Random Perturbations of Dynamical Systems
Many notions and results presented in the previous editions of this volume have since become quite popular in applications, and many of them have been “rediscovered” in applied papers. In the present 3rd edition small changes were made to the chapt
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The Multidimensional Case
1 Slow Component Lives on an Open Book Space In Chap. 8 we considered stochastic and deterministic perturbations of twodimensional dynamical systems with one first integral. Let us consider now an (m + n)-dimensional system ˙ X(t) = B(X(t))
(1.1)
with n first integrals z 1 (x), . . . , z n (x); i.e. smooth functions such that the scalar products ∇z 1 (x)·B(x), . . . , ∇z n (x)·B(x) are identically zero. (Our notations will be a little different: we’ll write the time argument in parentheses rather than a subscript, subscripts being reserved for coordinate numbers; and we’ll denote the identification mapping with a Gothic letter.) If, as in the case m = n = 1, we identify all points x within every connected component of m-dimensional level surfaces {x : z 1 (x) = const, . . . , z n (x) = const} we obtain a space Γ of n dimensions; Γ consists, typically, of some number of n-dimensional “faces” having the structure of a manifold, that join, sometimes several at a time, at “faces” of smaller dimensions. Such a space equipped with the natural topology is called an open book (see, e.g., Ranicki [1]); its n-dimensional faces are called pages, and the faces of smaller dimensions form the binding of the book. Let Y be the corresponding identification mapping. We can introduce (local) coordinates on Γ taking z 1 (x), . . . , z n (x) as the first n coordinates, and the number i(x) of the page containing the point Y(x) as its (n + 1)-st, discrete coordinate (local coordinates because one page can “flow” into another). The open book Γ is the space on which we should consider the “slow component” of the motion corresponding to small perturbations of system (1.1). Let us consider an example and draw some pictures. Let m = 1, n = 2, the system (1.1) having the form ⎧ 1 X˙ (t) = b1 (X 1 (t), X 2 (t), Z(t)), ⎪ ⎪ ⎨ (1.2) X˙ 2 (t) = b2 (X 1 (t), X 2 (t), Z(t)), ⎪ ⎪ ⎩ ˙ Z(t) = 0,
M.I. Freidlin, A.D. Wentzell, Random Perturbations of Dynamical Systems, Grundlehren der mathematischen Wissenschaften 260, c Springer-Verlag Berlin Heidelberg 2012 DOI 10.1007/978-3-642-25847-3_9,
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9. The Multidimensional Case
where, for every fixed z, the system X˙ i (t) = bi (X 1 (t), X 2 (t), z), i = 1, 2, has a 2 first integral H(x, z) = H(x1 , x2 , z) (that is, i=1 bi (x1 , x2 , z) · ∂H/∂xi ≡ 0), the function H(x, z) going to ∞ as |x| → ∞. The two first integrals of the system (1.2) are z and H. If Yz (x), for a fixed z, is the identification mapping associated with the first integral H(x, z), the identification mapping Y corresponding to the two first integrals z and H is given by Y(x, z) = (Yz (x), z). Suppose that for z1 < z < z2 the function H has two minima at the points x1 (z) and x2 (z), and the system X˙ i (t) = bi (X 1 (t), X 2 (t), z), i = 1, 2, three equilibrium points: centers x1 (z), x2 (z), and a saddle point x3 (z). After identifying all points in every connected component of a level set of the function H in every horizontal section we get a graph with three vertices Ok (z) = Yz (xk (z)) and thre
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