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Dynamics of a linear differential system with piecewise constant argument. (English) Zbl 0976.34062

The authors study the dynamics of a linear differential system with piecewise constant argument \((L)\) \(\dot x(t)=-\alpha R(\theta)x([t])\), with \(\alpha>0\) and \(R(\theta)= \left(\begin{smallmatrix}\cos\theta &-\sin\theta\\ \sin\theta &\cos\theta\end{smallmatrix}\right)\) with \(|\theta |< \frac{\pi}2\). In case \(\alpha =2\cos \theta \), it is shown that there exist star-shaped periodic solutions to (L) if \(\frac\theta\pi\in Q\) and any nontrivial trajectory \(x(t)\) to (L) is everywhere dense on some closed annular region if \(\frac{\theta }{\pi} \notin Q\). It is also proved that any solution to (L) converges to \(0\) as \(t\to \infty\) if and only if \(\alpha<2\cos\theta \).

MSC:

34K13 Periodic solutions to functional-differential equations
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