By Professor Alexander S. Mikhailov, Professor Alexander Yu. Loskutov (auth.)
This textbook is predicated on a lecture path in synergetics given on the college of Moscow. during this moment of 2 volumes, we speak about the emergence and houses of advanced chaotic styles in disbursed lively structures. Such styles may be produced autonomously by means of a method, or may end up from selective amplification of fluctuations because of exterior vulnerable noise. even though the fabric during this ebook is usually defined via subtle mathematical theories, we've got attempted to prevent a proper mathematical variety. rather than rigorous proofs, the reader will often be provided basically "demonstrations" (the time period utilized by Prof. V. I. Arnold) to inspire intuitive realizing of an issue and to give an explanation for why a selected assertion turns out believable. We additionally shunned detailing concrete purposes in physics or in different clinical fields, in order that the ebook can be utilized through scholars of alternative disciplines. whereas getting ready the lecture path and generating this e-book, we had extensive 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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Extra resources for Foundations of Synergetics II: Complex Patterns
1) . 2) exp(-iwT)g(T)dT -00 Suppose that x(t) is a periodic function with period TI. 3) , n=-CX) where WI = 211" ITI and C n are the Fourier coefficients. In this case the autocorrelation function is 1 (T g(T) =T 10 x(t)x(t+T)dt L L 00 = 00 CnCn' exp(-in'WIT) n=-oon=-oo I1T X - T L 0 exp[ -i(n + n')WI t] dt 00 = n=-(X) L IC I exp(-inw 00 CnC_nexp(-inWIT) = n where we have taken into account that C- n = We see that g( T) is periodic with period TI. 4) n=-oo C~ because the function x(t) is real. 5) .
If both eigenvalues Al and A2 are real and have the same sign, the fixed point is a node (Fig. 1). 1. Stable (a) and unstable (b) nodes 1 Usually a dynanIical system has a finite number of attractors. 10). 31 I Fil;. 2. A saddle point a) Fig. 3. Stable (a) and unstable (b) focuses Fig. 4. Center opposite signs, this unstable fixed point is a saddle (Fig. 2). Trajectories I and II, passing through the saddle, are separatrices. When both eigenvalues are complex (but not purely imaginary), the fixed point is a focus (Fig.
In other words, a phase trajectory of an ergodic system uniformly and densely covers all of the region fl. Note that the phase flow of Hamiltonian systems cannot be ergodic in the strict sense of the above definition. Since such systems conserve energy, the phase trajectory must lie on some energetic hypersurface H = E and the flow can be ergodic only within this hypersurface. It there are K additional integrals of motion, the motion of such a system will be restricted to a hypersurface of n - K - 1 dimensions.
Foundations of Synergetics II: Complex Patterns by Professor Alexander S. Mikhailov, Professor Alexander Yu. Loskutov (auth.)