Bistability, Period \({\text{n}}\) Solutions, Quasi-Periodic, Phase-Locked Solutions and Chaos in an Optoelectronic Reservoir Computing System with Two Delayed Feedback Loops

In this article, we consider the complicated dynamics and co-dimension two bifurcations of an optoelectronic reservoir computing (RC) system with two delayed feedback loops. RC is a bio-inspired paradigm that uses supervised training to analyze and predict time-dependent data. The dual feedback loops help stabilize the optical and electronic signals, reduce noise, and ensure consistent performance. In simple terms, an optoelectronic RC system with two feedback loops is essential for enhancing the accuracy, stability, and reliability of both optical and electronic systems. Feedback time delays and strength play important roles in the dynamics of optoelectronic RC systems.

We first use DDE-BIFTOOL to draw the bifurcation diagram with respect to two bifurcation parameters, i.e., feedback time delays, \({\tau }_{1}\) ${\tau }_1$ and \({\tau }_{2},\) ${\tau }_2,$ find some Hopf–Hopf bifurcation points. We then classify and unfold these Hopf–Hopf bifurcation points by the method of multiple scales (MMS).

We find three kinds of Hopf–Hopf bifurcation, where the bistability of two coexisting stable periodic solutions and the unusual one of coexisting stable equilibrium and quasi-periodic solution occur. Besides, other complicated dynamics are found, such as Period- \(n\) $n$ solutions, quasi-periodic solutions, phase-locked solutions, and chaos.

These complicated dynamical phenomena are of great significance to the enhancement of the computing performance of the delayed optoelectronic RC system with two feedback loops.