## Differential and Integral Equations

### Instability of rapidly-oscillating periodic solutions for discontinuous differential delay equations

#### Abstract

We study the equation $$(\star)\qquad \dot{x}(t)=- h(x(t-1)) + f(x(t)) \ \text{ for } \ t\ge 0, \ x_{|_{[-1,0]}}=x_0,$$ where $h$ is an odd function defined by $h(y)$ is equal to $a$ if $0 < y <c$, equal to $b$ if $y \ge c$, $a>b>0$ and $c>0$ and $f$ is an odd ${{\mathcal C}}^1$ function such that $\sup |f(x)| <b$. We first consider the equation $\dot{x}(t)=- h(x(t-1))$, corresponding to $f\equiv 0$. We find the admissible shapes of rapidly-oscillating symmetric periodic solutions and we show that these periodic solutions are all unstable. We then extend these results to our general equation $(\star)$.

#### Article information

Source
Differential Integral Equations, Volume 15, Number 1 (2002), 53-90.

Dates
First available in Project Euclid: 21 December 2012

https://projecteuclid.org/euclid.die/1356060883

Mathematical Reviews number (MathSciNet)
MR1869822

Zentralblatt MATH identifier
1021.34055

Subjects
Primary: 34K20: Stability theory
Secondary: 34K13: Periodic solutions

#### Citation

Akian, Marianne; Bismuth, Sophie. Instability of rapidly-oscillating periodic solutions for discontinuous differential delay equations. Differential Integral Equations 15 (2002), no. 1, 53--90. https://projecteuclid.org/euclid.die/1356060883