By Korsch H., Jodl H., Hartmann T.

This re-creation strives over again to supply readers with a operating wisdom of chaos thought and dynamical structures. It does so via parallel introductory reasons within the e-book and interplay with carefully-selected courses provided at the accompanying disk. The courses permit readers, specifically advanced-undergraduate scholars in physics, engineering, and math, to take on suitable actual platforms fast on their desktops, with out distraction from algorithmic info. For the 3rd version of Chaos: A application assortment for the computer, all of the earlier twelve courses is polished and rewritten in C++ (both home windows and Linux types are included). a brand new software treats kicked platforms, an enormous category of two-dimensional problems.

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IN). 46) where the Gauss bracket [ ] denotes the largest integer number less than or equal to the real number inside. 46) xn = x−1 n−1 − an , an+1 = [x−1 n ], n = 1, 2, . . 47) yields the series of the integers an . 45) at an defines a series of rational approximations x≈ r0 r1 rn , , ... , , ... , there is no rational number with s < sn and x− r rn < x− . 49) A well-known example of such an approximation in terms of continued fractions is the series 3 22 333 355 , , , ... 43). , x lies between two subsequent approximations.

X0 ) . )). 4 Special Topics 33 where the logarithm of the linearized map is averaged over the orbit x0 , x1 ,. . xn−1 . Negative values of the Lyapunov exponent indicate stability, and positive values chaotic evolution, where λL measures the speed of exponential divergence of neighboring trajectories. At critical bifurcation points the Lyapunov exponent is zero. For an interpretation of the Lyapunov exponent, it is instructive to note its relationship to the loss of information during the process of iteration.

It is investigated numerically in Chap. 9 . For increasing values of the parameter r, the fixed points of f 2 can also lose their stability at r2 and bifurcate again into period-four orbits (fixed points of f 4 ), and so on. 4 Special Topics 37 Fig. 13. Pitchfork bifurcation: A stable period-one fixed point x∗ loses stability at a critical value of the parameter r = r1 , where the slope |f (x∗ )| is unity, and a ∗ pair of period-two fixed points x∗ − and x+ is born. x*+ x* x*r1 r Fig. 14. Pitchfork bifurcation: A stable period-one fixed point x∗ loses stability at a critical value of the parameter r = r1 and a pair of period-two fixed ∗ and x∗ is born.

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