Deterministic and Random Evolution (Mathematics Research

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If we define b > 0 by b2 = a2 − c2 then we obtain 1= ( x )2 ( y )2 . a b This is the well–known equation for an ellipse with semi–axes a and b in Cartesian coordinates. Polar Coordinates. Denote polar coordinates centered at F1 by (r, θ) and let P = (x, y) denote a point on the ellipse with polar coordinates (r, θ). The distances of P to the foci are d1 = |F1 − P | = r, + d2 = |F2 − P | = 2a − r . If ϕ = π − θ then the cosine theorem in the triangle F1 P F2 yields (2a − r)2 = 4c2 + r2 − 4cr cos ϕ .

We will ignore the exact solution formula, however. Instead, we will derive an approximate solution. The process gives insight into the two different time scales that are present. 5) is u′ = εi u. Its solution uhom (t) = eit/ε = cos(t/ε) + i sin(t/ε) 1 and 0 ≤ t ≤ 2π, then uhom (t) goes through 50 full varies fast. For example, if ε = 50 oscillations. 5) is − i sin t ε which goes through only one oscillation for 0 ≤ t ≤ 2π. One says that uhom (t) = eit/ε varies on the fast time scale whereas the above forcing term varies on the slow time scale.

Countably infinite. All three sets Sj are dense in [0, 1). Now consider a trajectory yn = B n (y0 ) and its long time behavior. If y0 ∈ S1 then yn = 0 for all large n. The point y = 0 is the only fixed point of B. If y0 ∈ S2 \ S1 then yn becomes a periodic cycle for large n . If y0 ∈ S3 then yn does neither approach a fixed point nor a periodic cycle. Since all three sets are dense in the interval [0, 1) we see that the long time behavior of a trajectory depends, in some sense, discontinuously on the initial value y0 .