By John Guckenheimer (auth.), C. Marchioro (eds.)
Lectures: J. Guckenheimer: Bifurcations of dynamical systems.- J. Moser: a number of points of integrable.- S. Newhouse: Lectures on dynamical systems.- Seminars: A. Chenciner: Hopf bifurcation for invariant tori.- M. Misiurewicz: Horseshoes for non-stop mappings of an interval.
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Additional resources for Dynamical Systems
Example text
The map -1 f = noYos not a homeomorphism due to the kinks of @ we will discuss at some length. T is a map of the circle which is . It is this circle map f which We will want to see what the bifurcation theory of circle maps tell us about the behavior of the solutions to the forced van der Pol equation. The second example which we consider in this section is the Lorenz attractor. This is a global attractor for the flow of the following system of differential equations: jr = -lox 4; = 5 where U + ax -y - -8z/3 10y - xz + xy.
About qualitative properties of 4. We start with observations Consider an annulus A of moderate size surrounding the limit cycle of the van der Pol equation. properly, then 0 will map A annulus which is much thinner. parameter E increases. 2 illustrates A and its image under In passing to a map of the circle we want to approximate 0 0. by a map with an image of 0 image. , by a map Y of rank 1 with a one dimensional We assume that an approximating Y can be found with the property that each inverse image of a point is a smooth curves which connects the two boundary components of A.
If x then the coordinate of f(x) while if x E E 2i+l IZi, -I 1 n is given by dropping the first coordinate of x; [ 2 i , ~ ] , the coordinate of x is given by dropping the first coordinate of x and replacing each j in the expansion by (n-1)-j. image of x If no is a turning point, this allows the itinerary of x to be calculated from its base n expansion in such a way that the mth address is determined frcan the first m places in the base n expansion of x. from the behavior at the turning points, the map Apart f is topologically con- jugate to the shift m m on the space of (one-sided) sequences of n symbols in a particularly nice way.