2006, Turaev, Dmitry, Vladimirov, Andrei, Zelik, Sergey

A new type of stable dynamic bound state of dissipative localized structures is found. It is characterized by chaotic oscillations of distance between the localized structures, their phase difference, and the center of mass velocity.

2011, Turaev, Dmitry, Vladimirov, Andrei G., Zelik, Sergey

We study interaction of well-separated oscillating localized structures (oscillons). We show that oscillons emit weakly decaying dispersive waves, which leads to formation of bound states due to subharmonic synchronization. We also show that in optical applications the Andronov-Hopf bifurcation of stationary localized structures leads to a drastic increase in their interaction strength.

2021, Yanchuk, Serhiy, Wolfrum, Matthias, Pereira, Tiago, Turaev, Dmitry

An equilibrium of a delay differential equation (DDE) is absolutely stable, if it is locally asymptotically stable for all delays. We present criteria for absolute stability of DDEs with discrete timedelays. In the case of a single delay, the absolute stability is shown to be equivalent to asymptotic stability for sufficiently large delays. Similarly, for multiple delays, the absolute stability is equivalent to asymptotic stability for hierarchically large delays. Additionally, we give necessary and sufficient conditions for a linear DDE to be hyperbolic for all delays. The latter conditions are crucial for determining whether a system can have stabilizing or destabilizing bifurcations by varying time delays.