Directional Lyapunov Exponents – the first universal method designed for extended studies, experiments and control of dynamics of complex networks and coupled nonlinear systems investigatoins

Abstract

Abstract What is most fundamental in oscillations of Complex Networks of nonlinear coupled systems (CN) are phenomena connected with different types of their collective behavior. Patterns discernible in the dynamics of such systems, for instance different types of synchronization, chimera states and others, have attracted researchers since their discovery. However, they have remained invariably challenging in terms of designing methods with which to monitor and control them. In this article, the author presents the results obtained from applying a novel method—Directional Lyapunov Exponents (DLEs) for the purpose of addressing these challenges. Additionally, the application of DLEs in a wide range of contexts allows the demonstration of features of CNs that were previously unobservable using other methods. Thus far, for instance, the state of complete synchronization of CNs has been recognized as entirely unrelated to a chimera state. The use of DLEs has demonstrated that in the neighborhood of a synchronization manifold, the symmetry of the system dynamics is broken and there can be observed new symmetrical patterns that are shadowing the complete synchronization by chimera type dynamics. Moreover, within the range of chimera existence, DLEs have also revealed another symmetry with a symmetrical flow between desynchronized chimera parts. It has also been proved that in the case of two-node systems, the largest DLE is equivalent to the largest Lyapunov exponent – a metric which has been commonly applied in stability analyses in many branches of science and engineering. Consequently, DLEs have proved to be the most effective tool in investigating CNs, as they expose tendencies of oscillators to synchronize, or split their dynamics, long before the system stabilizes. As such, they have the potential to become the fastest method of scanning a system’s parameters or its initial conditions while looking for chimeras’ existence, as well as other interesting dynamical behaviors. Notably, DLEs represent a very universal approach and can be applied in any type of coupled systems, identical or not, complex or simple ones, as well as in monitoring relative behavior of complex subsystems. Another significant benefit of their application is connected with the way DLEs are arrived at. Since they are derived from the real-time state of a dynamical system and the values of the system variables, they can readily be applied in experiments and CN control.