Download PDF by Jens Lorenz: Deterministic and Random Evolution

By Jens Lorenz

ISBN-10: 1626180148

ISBN-13: 9781626180147

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Determines a trajectory in the state space [0, 1), which we denote by {yn }n=0,1,... or simply by yn . 1. Graph of the function B(y) To study the Bernoulli shift, the binary representation of numbers y is very useful. 1. Every 0 ≤ y < 1 can be written in the form y= ∞ ∑ bj 2−j where bj = 0 or bj = 1 . 1) j=1 This representation of y is unique if one forbids that bj = 1 for all large j. 1) determines a number y ∈ [0, 1). One can prove this result by modifying the arguments which lead to the familiar decimal representation y= ∞ ∑ dj 10−j where dj ∈ {0, 1, 2, .

Going into the coordinate frame where the center of mass rests, removes six variables. Equivalently, one obtains the one–body problem. The motion takes place in the plane orthogonal to the constant vector of angular momentum, reducing from six variables to four. Constancy of the energy and the size of the angular momentum leads to a system of two variables, r(t) and θ(t). 8). This system is difficult to solve, but one can eliminate time and obtain an equation for dr dr/dt = . dθ dθ/dt The solution of this equation finally leads to Kepler’s ellipse.

With these settings we claim that x0 = s(y0 ) implies x1 = s(y1 ). Proof: We have x1 = 4x0 (1 − x0 ) = 4s(y0 )(1 − s(y0 )) = 4 sin2 (πy0 ) cos2 (πy0 ) = sin2 (2πy0 ) = sin2 (π((2y0 ) mod 1)) = sin2 (πy1 ) = s(y1 ) ⋄ An application of this lemma and our results on the Bernoulli shift yields the following for the logistic map: a) There is a dense denumerable set T1 ⊂ [0, 1] so that x0 ∈ T1 implies xn → 0. 2 named after the American mathematical physicist Mitchel Feigenbaum (1944 – ) 50 Jens Lorenz b) There is a dense denumerable set T2 ⊂ [0, 1] so that x0 ∈ T2 implies that xn becomes a periodic cycle for large n.

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Deterministic and Random Evolution by Jens Lorenz


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