By Professor Alexander S. Mikhailov, Professor Alexander Yu. Loskutov (auth.)
This textbook is predicated on a lecture direction in synergetics given on the college of Moscow. during this moment of 2 volumes, we talk about the emergence and houses of advanced chaotic styles in disbursed energetic structures. Such styles might be produced autonomously through a method, or may result from selective amplification of fluctuations because of exterior susceptible noise. even supposing the fabric during this ebook is usually defined by means of subtle mathematical theories, we've attempted to prevent a proper mathematical variety. rather than rigorous proofs, the reader will often be provided in basic terms "demonstrations" (the time period utilized by Prof. V. I. Arnold) to motivate intuitive knowing of an issue and to give an explanation for why a specific assertion turns out believable. We additionally avoided detailing concrete purposes in physics or in different medical fields, in order that the ebook can be utilized by way of scholars of alternative disciplines. whereas getting ready the lecture direction and generating this ebook, we had in depth discussions with and requested the recommendation of Prof. V. I. Arnold, Prof. S. Grossmann, Prof. H. Haken, Prof. Yu. L. Klimontovich, Prof. R. L. Stratonovich and Prof. Ya.
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Additional info for Foundations of Synergetics II: Complex Patterns
Sample text
For t ~ tmix the behavior of a system can be predicted with sufficiently good accuracy (which obviously cannot exceed the accuracy € with which the initial position of the point is specified). 4) is an important property of a dynamical system. Today it is actively used as a practical criterion for chaotic motion. 1,26-27]. Generally, the value of h might also depend on the choice of the initial point :VI (0) of the trajectory. A dynamical system can simultaneously possess several different attractors with their own basins and.
Below we give definitions of several dimensions, together with a discussion of the relationship between them. 1-3]. Consider some set A in n-dimensional space.
4). Note that the last type of fixed point is permitted only in conservative systems. In phase spaces of higher dimensions, more complicated types of fixed points are possible. For instance, one can find combinations of the first and second, or of the second and third types. In these cases, the fixed points are the saddle-node (Fig. 5) and the saddle-focus (Fig. 6), respectively. Obviously, attractive fixed points, such as a stable node and a stable focus, are the attractors of dynamical systems.
