By E. B. Davies

Whereas the examine of the warmth equation is a classical topic, this e-book units a precedent because the first account of dramatic advancements made in contemporary years in our quantitative figuring out of a subject principal to differential equations.

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**Extra resources for Heat Kernels and Spectral Theory**

**Example text**

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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