Theoretical and Computational Advances in Nonlinear Dynamical Systems
Advances in Mathematical Physics
After the success of last year's special issue, we have received more than 30 submissions for TCAN 2018. Twelve articles, containing the studies in chaos and synchronization, nonlinear evolution equations, and applied dynamical systems, are accepted after strict review process. A new Guest Editor, Christos Volos, from Aristotle University of Thessaloniki, was invited to serve in the areas of chaotic systems and synchronization. We believe these articles, including their bibliographic resources,
... will substantially improve the quality of our special issue and show wide interest to the readers in nonlinear communities. Chaos belongs to the field of "Nonlinear Oscillations Theory," which was initiated in the previous century. The experiment that boosted the consideration of chaotic behavior was due to Lorenz . In 1961, working in a simplified model of atmospheric transfer with three nonlinear differential equations, he observed numerically that when making very small changes in the initial conditions he got a huge effect on their solutions. It was one evidence of the main properties of chaotic dynamics which was later known as sensitive dependence on initial conditions, also known as "Butterfly Effect." This property in fact had already been investigated from the topological point of view by Poincaré who described it in his monograph "Science and Method"  . For many years the property of chaos became undesirable, since it reduced the predictability of the chaotic system over long time periods. However, the scientific community was gradually becoming aware of this type of dynamical behavior. Some experiments, where abnormal results had been previously explained in terms of experimental error or additional noise, were evaluated for an explanation in terms of chaos. In the mid-70s, the term deterministic chaos was introduced by Li and Yorke in a famous paper entitled "Period Three Implies Chaos"  . Since then, a huge number of studies in chaotic phenomena and dynamical systems that produce chaos have been published. The dynamics of a system displays chaotic behavior; when it never repeats itself, and even if initial conditions are correlated by proximity, the corresponding trajectories quickly become uncorrelated. As such, the possibility of two (or more) chaotic systems oscillating in a coherent and synchronized way seems to be not an obvious phenomenon. However, there are sets of coupled chaotic oscillators in which the attractive effect of a sufficiently strong coupling can counterbalance the trend of the trajectories to diverge. As a result, it is possible to reach full synchronization in chaotic systems since they are coupled by a suitable dissipative coupling. Chaos synchronization began in the mid-80s about coupling of discrete and continuous identical systems, evolving from different initial conditions      . These works immediately received a great deal of attention from the scientific community and opened up a wide range of applications outside the traditional scope of chaos and nonlinear dynamics research. Since then, various synchronization methods and several new concepts necessary for analyzing synchronization have been developed. In this special issue, three articles are dedicated to the investigation of chaotic systems and their synchronization.