Andrzej L. Kawczyński

Period adding and broken Farey tree sequence of bifurcations for mixed-mode oscillations and chaos in the simplest three-variable nonlinear system

A detailed study of the simplest three-variable model exhibiting mixed-mode oscillations and chaos is presented. We show that mixed-mode oscillations appear due to a sequence of bifurcations which is characterized by a combination of the Farey tree that is broken by chaotic windows and period adding. This scenario is supported by a family of one-dimensional return maps. The model also exhibits hysteresis between stable steady state and mixed modes.

Complex mixed-mode periodic and chaotic oscillations in a simple three-variable model of nonlinear system.

A detailed study of a generic model exhibiting new type of mixed-mode oscillations is presented. Period doubling and various period adding sequences of bifurcations are observed. New type of a family of 1D (one-dimensional) return maps is found. The maps are discontinuous at three points and consist of four branches. They are not invertible. The model describes in a qualitative way mixed-mode oscillations with two types of small amplitude oscillations at local maxima and local minima of large amplitude oscillations, which have been observed recently in the Belousov-Zhabotinsky system. (c) 2000 American Institute of Physics.

Molecular dynamics simulations of nonequilibrium interactions between two model thermally activated reactions

The results of molecular dynamics simulations of nonequilibrium interactions between two thermally activated reactions which use the same reactant are presented. It is shown that a fast reaction with low activation energy may significantly reduce the rate of reaction with higher activation energy. Our results indicate that the nonequilibrium interactions between reactions may be important for modeling the behavior of complex systems.

Application of the molecular‐dynamics technique for simulations of an oscillating thermochemical system

The results of molecular‐dynamics simulations of transient thermochemical oscillations in closed, nonadiabatic systems are presented. A system of 600 molecules has been studied. The simulations confirm qualitatively oscillatory behavior predicted by the phenomenological model. The quantitative agreement fails because far from equilibrium properties of the system appearing in simulations are missing in the phenomenological picture.

New type of the source of travelling impulses in two-variable model of reaction–diffusion system

A two-variable model of a one-dimensional, open, excitable, finite reaction–diffusion system describing time–space evolution of traveling impulses is investigated. It is shown that depending on the size of the system, the traveling impulse after reflection can generate either a source of decaying traveling impulses or a stationary periodical structure. A continuous increase of the size of the system causes periodical repetitions of these patterns. The chemical model is realistic and can become a stimulus for seeking the described effect in experiments.

Stationary periodical structure emitting an infinite number of traveling impulses in a model of a one-dimensional infinite excitable reaction-diffusion system.

A two-variable model of a one-dimensional infinite excitable reaction-diffusion system describing an expanding stationary periodical structure emitting traveling impulses is presented. The model is based on two coupled catalytic (enzymatic) reactions. The chemical scheme consists of mono- and bimolecular reactions.

Scalings of mixed-mode regimes in a simple polynomial three-variable model of nonlinear dynamical systems

We describe scaling laws for a control parameter for various sequences of bifurcations of the LSn mixed-mode regimes consisting of single large amplitude maximum followed by n small amplitude peaks. These regimes are obtained in a normalized version of a simple three-variable polynomial model that contains only one nonlinear cubic term. The period adding bifurcations for LSn patterns scales as 1/n at low n and as 1/n2 at sufficiently large values of n. Similar scaling laws 1/k at low k and 1/k2 at sufficiently high values of k describe the period adding bifurcations for complex k(LSn)(LS(n + 1)) patterns. A finite number of basic LSn patterns and infinite sequences of complex k(LSn)(LS(n + 1)) patterns exist in the model. Each periodic pattern loses its stability by the period doubling bifurcations scaled by the Feigenbaum law. Also an infinite number of the broken Farey trees exists between complex periodic orbits. A family of 1D return maps constructed from appropriate Poincaré sections is a very fruitful tool in studies of the dynamical system. Analysis of this family of maps supports the scaling laws found using the numerical integration of the model.

Stochastic transitions between attractors in a tristable thermochemical system: competition between stable states

We present the thermochemical model in the tristable regime which is characterized by three distinct attractors: two stationary states and a limit cycle. Transitions between the attractors are studied on the basis of the stochastic dynamics at the mesoscopic level, in which internal fluctuations are described by means of the master equation. Simulations of the stochastic dynamics follow the master equation, which has the specific form due to the necessary description of a continuous spectrum of temperature changes related to the exchange of energy with a thermostat. The mean first passage times are calculated for transitions from the limit cycle to each stable steady state. The effect of competition between the states is demonstrated: the duration of the passage can be strongly increased if the system during the transition to one state is also visiting the second competitive one. The variation of the passage times is studied for systems with different reaction heat, and the obtained dependence is explained by the location of the attractors and repellers in the phase space of system temperature and composition. The proportion between the mean passage times for the two states can be related to their relative strength of attraction.

Oscillons localized inside breathing periodical structures in a two-variable model of a one-dimensional infinite excitable reaction-diffusion system.

A two-variable model of a one-dimensional (1D), infinite, excitable, reaction-diffusion system describing oscillons localized inside an expanding breathing periodical structure emitting traveling impulses is presented. The model is based on two coupled catalytic (enzymatic) reactions.

Periodic perturbations of chaotic dynamics

Small periodic perturbations of a chemical model exhibiting infinite number of period adding bifurcations are investigated. At some periods of the perturbations chaotic trajectories become periodic.