Dynamics of equations with delayed sampled-data feedback (Q2703958)

The author studies the delay-differential equation NEWLINE\[NEWLINE{dx\over dt}+ x= F(x(t- T))NEWLINE\]NEWLINE under the assumption that \(F\) is a sampled-data function, that is, \(F\) is a \(\delta\)-function concentrated at some point \(\gamma> 0\): NEWLINE\[NEWLINEF(x)= \begin{cases} 0\quad &\text{for }x\neq\gamma,\\ \int^{\tau_2}_{\tau_1} F(x)\,dx= \alpha\;(0<\alpha\leq 1)\quad &\text{for any }\tau_1,\tau_2> 0.\end{cases}NEWLINE\]NEWLINE Under the condition \(\exp(- T)+\alpha j^{-2}> 1\) \((< 1)\) he proves the existence (nonexistence) of a slowly oscillating periodic solution. A condition for the existence of a rapidly oscillating periodic solution is also given. The dynamics of the equation NEWLINE\[NEWLINE{d\over dt} [x(t)- F(x(t- T))]+ x(t)= 0NEWLINE\]NEWLINE is also investigated in the case that \(F\) is a relay function, that is NEWLINE\[NEWLINEF(x)= \begin{cases} 1\quad &\text{for }x< \gamma,\\ 0\quad &\text{for }x\geq j,\end{cases}NEWLINE\]NEWLINE where \(\gamma\) is not negative. Here, the author constructs stable periodic solutions and families of periodic solutions of differential structure.