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 $(\star)$.
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Disclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.