Emergence of extreme events in coupled systems with time-dependent interactions
Dynamics of two or more nonlinear interacting systems being a subject of this paper is a very interesting topic. Usually a general solution to equations of motion of such a system cannot be found in symbolic form and so numerical methods are applied. In this work, the system of two Stuart-Landau (SL) oscillators in the presence of time-dependent interactions is studied and Mathematica software is used to find numerical solutions to the equations of motion for some chosen values of the system parameters. The authors consider two cases of time-varying coupling strengths. In the first case parameter determining the interactions is a periodic function of time and so the interaction depends on time explicitly. In the second case parameter determining the interactions depends on the distance or coordinates which are functions of time and so such a case is considered as ``implicit time dependence''. Note that in the first case interaction of two SL oscillators depends on the parameter $d=1+\gamma$ that is a function of time because $\gamma=f\cos (\Omega t)$. As the parameter \(d\) appears in the denominator of the term describing this interaction becomes very large for $f=0.999$ when $\cos (\Omega t)=-1$ or $\gamma=-0.999$. So it is quite natural that large amplitude oscillations may occur that are considered as extreme events, and this results are confirmed by the corresponding numerical solutions of the equations of motion. At the same time the authors introduce the ``quasi-stable equilibrium points'' a sense of which is not clear. It seems such equilibrium points are determined as solutions of the equations which are obtained if the right-hand sides of equations (3) are equated to zero. But these equations contain function of time $\gamma=f\cos (\Omega t)$ and so the corresponding ``equilibrium points'' are functions of time, as well. The question arises what is the sense of such solutions and what does it mean stability of such ``equilibrium points''? The authors equate the Jacobian matrix (5) to zero and state that ``solving this equation, we get the stability of the equilibrium points in the systems''. This statement is very strange because the Jacobian matrix is a function of gamma and so it is a function of time. So it would be good to give a definition of stability which is used to define the corresponding criteriums of stability. This can help to understand the results and to reproduce all the calculations if a reader will be interested in their checking. Summarizing, one can state that the problem is very interesting but the results obtained are very questionable and must be checked carefully.
scientific article; zbMATH DE number 7469352
scientific article