Hopf–Hopf bifurcation and chaos in delay-coupled reservoir computing system with two delays

Abstract

Delay-coupled systems have recently made progress in the field of neuromorphic computing, which can effectively process unprecedented amount of data at high speed. It is owing to the nonlinear relationship between input and output to a certain extent. That is, more complicated dynamic system may bring higher performance. Research shows that the performance of delay-coupled reservoir computing system with two delays can surpass single delayed reservoir computing system. In this article, we mainly study Hopf–Hopf bifurcation and chaos of delay-coupled reservoir computing system. Then we analyze the performance difference caused by the different number of delays from the dynamics perspective. For the analysis of Hopf–Hopf bifurcation, we firstly draw the critical curves by DDE-BIFTOOL and obtain Hopf–Hopf bifurcation points. Subsequently, we unfold and classify aforesaid bifurcation points by amplitude equations of its normal form, which is obtained via method of multiple scales. Finally, we verify our theoretical analysis by numerical simulations. For the analysis of chaos, by plotting the bifurcation diagram, we prove the existence of chaos and determine how the system enters the chaotic state with a specific parameter changing. In conclusion, we find two different types of Hopf–Hopf bifurcation and the route to chaos through Period-Doubling bifurcation. From our analysis, we can determine which parameter has a greater impact on the system and guide us to set the parameters in practical applications.