Dynamics Reconstruction In The Presence Of Bistability By Using Reservoir Computer

Reservoir computing is an important framework in machine learning.It is simple in structure,fast in training,and has been wide applied in the field of nonlinear dynamics,such as predicting the short-term evolution of chaotic systems and reconstructing the long-term dynamic characteristics of chaotic systems.However,most of previous works are using reservoir computing to learn the dynamic system with fixed parameters,and rarely involve the dynamic evolution when system parameters change.This complex and interesting phenomenon has always been the focus of nonlinear dynamics.In this paper,we take coupled chaotic oscillators as an example which usually display rich dynamics such as synchronizations and bifurcations when the coupling strength changes.Dynamics reconstruction of the coupled system without relying on a model is a difficult problem in the field of nonlinear dynamics,especially in the presence of bistability.Following the previous works,we design a reservoir computer with parameter input channel consisting of the coupling strength and an indicator parameter.The indicator parameter is used to distinguish the possible coexisting dynamical states.Using a ring of coupled R?ssler oscillators,we demonstrate the power of the reservoir computer in dynamics reconstruction and predicting the transitions between different dynamical states.Specifically,reservoir computers are trained by time series of several coupled strengths and successfully reconstruct the complete bifurcation diagrams and phase relationship of the original coupled system.We find that a single one reservoir computer is enough to reconstruct the dynamics in the presence of bistability.In addition,we also investigate the robustness of reservoir computers when reconstructing different dynamics,which shows the nontrivial dependence of the robustness on parameters and dynamical states.