in conducting media. For substantial classes of magnetic fields, we consider well-known physical models allowing us to reduce investigation of such fields to study of vector fields and Morse–Smale diffeomorphisms as well as diffeomorphisms with nontrivial basic sets satisfying the A axiom introduced by Smale. For the point–charge magnetic field model, we consider the problem of the separator playing an important role in the reconnection processes and investigate relations between its singularities. We consider the class of magnetic fields in the solar corona and solve the problem of topological equivalency of fields in this class. We develop a topological modification of the Zeldovich funicular model of the nondissipative cinematic dynamo, constructing a hyperbolic diffeomorphism with chaotic dynamics that is conservative in the neighborhood of its transitive invariant set.

The role of individual neuron ion conductances in the synchronization processes of neuron networks

The partial phase synchronization (sometimes called cooperation) of neurons is fundamental for the understanding of the complex behavior of the brain. The lack or the excess of synchronization can generate brain disorders like Parkinson’s disease and epilepsy. The phase synchronization phenomenon is strongly related to the regular or chaotic dynamics of individual neurons. The individual dynamics themselves are a function of the ion channel conductances, turning the conductances into important players in the process of neuron synchronized health depolarization/repolarization processes. It is well known that many diseases are related to alterations of the ion-channel conductance properties. To normalize their functioning, drugs are used to block or activate specific channels, changing their conductances. We investigate the synchronization process of a Hodgkin–Huxley-type neural network as a function of the values of the individual neuron conductances, showing the dynamics of the neurons must be taken into account in the synchronization process. Particular sets of conductances lead to non-chaotic individual neuron dynamics allowing synchronization states for very weak coupling and resulting in a non-monotonic transition to synchronized states, as the coupling strength among neurons is varied. On the other hand, a monotonic transition to synchronized states is observed for individual chaotic dynamics of the neurons. We conclude the analysis of the individual dynamics of isolated neurons allows the prediction of the synchronization process of the network. We provide alternative ways to achieve the desired network state (phase synchronized or desynchronized) without any changes in the synaptic current of neurons but making just small changes in the neuron ion-channel conductances. The mechanism behind the control is the close relation between ion-channel conductance and the regular or chaotic dynamics of neurons. Finally, we show that by changing at least two conductances simultaneously the control may be much more efficient since the second conductance makes the synchronization possible just by performing a small change in the first. The study presented here may have an impact on new drug development research.

Dynamic Hydraulics in a Drinking Water Distribution System Influence Suspended Particles and Turbidity, But Not Microbiology

Spatial and short-term temporal changes in water quality as a result of water age and fluctuating hydraulic conditions were investigated in a drinking water distribution system. Online measurements of total and intracellular adenosine tri-phosphate (ATP), total and intact cell concentrations measured with flow cytometry (FCM), turbidity, and particle counts were performed over five weeks at five subsequent locations of the distribution system. The high number of parallel FCM and ATP measurements revealed the combined effect of water age and final disinfection on spatial changes in microbiology in the system. The results underlined that regular daily dynamics in flow velocities are normal and inevitable in drinking water distribution systems, and significantly impact particle counts and turbidity. However, hydraulic conditions had no detectable impact on the concentration of suspended microbial cells. A weak correlation between flow velocity and ATP concentrations suggests incidental resuspension of particle-bound bacteria, presumably caused by either biofilm detachment or resuspension from sediment when flow velocities increase. The highly dynamic hydraulic conditions highlight the value of online monitoring tools for the meaningful description of short-term dynamics (day-scale) in drinking water distribution systems.

Power spectrum and form factor in random diagonal matrices and integrable billiards

Triggered by a controversy surrounding a universal behavior of the power spectrum in quantum systems exhibiting regular classical dynamics, we focus on a model of random diagonal matrices (RDM), often associated with the Poisson spectral universality class, and examine how the power spectrum and the form factor get affected by two-sided truncations of RDM spectra. Having developed a nonperturbative description of both statistics, we perform their detailed asymptotic analysis to demonstrate explicitly how a traditional assumption (lying at the heart of the controversy) – that the power spectrum is merely determined by the spectral form factor – breaks down for truncated spectra. This observation has important consequences as we further argue that bounded quantum systems with integrable classical dynamics are described by heavily truncated rather than complete RDM spectra. High-precision numerical simulations of semicircular and irrational rectangular billiards lend independent support to these conclusions.

