By Professor Dr. H. J. Korsch, Professor Dr. H.-J. Jodl (auth.)
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Extra resources for Chaos: A Program Collection for the PC
It is important to realize that this simple criterion also applies to the stability of periodic k-cycles Xo, XI, . . , Xk = Xo with Xn = f(xn-I, r). Each member of this k-cycle is a fixed point x* of th k-times iterated map x* = fk(X*, r) = f(1( ... f (x*, r) ... 13. Pitchfork bifurcation: A stable period-one fixed point x* loses stability at a critical value of the parameter T = Tl, where the slope 1f'(x*)1 is unity, and a pair of period-two fixed points x~ and x~ is born. 83) for x* = X n , n = 0, ...
In coordinate space the former fill a hyperbolically bounded region, whereas the latter an annular region bounded by an elliptical caustic. 3] shows that these caustic curves are indeed conic sections, confocal with the boundary curve. For increasing values of Ipl, the area covered by the trajectory contracts to a tiny annular region elose to the boundary, and we observe behavior similar to that of a whispering gallery. Trajectories on the separatrix pass exactly through the focal points. Another interesting aspect of the ellipse is revealed by examining elosed diametric two-bounce orbits.
Another interesting aspect of the ellipse is revealed by examining elosed diametric two-bounce orbits. The impacts occur at opposite sides of the boundary curve at normal incidence. Two such orbits can be distinguished. The first, along the short diameter, is stable. This means that an orbit whose initial conditions 50 3. ;l~~'R~f:~~;";~~'~r:;;~ -H;ii~~~:: '~" ""):'I':~! 'i~;' . :t:. ~. :i;(~t:-ti';~'.. •.. ; ': ~l\:J'/;-;~'~. 5. Stadium billiard: Poincare section of a single trajectory. DAT). vary slightly from the elosed orbit always remains elose to it.
Chaos: A Program Collection for the PC by Professor Dr. H. J. Korsch, Professor Dr. H.-J. Jodl (auth.)