By Milos Marek, Igor Schreiber

Surveying either theoretical and experimental elements of chaotic habit, this e-book offers chaos as a version for lots of doubtless random methods in nature. simple notions from the idea of dynamical structures, bifurcation thought and the homes of chaotic ideas are then defined and illustrated by way of examples. A evaluate of numerical tools used either in reports of mathematical types and within the interpretation of experimental info can also be supplied. additionally, an in depth survey of experimental remark of chaotic habit and techniques of its research are used to emphasize common gains of the phenomenon.

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**Example text**

35 Example of non-uniqueness For the differential equation of second order and second degree or y =f(x,Y,Y')= 1 +y y"(1 +y')= 1 the conditions of boundedness and continuity are already infringed for variable y'. (2x)3 go through the point x = 0, y = 1 with the slope y' = - 1. , if the differential equation is of the form Pu(x)Y(°)(x) = r(x) with pn(x) = 1, and if we rewrite the differential equation as a system with n-1 Y(q) = Y(q+1) (q = 1, ... , n - 1), Y(n) = f = r(x) - Z PU(x)Y("11), V=0 then the partial derivatives of f with respect to the y(q) are precisely the coefficient functionspq _ i (x) for q = 1, 2, ...

We next check whether T is a contraction operator. 56) z°(E) - OI <_ P( ) II Z_ 211. Further we can now choose p(x) to be a positive continuous function on J. For p(x)=e-"I x-xoI with a>Ls we have P(x) di J XoP( ) Hence it follows altogether that II TZ - TZII <- KIIZ - ZII with K=Ls< 1. a Therefore the operator T is indeed a contraction in the domain considered. 22. BANACH'S FIXED-POINT THEOREM AND THE EXISTENCE THEOREM FOR ORDINARY DIFFERENTIAL EQUATIONS We have now introduced the concepts needed for formulating a general fixed- point theorem, from which the existence theorem for ordinary differential equations can easily be deduced.

Fig. 15. The two-valued direction field for the differential equation y'= x± 2+x 2 + y 2 23 Fig. 16. The infinitely-many-valued direction field of the tangents to a sine curve Eliminating the parameter by using cos t;= y', sin E=1-y'2 we obtain the differential equation y - j 1 - y' 2 = xy' - y' arc cosy' . 15. NON-UNIQUENESS OF THE SOLUTION For physical applications it is important to know sufficient conditions for uniqueness of the solution of a differential equation. That it is not sufficient for the slope function y' = f(x, y) to be singlevalued and continuous to ensure uniqueness of the solution is shown by the following example.