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Explicit Periodic Solutions in a Delay Differential Equation

Anatoli Ivanov, Sergiy Shelyag

Abstract

We construct stable periodic solutions for a simple form nonlinear delay differential equation (DDE) with a periodic coefficient. The equation involves one underlying nonlinearity with the multiplicative periodic coefficient. The well-known idea of reduction to interval maps is used in the case under consideration, when both the defining nonlinearity and the periodic coefficient are piece-wise constant functions. The stable periodic dynamics persist under a smoothing procedure in a small neighborhood of the discontinuity set. This work continues the research in recent paper [7] on stable periodic solutions of differential delay equations with periodic coefficients.

Explicit Periodic Solutions in a Delay Differential Equation

Abstract

We construct stable periodic solutions for a simple form nonlinear delay differential equation (DDE) with a periodic coefficient. The equation involves one underlying nonlinearity with the multiplicative periodic coefficient. The well-known idea of reduction to interval maps is used in the case under consideration, when both the defining nonlinearity and the periodic coefficient are piece-wise constant functions. The stable periodic dynamics persist under a smoothing procedure in a small neighborhood of the discontinuity set. This work continues the research in recent paper [7] on stable periodic solutions of differential delay equations with periodic coefficients.
Paper Structure (7 sections, 2 theorems, 11 equations, 1 figure, 1 table)

This paper contains 7 sections, 2 theorems, 11 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Main Results
  4. Explicit Periodic Solutions
  5. Smoothing of Nonlinearities
  6. Numerical Demonstration
  7. Discussion

Key Result

theorem 1

Suppose that the parameters $a_1,a_2,p_1,p_2$ are such that the inequality (b) is satisfied and $a_1>a_2$. Then DDE (DDE) has an asymptotically stable slowly oscillating periodic solution. The periodic solution is generated by the initial function $\phi(s)\equiv h_*, s\in[-1,0],$ where $h_*$ is give

Figures (1)

  • Figure 1: Slowly oscillating piece-wise affine solution

Theorems & Definitions (2)

  • theorem 1
  • theorem 2