The Applications Of Singular Perturbation Method To Slow-fast Systems

As physicists and applied Mathematicians confront with a lot of difficulties concerned with nonlinear controlling equation, variable coefficient and complex known or unknown nonlinear boundary conditions-perturbation problem of higher derivative with a small pa-rameter. To solve it, we have to use singularly perturbed methods as approximation, value solution or combined. In my paper, we mainly introduce some basic conceptions and the-ory to discus the solution's nature of the slow-fast system by the method and technique in Singular perturbation natures of slow-fast systems. We generally introduce the perturbation theory and the research backgrounds of slow-fast systems, skills and methods in part 1, while the basic theory in slow-fast systems and the applications of singular perturbation method to slow-fast systems in part 2, included regular solutions near slow-fast curves, generalized normal linearization theorem of Bonckaert, approximation solution of slow-fast systems, and the periodic solution of van der Pol equation, and so on.Slow-fast systems is a kind of important singular perturbation problems, which play an essential role in Math and Biology such as relaxation oscillation model, beating heart model, nerve impulse model,closure model leaves, storage and release of Calcium in cytoplasm, electronic circuit model.Slow-fast systems are vector fields of the form: where f:Rn×Rm×R+(?)Rn,g:Rn×Rm×R+(?)Rm are C∞maps, f(x,y,0)≠0,ε> 0 is a small parameter.To study the behavior of solutions, we introduce a few definitions,Definition 1 (Limit periodic set) Let Xμbe aμ-family of vector fields on a manifold M, withμ∈U (U can also be a manifold, but think of U (?) Rp+1 withμ= (ε,λ),Γ(?) Mis a limit periodic set of Xμforμ=μ0 if there is (μn)n≥1 withμn→μ0, n→∞and (?)n≥1 there is a limit cycleΓn, so thatΓn→Γ, n→∞for the Hausdorff topology.Definition 2 A slow-fast cycle of family (2.8) is a limit periodic set of a layer equation, which we get by takingε= 0 in (2.8).However we also use some important theorems as Poincare theorem and nonlinear Tak-ens theorem. Let us introduce these theorems.Theorem 1 (Normal Linearization Theorem of Takens) Let p=(u0,λ0)∈γλ0, ifD(X0,λ0,) has a non-zero eigenvalue at p, so for every k∈N and each Ck center manifold W, there is a localCk coordinates (u, v)near at point p, where p= (0,λ0),W=v=0,X can be written as: with k, l∈Ck, kis always positive, if the D(X0,λ0) eigenvalue is negative at p, The coordinate change depends in a Ckway on (ε,λ).Theorem 2 (Generalized Normal Linearization Theorem of Bonckaert)Let X(x, y,ε,λ) be a smoothλ-family of vector fields on R3,withλ∈∧(?) Rp. Suppose that, for eachλ∈∧, Xλhas the following properties:ⅰ) Xλ{x, y,ε)-X(x, y,ε,λ) has the function F as a first integral (i.e. Xλis tangent to the foliation dF(x,ε)= 0), where F(x,ε)= xpεq,with (p,q)= (0,1) or p,q∈N1 and relatively prime.ⅱ) Xλ(0,0,0)= 0 and D(Xλ)0has exactly one non-zero eigenvalue of which the related eigenspace is given by x=ε=0. Let Wλbe a Ck family of center manifolds of respectively Xλat 0,withk∈N1.Then there exists a A-dependent local Ck coordinate changeφof the form with and a non-zero Ck function f(x,y,ε,λ) such that.φ*X is given by withY of class Ck, Y·λ= 0 (i.e. Y is a smooth A-family of two-dimensional vector fields), Y·F= 0 andφ(W)= y=0.Theorem 3 Consider the initial value problem For f and g, we take sufficiently smooth vector functions in x,y and t; the dots represent (smooth) higher-order terms inε.a). We assume that a unique solution of the initial value problem exists and suppose this holds also for the reduced problem with solutions x(t),y(t).b. Suppose that 0= g(x, y, t) is solved by y=φ(x, t), whereφ(x, t) is a continuous function and an isolated root. Also, suppose that y=φ(x, t) is an asymptotically stable solution of the equation dy d(?)= g(x,y, t) that is uniformin the parameters x∈D and t∈R+.c. y(0) is contained in an interior subset of the do-main of attraction of y=φ(x, t) in the case of the parameter values x= x(0), t= 0. Then, we have with d and L constants independent ofε.At last,we construct an asymptotic periodic solution of van der Pol equation by singular perturbation method.