Further remarks on Markus-Yamabe instability for time-varying delay differential equations∗

We study the stability of time-varying delay differential equations of type x?(t)= Ax(t - t(t)), when the delay t (t) takes values in an interval [0, m], for some tm > 0, and A is a n × n real matrix. The fundamental question that we consider is the following: is the system exponentially stable when every individual (constant-delay) system x?(t)= Ax(t - t), for t ? [0, m], is exponentially stable? This is nothing else than the so-called Markus-Yamabe instability. The answer to this question is ‘no’ for one-dimensional systems, as illustrated in the literature. The situation is more complicated for n-dimensional systems, n = 2, and the previous question remains open for a general matrix A and a general tm as above. Nevertheless in this paper we show that the answer is still ‘no’ for particular classes of A and tm.