Delay-Induced Transient Oscillation (DITO) and Metastable Behavior
Delay-induced transient oscillations (DITOs) are oscillations that occur in bi- or multi-stable system of delayed differential equations as it converges to one of its stable equilibria, oscillations that do not exist when the delay vanishes.
Metastable oscillations are transient oscillatory patterns (or states) that attract neighboring initial conditions and last for a “long time” almost without change. These oscillatory patterns seem to be stable: to detect their instability, it is necessary to observe the system over a seemingly endless time window. Such long-lasting transients would be undistinguishable from (nearly) periodic solutions. A mathematical definition of metastability needs a quantification of “long time”. The most common approach is to consider a one-parameter family of systems and to compare the transient time duration for different values of the parameter. For instance, if ε >0 is a real parameter, then an oscillatory pattern is said to be metastable if...