Multiple periodic orbits of high-dimensional differential delay systems

Zhongmin Sun,Lin Li,Weigao Ge

doi:10.1186/s13662-019-2427-3

Zhongmin Sun, Lin Li + Show 1 more

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https://doi.org/10.1186/s13662-019-2427-3

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Abstract

In this paper, we consider differential delay systems of the formx′(t)=−∑s=12k−1(−1)s+1∇F(x(t−s)),\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$x'(t)=-\\sum_{s=1}^{2k-1}(-1)^{s+1} \\nabla F \\bigl(x(t-s) \\bigr), $$\\end{document} in which the coefficients of the nonlinear terms with different hysteresis have different signs. Such systems have not been studied before. The multiplicity of the periodic orbits is related to the eigenvalues of the limit matrix. The results provide a theoretical basis for the study of differential delay systems.

Highlights

In the past several decades, many papers [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16] have studied the existence of periodic solutions of delay differential equations
Proof Let Π, Λ, and Γ be the orthogonal mappings from X to X∞+, X∞, and X∞0, respectively
We can see that the dimension of X∞0 is finite, so we can suppose that zn → φ as zn is bounded

Summary

Introduction

In the past several decades, many papers [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16] have studied the existence of periodic solutions of delay differential equations. By transforming them respectively into associated systems of ordinary differential equations and making analysis by qualitative approaches. They guessed that there should exist 2(n + 1)-periodic solutions to the equation n x (t) = – f x(t – i) , i=1. On the basis of this work, Fei [3, 4] studied the multiple periodic solutions of differential delay equations via Hamiltonian systems. (f1) F satisfies (1.2) and (1.3), (f2) there are M > 0 and a function r ∈ C0(R+, R+) satisfying r(s) → ∞ and r(s)/s → 0 as s → ∞ such that

Variational structure Let

Division of space X and lemmas Let

Main results Denote