From Discrete Homoclinic To Continuous Homoclinic

we will consider a parameterized dynamical systemand it's one-step method:where e is the step size.(H1) are fixed points of has no eigenvalue on the unit circle. By k_±s,u we denote the numbers of stable eigenvalue and unstable eigenvalue of then there existsδ> 0, for |λ-λ|-|<δwe have(H2)At A = A, the one-step method x_n+1 =ψ(x_n,λ,ε) possesses a transversal connecting orbit (x|-)_Z = ((x|-)_n)_n∈Z which satisfiesIn case of x|-₊ = x|-_-, such an orbit is called homoclinic orbit, otherwise it is called hctcrclinic orbit.In this paper, we only discuss the case of homoclinic orbit. According to bibliography [23], we can obtain the following property.Corollary 1.1. Assume (H1)-(H2), forε> 0, there exists interval (λ₀^-(ε),λ₀⁺(ε)), such that for anyλ∈(λ₀^-(ε),λ₀⁺(ε)), the map (2.2) possesses a transversal homoclinic orbit (x|-_z(λ),λ), and satisfies x|-_Z(λ|-) = x|-_z, whenλ=λ₀^±(ε), the map (2.2) has a one-tangential homoclinic orbit respectively. See figurel we define the operatorΓ| as follows:and thenand the corresponding variational equation:Now we have the following conclusion.Theorem 1.1. Assume (H1) - (H2). Then for 0 <ε<ε₁, for anyλ|-∈(λ₀^- (ε),λ₀⁺(ε)), x|-_Zare ragular solutions of the equationΓ|^{·,λ|,ε} = 0, i.e.Γ|^<sub>x_Z(x|^<sub>Z,λ|^,ε) are homeomorphisms.n=+oo(H3) Assume , where are theunique solution of the equationThen we also define the operator:With this and the operatorΓ|^, we can define the operator F|^:The following theorem contains the main properties of the operator F|^. Theorem 1.2. Assume (H1) - (H3). Then there existsε₁ > 0, such that 0 <ε<ε₁,for anyare regular solutions of the equation F = 0, i.e..are homeomorphisms.(H4) we assume that there existsβ> 0, such thatsatisfies thefollowing estimates:Lemma 1.1. Let F : Y×∧→Z be a C^l mapping from a Banach space Y×∧into some Banach space Z. Assume there exists a Junction y|^<sub>0 :∧→Y such that F_y (y|-(μ),μ) are homeomorphisms for all |μ|<δ₂, and there exist some constantsκ=κ(μ) > 0,σ=σ(μ) > 0, such that for |y-y|-₀(μ)|≤δ₁,|μ|≤δ₂ we havethen for any |μ| <δ₂, F(·,μ) has a unique C^r - smooth zero y|- = y|-(μ) near y|^<sub>0(μ), and the following estimates hold for |y_i-y|^<sub>0(μ)|≤δ₁, i = 1,2:Next we consider the operator:andFor the operator F, the following conclusion holds.Theorem 1.3. Assume (H1) - (H4). Then there exists constantε₃,δ> 0, such that for0 <ε<ε₃, for anyλ|^∈(λ₀^-(ε),λ₀⁺(ε)), the function F has a unique zero (x|-_z(ε),λ|-(ε)) in aδneighborhood . This orbit satisfies the following estimates:Remark 1.1. Define ,let it is the homoclinic orbitφ(·,λ|-(ε),ε). y(t) are the solution of equation (2.1), and the interplation of {y_n}. We hope that y(t) is the homoclinic orbit of equation (2.1), but to date, we unable to complete this proof We hope to carry on this study in our future work.