Basic Principles of Nonlinear Dynamics
Fundamental concepts of nonlinear dynamics are described in terms of outline mathematical theory of stability of the ordinary differential equations used in analysis of chemical and electrochemical kinetics. The bistable and oscillatory instabilities are
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Basic Principles of Nonlinear Dynamics
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Concise Vocabulary of Nonlinear Dynamics
In this section, we shall briefly explain the most fundamental terms necessary to understand the concepts of nonlinear dynamics. As stated in Introduction, the term “dynamic self-organization” means generally the spontaneous formation of order in the domain of time and/or space, when the system is maintained sufficiently far from equilibrium. There are two reasons for this condition: the thermodynamic and kinetic one. The first one is justified by the second law of thermodynamics: since the creation of any order, including dynamical self-organization phenomena, is associated with the decrease in entropy, there must exist, in the same system, a dissipative process characterized with entropy production at least compensating this decrease. This condition was the basis for the new term, introduced by the Nobel prize winner, I. Prigogine: dissipative structures [1]. In fact, all the manifestations of self-organization under nonequilibrium conditions, like the formation of temporal or spatiotemporal patterns, are dissipative structures, i.e., they emerge only as long as the sufficient dissipation of energy occurs. The formation of the dissipative structures is thus a dynamic phenomenon. In order to exhibit selforganization, the kinetic characteristics of this process must however meet also the following additional conditions: (1) the (usually) nonlinear dependence between the driving forces and the resulting flows1; (2) the presence of feedback loops, i.e., of autocatalysis and/or autoinhibition in the kinetic mechanism. Mathematically, the dynamical systems are defined in terms of differential equations which can be
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In spite of nonlinear nature of dynamics of vast majority of processes leading to selforganization, in rare cases it can occur also due to linear nonequilibrium phenomena; an example is the concentration gradient appearing in the multicomponent system in the presence of imposed temperature gradient, due to thermal diffusion which is a process of linear characteristics. In the following, we shall however avoid these rare cases and focus on nonlinear phenomena. M. Orlik, Self-Organization in Electrochemical Systems I, Monographs in Electrochemistry, DOI 10.1007/978-3-642-27673-6_1, # Springer-Verlag Berlin Heidelberg 2012
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1 Basic Principles of Nonlinear Dynamics
ordinary ones (abbreviated as ODE), if the dynamic variables depend solely on time, or partial ones (PDE) if the variables depend also on the spatial coordinates. In this chapter we shall focus on the ODEs properties, while the systems requiring analysis in terms of PDEs will be described in Chap. 1 of volume II. Let us now repeat those three fundamental conditions of self-organization: the irreversible course of the process, the (usually) nonlinearity of its dynamic characteristics, and the presence of appropriate feedback loops in its mechanism. Let us assume that the differential equations meeting all these conditions were formulated in terms of appro
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