Deterministic and Random Evolution (Mathematics Research Developments)

Deterministic and Random Evolution (Mathematics Research Developments)

Jens Lorenz

Language: English

Pages: 188

ISBN: 1626180148

Format: PDF / Kindle (mobi) / ePub

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Kepler’s equation. More than one hundred years after Newton, the astronomer F.W. Bessel introduced a new class of functions, now called Bessel functions, to attack this problem. This is our second subject in this chapter. The equations for planetary motion are deterministic, and Newton’s success contributed to a deterministic world view. Does the whole universe evolve deterministically? Is free will an illusion? We end the chapter with a short discussion. 22 Jens Lorenz 1. Outline Because

equation (4.13) becomes cj = u(µ) = µ + 2 ∞ ∑ 1 n=1 8. j Jj (εj) sin(jµ) . (4.18) Bessel Functions via a Generating Function: Integral Representation In most of today’s texts, the Bessel functions Jj (x) are not defined via the integral representation (4.17). Let us derive the formula (4.17) if one defines the Bessel functions via the generating function ∞ ∑ 1 ) Jn (z)tn . g(z, t) := exp (t − ) = 2 t n=−∞ (z (4.19) Here z and t are complex numbers, t ̸= 0. For fixed z, the function t →

(4.18) and other formulas of this chapter, one obtains the position of a planet as a function of time as a convergent series. Of course, it was tried to go from two bodies to three and more. One can write down a system of equations, extending (4.2) and (4.3), which determines the motion of n bodies. However, it turns out that one can no longer find solutions in terms of convergent series if n ≥ 3. The interesting history of this subject — one of the starting points of chaos theory — is described

The last equation is true since j 2 + 2j(N − j) + (N − j)2 = (j + N − j)2 = N 2 . N j )2 . 178 Jens Lorenz Thus we have shown that (P π)j = πj for 1 ≤ j ≤ N . The special cases j = 0 and j = N are easy to check, and the vector equation P π = π is proved. 2) It remains to prove the uniqueness statement. Assume, then, that there exists a vector β ∈ RN +1 , β ̸= π, with P β = β, ∑ βj = 1, βj > 0 for j = 0, 1, . . . , N . j If we consider the matrix P 2 it is easy to see that all its

wave function, 107 Schr¨ odinger equation, 107 scripts for logistic growth, 79 sensitive dependence on initial conditions, 14 stability and bifurcations, 61 asymptotically stable, 61 asymptotically stable fixed point, 61 fixed points, 61 Jacobian of a fixed point, 62 stable fixed point, 62 stochastic matrix, 8, 105 definition, 105 stochastic model of a simple growth process, 121 Taylor, 15 Taylor’s formula, 15 theory of evolution, 10 time average equal space average for a circle map, 91 time

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