Generalized synchronization: Master-slave relationship in three coupled systems

Abstract

Synchronization is an important phenomenon for complex, biological, and physical systems such as the brain, i.e., Parkinson’s disease, heart beating, hand-clapping, power grids, lasers, and many others. Intuitively, we can express synchronization as strong correlations between coupled systems. We can state two scenarios in this manner. One is synchronization between identical systems, which is called complete synchronization; the other is the synchronization between the non-identical systems, called generalized synchronization. In this thesis, initially, we considered the two coupled systems and calculated the critical coupling value for the generalized synchronization analytically. More precisely, the Lorenz system drives two R¨ossler systems. We investigated the critical coupling value for synchronization numerically. However, real-world examples are much more complex. The most straightforward case was the two coupled systems for the generalized synchronization, and next, we focus on three coupled systems. In particular, suppose that we have three coupled one Lorenz and two R¨ossler systems. In our example, the Lorenz system drives the first R¨ossler system, and first R¨ossler system drives the second R¨ossler system, and finally, the second R¨ossler system also drives the Lorenz system. We calculated the critical coupling of the whole system for generalized synchronization and analyzed the time series for each system.