Proportional and Derivative Coupling: a Way to Achieve Synchronization for Coupled Oscillators

In order to achieve synchronization, the concept of proportional coupling and derivative coupling, as analogue to PD control theory, is proposed. The author analyzes from the coupled simple harmonic oscillators to Kuromoto model, and to the coupled Van der Pol oscillators. It is conjectured that only proportional coupling for two or multiple linear and nonlinear coupled oscillators is not sufficient most of the time to achieve synchronization, whereas derivative coupling is the dominant factor for two or multiple linear and nonlinear coupled oscillators to get synchronization. In addition, it is noticed that if there is no coupling, there is no synchronization if starting from different initial conditions. In order to achieve synchronization, coupling is required. However, coupling does not necessary mean it will achieve synchronization, it has to be coupled in a particular way, and otherwise it will not get synchronization. As a result of this study, two major questions for the linear and nonlinear coupled oscillators to get synchronization are proposed. Generally, only proportional coupling ( i.e. spring coupling) is earlier to analyze as it is a simple physics problem. Once we couple them using the derivative coupling approach, it will get major complications. Copyright (c) 2023 The Authors.