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Extra resources for Advanced Vector Analysis with Application to Mathematical Physics
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The set (x1 ≥ 0, x2 ≥ 0) is invariant for this system, and there are two equilibria: the coexistence equilibrium (x∗1 = γ/δ, x∗2 = α/β) and the extinction equilibrium (x1 = 0, x2 = 0). There is a first integral for this system, namely, φ(x1 , x2 ) = xγ1 xα 2 exp (−δx1 − βx2 ). Thus, any solution starting away from an equilibrium is periodic with a period T (c), where c = φ(x1,0 , x2,0 ), and T (c) = o(− log c) as c → 0. Random perturbations of the data in this system take the form x˙ 1,ε = α(y(t/ε))x1,ε − β(y(t/ε))x1,ε x2,ε , x˙ 2,ε = γ(y(t/ε))x2,ε + δ(y(t/ε))x1,ε x2,ε , where y is an ergodic Markov process in a measurable space (Y, C).
Bottom: A histogram of 100 samples of randn, the MATLAB function for normal random variables, is plotted for comparison. The solution to equation (15) can be found using the variation–of–constants formula as xε (t) = xε (0)e t 0 a1 (s,y(s/ε)) ds t + e t u a1 (s,y(s/ε)) ds 0 a0 (u, y(u/ε)) du. (16) Consider the stochastic processes t γ1ε (t) = t a1 (s, y(s/ε)) ds, γ2ε (t) = 0 0 a2 (s, y(s/ε)) ds. If the limits 1 T →∞ T T lim 0 ak (s, y)ak (s, y )R(y, dy )ρ(dy) ds = bkk , 20 Introduction for k = 0, 1, and 1 T →∞ T T lim 0 [a0 (s, y)a1 (s, y ) + a1 (s, y)a0 (s, y )]R(y, dy )ρ(dy) ds = b01 exist, then the two–dimensional process (γ0ε (t), γ1ε (t)) converges weakly in C to a two–dimensional Gaussian process with independent increments, say (γ0 (t), γ1 (t)), for which Eγk = 0, Eγk2 (t) = 2 bkk , and Eγ0 (t)γ1 (t) = 2 b01 .
Applying noise can uncover a great deal of information about the underlying system. As we’ve seen, all possible static states will (probably) be visited by any trajectory of a gradient system perturbed by noise, but only one will be visited otherwise. , F (x, α + 1) ≡ F (x, α) for all α. We consider the case where the parameter α is slowly changing in time, say α = t/T (ε), where log T (ε) = O(1/ε) and ε 1. 9 1 Figure 10. g1 and g2 as functions of α. Note that g1 ( 12 ) = g2 (0) = g. are of the form shown in Figure 10.