Rotating-periodic Solutions For Nonlinear Differential Equations

At the end of the 19th century, H. Poincare introduced the concept of periodic solutions in the study of Three-Body Problem and developed the qualitative theory of differential equations. Since then, researches on peri-odic solutions have always been a central topic in the study of the qualita-tive theory. On basis of periodic solutions, people introduced the concepts of almost-periodic and almost automorphic functions to describe those natural phenomena which are nearly periodic. But not all the natural phenomena can be described by alone periodicity. In fact, some system often exhibit certain symmetry rather than periodicity. In the study of these systems, Y. Li et al. introduced a new concept of periodic system and named it as affine-periodic system[41,54,55,61]. In the present thesis, we consider a special kind of affine-periodic systems-rotating-periodic system, and discuss the existence of rotating-periodic solutions for nonlinear differential equations.In Chapter 2, we consider a(Q,T)-rotating-periodic system x’=f(t,x), (0.0.5) and the auxiliary system of (0.0.5) x’=λf(t,x),(0.0.6) where the function f(t,x):R1×Rn�'Rn is continuous and ensures the uniqueness of solutions with respect to initial values,Q∈O(n),λ∈[0,1]. In Section 2.2, we introduce an existence theorem for rotating-periodic solutions of system 0.0.5.Theorem 0.0.1 Let D(?)Rn be a bounded open set. Assume the follow-ing hypotheses hold for equation (0.0.6). (H1):For each λ∈(0,1], every possible affine-periodic solution x(t) of equa-tion (0.0.6) satisfies x(t)(?)D (?)t; (H2):The Brouwer degree deg(g, D∩Ker(I-Q),0)≠ 0, if Ker(I-Q)≠{0}, where P:Rn�'Ker(I-Q) is an orthogonal projection.Then equation (0.0.5) has at least one (Q, T)-affine-periodic solution x*(t)∈ D for all t.Via the topological degree theory, we proved Theorem 0.0.1 in Section 2.2. This theorem offers a topological method to study the existence of rotating-periodic solutions in theory.When dealing with specific problems,we hope to have a more directly method. Hence in Section 2.3, we proved two different results on basis of Lyapunov function method as follows by using Theorem 0.0.1.Corollary 0.0.2 Consider equation (0.0.5), where the affine-periodic func-tion f:R1×Rn�'Rn is continuous and ensures the uniqueness of solutions with respect to initial values. Assume that there exist C1 functions Vi(x), i=0,1,…, m and σ>0, such that: (H3): For Mi large enough, |<▽Vi(x),f{t,x)>|≥σ (?)|x|≥Mi,i=0,1,…,m, t∈R1. And if Ker(I-Q)≠{0}, |<▽Vi(x),Pf(t,x))}≥σ>0 (?)x∈Ker(I-Q) and |x|≥Mi, i= 0,1, ..., m, t∈R1, where P: Rn�'Ker(I-Q) is an orthogonal projection; (H5): The Brouwer degree deg(▽V0, BM0∩Ker(I-Q), 0)≠ 0, if Ker(I-Q)≠{0}, where Bρ = {p ∈Rn :|p| < ρ}. Then equation (0.0.5) has at least one (Q, T)-affine-periodic solution x*(t).Theorem 0.0.3 Consider equation (0.0.5), where the affine-periodic func-tion f : R1×Rn�'Rn is continuous and ensures the uniqueness of solutions with respect to initial values. Let V : D�' R be a C1 function such that (H6): D is a bounded open set; (H7): There exists a constant a > 0 such that |{▽V{x),f(t,x))|≥σ (?)(t,x)∈R1×(?)D. And if Ker(I-Q)≠{0}, |(▽V(x),Pf{t,x)>|≥σ (?)(t,x)∈R1×*(?)D∩ Ker(I-Q)). where P : Rn�'Ker(I-Q) is an orthogonal projection; (H8): deg(▽V, D∩Ker(I- Q), 0)≠ 0, if Ker(I- Q) ≠{0}. Then equation (0.0.5) has at least one (Q,T)-affine-periodic solution x*(t)∈ D for all t.Corollary 0.0.2 has been proved first by H. Wang et al.[55], hence we give a new proof in Section 2.3 as an application of Theorem 0.0.1. Theorem 0.0.3 is new. Each of these two results offers a method to prove the existence of rotating-periodic solutions by Lyapunov functions. Also, an application of Corollary 0.0.2 has been shown in this section.In Section 2.4, we proved an invariant region principle in rotating-periodic systems. Although this kind of principle has become popular in the study of periodic solutions, it is unknown wether similar results hold in almost periodic or quasi-periodic solutions. Our rotating-periodic solutions might be quasi-periodic. In some sense, this theorem improved the former principle.Theorem 0.0.4 Consider equation (0.0.5), where f:R1×Rn�'Rn is continuous and(Q,T)-affine-periodic. Let D (?) Rn be a bounded open and simply connected set, such that (?)D is piecewisely smooth. Let H(f)denote the hull of f. Assume that (H9):For every (t,p)∈R1×(?)D and h∈H(f), h(t,p) is inward to D; (H10):g(a)≠0 for all a ∈(?)D,where P:Rn�'Ker(I-Q)is an orthogonal projection.Then equation(0.0.5) has at least one (Q, T)-affine-periodic solution x*(t)∈ D for all t.Not all the natural phenomena can be described by continuous systems. Sometimes,the system admits jumps or disconnections.Hence in Chapter 3, by using the theory of time scales, we considered a(Q,T)-rotating-periodic system on time scale T xΔ=f{t,x), (0.0.7) and its auxiliary system xΔ), (0.0.8) where f(t,x) : T×Rn�'Rn is a rd-continuous function and ensures the uniqueness of solutions with respect to initial values, T is a time scale, Q∈ O(n),λ∈[0,1]. In Section 3.2, we proved an existence theorem of rotating-periodic solutions on system (0.0.7).Theorem 0.0.5 Let D C Rn be a bounded open set. Assume the follow-ing hypotheses hold for system (0.0.7). (H11): For each λ∈(0,1], every possible affine-periodic solution x(t) of the auxiliary equation xΔ =λf(t,x) satisfies that if x(t)∈D, then (H12): The Brouwer degree deg(g,D∩Ker(I-Q), 0)≠0, if Ker(I-Q)≠{0}, where P:Rn�' Ker(I-Q) is an orthogonal projection. Then equation (0.0.7) has at least one (Q, T)-affine-periodic solution x*(t) € D for all t∈[0,T]T.As an application of Theorem 0.0.5, we proved the following corollary in Section 3.3 and showed two examples.Corollary 0.0.6 Consider system (0.0.7). Assume that there exists a constant M > 0, such that (x(t),f(t,x(t))> +1/2μ(t)|f(t,x(t))|2≥δ >0 (?)|x(t)|≥M,t∈T. And if Ker(I-Q)≠{0},for all x∈ Ker(I-Q) and {x(t)|≥M,t∈T |(x,Pf{t,x)>|≥δ>0, where P :Rn�'Ker(I-Q) is an orthogonal projection.Then system (0.0.7) has at least one (Q,T)-affine-periodic solution x*(t).