Mathematical model of chaotic oscillations and oscillatory entrainment in glycolysis originated from periodic substrate supply.

We study the influence of periodic influx on a character of glycolytic oscillations within the forced Selkov system. We demonstrate that such a simple system demonstrates a rich variety of dynamical regimes (domains of entrainment of different order (Arnold tongues), quasiperiodic oscillations, and chaos), which can be qualitatively collated with the known experimental data. We determine detailed dynamical regimes exploring the map of Lyapunov characteristic exponents obtained in numerical simulations of the Selkov system with periodic influx. In addition, a special study of the chaotic regime and the scenario of its origin in this system was evaluated and discussed.

Nonspectral relaxation in one dimensional Ornstein-Uhlenbeck processes.

The relaxation of a dissipative system to its equilibrium state often shows a multiexponential pattern with relaxation rates, which are typically considered to be independent of the initial condition. The rates follow from the spectrum of a Hermitian operator obtained by a similarity transformation of the initial Fokker-Planck operator. However, some initial conditions are mapped by this similarity transformation to functions which grow at infinity. These cannot be expanded in terms of the eigenfunctions of a Hermitian operator, and show different relaxation patterns. Considering the exactly solvable examples of Gaussian and generalized Lévy Ornstein-Uhlenbeck processes (OUPs) we show that the relaxation rates belong to the Hermitian spectrum only if the initial condition belongs to the domain of attraction of the stable distribution defining the noise. While for an ordinary OUP initial conditions leading to nonspectral relaxation can be considered exotic, for generalized OUPs driven by Lévy noise, such initial conditions are the rule.

Phase reversal in the Selkov model with inhomogeneous influx.

The dynamical reaction-diffusion Selkov system as a model describing the complex traveling wave behavior is presented. The approximate amplitude-phase solution allows us to extract the base properties of the biochemical distributed system, which determines such patterns. It is shown that this relatively simple model could describe qualitatively the main features of the glycolysis waves observed in the experiments.

Density fluctuations and random walks in an overdamped and supercooled simple liquid.

In this work, the short-time dynamics of simple liquid is explored both analytically and numerically with the focus on the interplay between the density fluctuations in a volume surrounding a chosen particle and its random walk motion. The particles interact via the Lennard-Jones potential with parameters corresponding to liquid argon. For large times, analytical calculations based on the fluctuation theory provides an explicit expression reproducing isothermal change of the self-diffusion coefficient in liquid argon corresponding to the experimental data. These results lead to the conclusion that such behavior is based on the reduced mobility of particles reflected in their density fluctuations that can be equivalently achieved in the cases of either low temperatures and pressures (supercooling) or moderate temperatures and high pressures (overdamping).

Nonspectral modes and how to find them in the Ornstein-Uhlenbeck process with white µ-stable noise.

We consider the Ornstein-Uhlenbeck process with a broad initial probability distribution (Lévy distribution), which exhibits so-called nonspectral modes. The relaxation rate of such modes differs from those determined from the parameters of the corresponding Fokker-Plank equation. The first nonspectral mode is shown to govern the relaxation process and allows for estimation of the initial distribution's Lévy index. A method based on continuous wavelet transformation is proposed to extract both (spectral and nonspectral) relaxation rates from a stochastic data sample.

Analytical properties of a three-compartmental dynamical demographic model.

The three-compartmental demographic model by Korotaeyv-Malkov-Khaltourina, connecting population size, economic surplus, and education level, is considered from the point of view of dynamical systems theory. It is shown that there exist two integrals of motion, which enables the system to be reduced to one nonlinear ordinary differential equation. The study of its structure provides analytical criteria for the dominance ranges of the dynamics of Malthus and Kremer. Additionally, the particular ranges of parameters enable the derived general ordinary differential equations to be reduced to the models of Gompertz and Thoularis-Wallace.

Traveling glycolytic waves induced by a temperature gradient and determination of diffusivities for dense media.

Here we consider the spatially extended model incorporating the temperature-dependent autocatalytic coefficient into the Merkin-Needham-Scott version of the Selkov system and show that this model with temperature gradient quite reasonably explains the experimentally detected traveling glycolytic nonstationary waves, which can be attributed as kinematic ones. Additionally, we analyze the influence of possibly incorporating diffusion terms into the equations. It is shown that the value of diffusivity influences the timetable for the birth of new wave and their further evolution. This result could be used as a method for the determination of diffusivity.

Simple model for temperature control of glycolytic oscillations.

We introduce the temperature-dependent autocatalytic coefficient into the Merkin-Needham-Scott version of the Selkov system and consider the resulting equations as a model for temperature-controlled, self-sustained glycolytic oscillations in a closed reactor. It has been shown that this simple model reproduces key features observed in the experiments with temperature growth: (i) exponentially decreasing period of oscillations; (ii) reversal of relative duration leading and tail fronts. The applied model also reproduces the modulations of oscillations induced by the periodic temperature change.

Decomposition of strong nonlinear oscillations via modified continuous wavelet transform.

We propose the modification of the complex wavelet transform adapted for an analysis of strong nonlinear oscillations with a shape far from sinusoidal. It is based on the rotation of transform modulus obtained via usage of the Morlet wavelet in such a way that higher harmonics are merged with a main one. The method is illustrated by application to the analysis of regular and chaotic oscillations generated by the Rössler system.

Analysis of patterns formed by two-component diffusion limited aggregation.

We consider diffusion limited aggregation of particles of two different kinds. It is assumed that a particle of one kind may adhere only to another particle of the same kind. The particles aggregate on a linear substrate which consists of periodically or randomly placed particles of different kinds. We analyze the influence of initial patterns on the structure of growing clusters. It is shown that at small distances from the substrate, the cluster structures repeat initial patterns. However, starting from a critical distance the initial periodicity is abruptly lost, and the particle distribution tends to a random one. An approach describing the evolution of the number of branches is proposed. Our calculations show that the initial pattern can be detected only at the distance which is not larger than approximately one and a half of the characteristic pattern size.

Self-sustained biochemical oscillations and waves with a feedback determined only by boundary conditions.

We discuss the biochemical three-dimensional reaction-diffusion model, which does not provide temporal self-sustained oscillations via reaction terms. However, the self-sustained oscillations and waves could be obtained using the proper boundary conditions for systems with a finite thickness. We have carried out in our numerical simulation the results quite corresponding to the experimental ones. We discuss the range of models for which our approach is applicable.

Wavelet phase synchronization and chaoticity.

It has been shown that the so-called "wavelet phase" (or "time-scale") synchronization of chaotic signals is actually synchronization of smoothed functions with reduced chaotic fluctuations. This fact is based on the representation of the wavelet transform with the Morlet wavelet as a solution of the Cauchy problem for a simple diffusion equation with initial condition in a form of harmonic function modulated by a given signal. The topological background of the resulting effect is discussed. It is argued that the wavelet phase synchronization provides information about the synchronization of an averaged motion described by bounding tori instead of the fine-level classical chaotic phase synchronization.

Hierarchical mean-field model describing relaxation in a small-world network.

This Brief Report presents the hierarchical reaction-diffusion partial differential equations (PDE) system, which reproduces a mean-square displacement and a density relaxation process corresponding to the anomalous diffusion on a small-world network. These results are confirmed by the comparison with the known direct numerical simulations.