NON-TRIVIAL DYNAMIC REGIMES OF SMALL (NANO-SCALE) QUANTUM SYSTEMS

Small (but still containing many atoms) quantum systems (traditionally termed nano-systems) are dramatically different from their macroscopic or genuine microscopic (atomic) cousins. Microscopic molecular systems (with a few atoms) obey a regular quantum dynamics (described by time dependent Schrodinger equation), whereas in macroscopic systems with continuous energy spectra, one can expect, also regular, although typically relaxation, dynamic behavior. The topic of our paper is in-between these limits. System behavior becomes non-trivial and manifests a sort of transitions between regular and chaotic dynamics. We show that such dynamic transitions occur when the Loschmidt echo time of life exceeds the typical recurrence cycle period. We illustrate this behavior in the frame work of a few versions of the exactly solvable quantum problem, proposed long ago by Zwanzig. It is based on the study of time evolution of the initially prepared vibrational state coupled to a reservoir with dense spectrum of its vibrational states. In the simplest version of the Zwanzig model, the reservoir has an equidistant spectrum, and the system - reservoir coupling matrix elements are independent of the reservoir states. We generalize the model to include into consideration the coupling of the initially prepared single state to system phonon excitations. The coupling results to temperature dependent broadening and decay of the echo components. Another generalization is to replace a single level by two states coupled to the Zwanzig reservoir. We anticipate that the basic ideas inspiring our work can be applied to a large variety of interesting for the applications nano-systems (e.g., dissipative free propagation of excitations along molecular chains, or as a model for exchange reactions).

О ТЕРМИЧЕСКОМ РЕЖИМЕ ПРИЗЕМНОГО СЛОЯ ВОЗДУХА В БАССЕЙНЕ РЕКИ ДЕБЕД (АРМЕНИЯ)

The features of the thermal regime of the surface air layer in the Debed river basin are considered. A statistical analysis of the average annual and average seasonal values of air temperature from 1964 to 2018 was carried out, two periods were identified, their time course was shown. The analysis was carried out using data from six meteorological stations representing the lowland, mountain and high-mountain climatic zones of the Debed river basin. A correlation was obtained between the absolute altitude and the monthly average values of air temperature for January and July, which can be used to assess the thermal conditions of unexplored or poorly studied territories and for cartography. The time course of average values of air temperatures for the seasonal period has been studied. Analysis of trend lines of temporal changes in air temperatures shows that in all situations on the territory of the basin as a whole, there is a tendency of temperature growth. Moreover, with a range of interannual fluctuations, a break in the course of temperatures in the early to mid 1990 is clearly visible, after which their significant increase began. It turned out that a significant increase in seasonal temperatures is observed especially over the period 1993-2018, which means that the annual warming after the mid 1990 occurred primarily due to summer and spring seasons. The regular dynamics indicates that in the studied area in terms of temperatures, a tendency of softening winters, a decrease in the water content of rivers, aridization of the climate. The results obtained can be used to assess the regularities of the spatial-temporal distribution of the temperature of the study area, to clarify the thermal balance, for the rational use of heat resources, as well as in the development of strategic programs for longterm analysis.

Regular Collective Dynamics of Research Collaboration

Regular Collective Dynamics of Research Collaboration

REGULAR DYNAMICS AND BOX-COUNTING DIMENSION FOR A RANDOM REACTION-DIFFUSION EQUATION ON UNBOUNDED DOMAINS

In this article, we study a random reaction-diffusion equation driven by a Brownian motion with a wide class of nonlinear multiple. First, it is exhibited that the weak solution mapping $ L^{2}( {\mathbb{R}}^N) $ into $ L^{p}( {\mathbb{R}}^N) \cap H^{1}( {\mathbb{R}}^N) $ is Hölder continuous for arbitrary space dimension $ N\geq 1 $, where $ p&gt;2 $ is the growth degree of the nonlinear forcing. The main idea to achieve this is the classic induction technique based on the difference equation of solutions, by using some appropriate multipliers at different stages. Second, the continuity results are applied to investigate the sample-wise regular dynamics. It is showed that the $ L^{2}( {\mathbb{R}}^N) $-pullback attractor is exactly a pullback attractor in $ L^{p}( {\mathbb{R}}^N) \cap H^{1}( {\mathbb{R}}^N) $, and furthermore it is attracting in $ L^{\delta}( {\mathbb{R}}^N) $ for any $ \delta\geq2 $, under almost identical conditions on the nonlinearity as in Wang et al [<span class="xref"><a href="#b31" ref-type="bibr">31</a></span>], whose result is largely developed in this paper. Third, we consider the box-counting dimension of the attractor in $ L^{p}( {\mathbb{R}}^N) \cap H^{1}( {\mathbb{R}}^N) $, and two comparison formulas with $ L^2 $-dimension are derived, which are a straightforward consequence of Hölder continuity of the systems

On regular and chaotic dynamics of a non--symmetric Hamiltonian system of a coupled Duffing oscillator with balanced loss and gain

