The paper presents a simple mathematical model called a coupled map lattice (CML). For some range of its parameters, this model generates complex, spatiotemporal behavior which seems to be chaotic. The main purpose of the paper is to provide results of stability analysis and compare them with those obtained from numerical simulation. The indirect Lyapunov method and Lyapunov exponents are used to examine the dependence on initial conditions. The net direction phase is introduced to measure the symmetry of the system state trajectory. In addition, a real system, which can be modeled by the CML, is presented. In general, this article describes basic elements of environment, which can be used for creating and examining methods of chaos controlling in systems with spatiotemporal dynamics.

Many real phenomenona preserves the properties of chaotic dynamics. However, unambiguous determination of belonging to a group of chaotic systems is difficult and complex problem. The main purpose of this paper is to present compound method of time series classification which is basically directed to the detection of chaotic behaviors. The method has been designed for differentiation of three types of time series: chaotic, periodic and random. Our approach assumes, that more reliable information about the dynamics of the system will provide the compilation of several methods, than any individual. This paper focuses on choosing a good set of methods and analysis of their results. In our investigation, we used the following methods and indicators: time delay embedding, mutual information, saturation of system invariants, the largest Lyapunov exponent and Hurst exponent. We checked the validity of the methods applying them to three kinds of basic systems which generate chaotic, periodic and random time series. As a summary of this paper, all selected methods and indicators computed for generated times series have been summarized in the table, which gives the authors a possibility to conclude about type of observed behavior.

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This paper presents a mathematical model for a piston flow reactor based on the material balance law using partial differential equations. A more general, nondimensional variant of the model is also derived. The finite difference method and coupled map lattice are used to create numerical algorithms to simulate spatio-temporal behavior in the studied system. The paper also includes a stability analysis of the proposed algorithms and results of numerous numerical simulations, done in order to compare both methods and to visualize the spatio-temporal behavior of the reactor and the effects of different model parameters. Discussion of the obtained results and comparison of both algorithms is also provided.