A non--symmetric Hamiltonian system of a Duffing oscillator coupled to an anti-damped oscillator with a variable angular frequency is shown to admit periodic solutions. The result implies that -symmetry of a Hamiltonian system with balanced loss and gain is not necessary in order to admit periodic solutions. The Hamiltonian describes a multistable dynamical system—three out of five equilibrium points are stable. The dynamics of the model is investigated in detail by using perturbative as well as numerical methods and shown to admit periodic solutions in some regions in the space of parameters. The phase transition from periodic to unbounded solution is to be understood without any reference to -symmetry. The numerical analysis reveals chaotic behaviour in the system beyond a critical value of the parameter that couples the Duffing oscillator to the anti-damped harmonic oscillator, thereby providing the first example of Hamiltonian chaos in a system with balanced loss and gain. The method of multiple time-scales is used for investigating the system perturbatively. The dynamics of the amplitude in the leading order of the perturbation is governed by an effective dimer model with balanced loss and gain that is non--symmetric Hamiltonian system. The dimer model is solved exactly by using the Stokes variables and shown to admit periodic solutions in some regions of the parameter space.

Deep learning of chaos classification

We train an artificial neural network which distinguishes chaotic and regular dynamics of the two-dimensional Chirikov standard map. We use finite length trajectories and compare the performance with traditional numerical methods which need to evaluate the Lyapunov exponent (LE). The neural network has superior performance for short periods with length down to 10 Lyapunov times on which the traditional LE computation is far from converging. We show the robustness of the neural network to varying control parameters, in particular we train with one set of control parameters, and successfully test in a complementary set. Furthermore, we use the neural network to successfully test the dynamics of discrete maps in different dimensions, e.g. the one-dimensional logistic map and a three-dimensional discrete version of the Lorenz system. Our results demonstrate that a convolutional neural network can be used as an excellent chaos indicator.

Atomistic hybrid particle-field molecular dynamics combined with slip-springs: Restoring entangled dynamics to simulations of polymer melts.

In hybrid particle-field (hPF) simulations (J. Chem. Phys., 2009 130, 214106), the entangled dynamics of polymer melts is lost due to chain crossability. Chains cross, because the field-treatment of the nonbonded interactions makes them effectively soft-core. We introduce a multi-chain slip-spring model (J. Chem. Phys., 2013 138, 104907) into the hPF scheme to mimic the topological constraints of entanglements. The structure of the polymer chains is consistent with that of regular molecular dynamics simulations and is not affected by the introduction of slip-springs. Although slight deviations are seen at short times, dynamical properties such as mean-square displacements and reorientational relaxation times are in good agreement with traditional molecular dynamics simulations and theoretical predictions at long times.

Regular dynamics for stochastic Fitzhugh-Nagumo systems with additive noise on thin domains

<p style='text-indent:20px;'>This paper is devoted to bi-spatial random attractors of the stochastic Fitzhugh-Nagumo equations with additive noise on thin domains when the terminate space is the Sobolev space. We first established the existence of random attractor on regular space and then show that the upper semi-continuity of these attractors under the Sobolev norm when a family of <inline-formula><tex-math id="M1">\begin{document}$ (n+1) $\end{document}</tex-math></inline-formula>-dimensional thin domains degenerates onto an <inline-formula><tex-math id="M2">\begin{document}$ n $\end{document}</tex-math></inline-formula>-dimensional domain.

Features of reproduction of the shrub vole &lt;i&gt;Microtus majori&lt;/i&gt; Thomas, 1906 in natural conditions

Aim . The purpose of this research based on the study of the ecological and biological characteristics of the shrub vole is to expand knowledge about patterns of microevolution in space, ecology and biology on the basis of which to assess the adapted characteristics of the Central North Caucasian population of the shrub vole ( Microtus majori Thomas, 1906) in natural conditions. Material and Methods. The material for this work was obtained during expeditions and stationary studies carried out by the authors in the Russian Central Caucasus from 2013 to 2017. During this time, a survey of various biotopes of the broad- leaved forest belt was carried out, taking into account both anthropogenic loads and the season of the year. Results . Stationary studies were carried out, taking into account the season of the year, in deciduous forest belt. Reproduction of shrub voles in the population studied takes place over five months (April ‐ August). Overwintered voles reproduce within four months (from early April to late July), females bearing three litters. Accordingly, we can say that the greatest importance in maintaining the viability of the population of this vole species falls on the overwintered yearlings of the first spring cohorts. Conclusions . The data obtained indicate the regular dynamics of the (seasonal) age structure of the population, associated with the reproduction and death of individuals of certain age groups. When conditions of existence in the population of the shrub vole studied change, it reacts with an increase in the intensity of reproduction, a seasonal change in the number of embryos in a litter, a change in sex and age structure: it is sensitive to any fluctuations in its habitat.