Study of a tri-ring Er-doped fiber laser system: Stability, bifurcation, hyper-scroll attractor, and a path to hyperchaos

To develop a hyperchaotic laser generator, we designed and investigated a novel tri-ring Er-doped fiber laser system. The system was assembled from three single-ring Er-doped fiber lasers with two couplers, and a mathematical model was established using a set of six-dimensional nonlinear coupling equations. We mathematically deduced the function of the stable field of each single-ring laser as the pump varied, and presented distributions of the lasers’ stable outputs. We theoretically analyzed the instability of the system using three sets of cubic relation expressions, which were verified by nonlinear function curves. It demonstrated the possible existence of a twin-scroll strange attractor in each laser ring and a hyper-scroll strange attractor in the assembled tri-ring laser system, which was consistent with our numerical result. We found that, as a subsystem, each single-ring laser could maintain its nonlinear dynamics, while the assembled tri-ring system exhibited rich nonlinear dynamic behaviors, such as quasi-periodicity, bifurcation, chaos, and hyperchaos. Lyapunov exponents were used to characterize the system’s dynamic behavior, while fractal dimensions were used to investigate the spatial construction of the dynamics within the system. In the numerical analysis, we evaluated the evolution of the system, starting at a stable state, passing from a quasi-period state, and developing into chaos. This revealed a path to chaos and hyperchaos through a bifurcation scenario by shifting one parameter of the system. Chaotic, stable, and double-periodic bifurcation regions were found after exploring a path to chaos after bifurcation by adjusting the pump level of each laser ring. Two chaotic regions and double-periodic regions were observed when exploring a path toward or away from chaos by adding the coupling level of two rings. Chaotic, stable, and double-periodic bifurcation regions were found after exploring a path toward or away from chaos by varying the gain coefficient and decay rate, respectively. We also found that hyperchaotic waves were accompanied by hyperchaotic moving orbits in the dynamic phase. The strange attractor was characterized by ergodicity, the real-time wave by both complexity and randomness, and the hyperchaotic signal by the wide-band spectrum. The power spectrum clearly revealed the hyperchaotic response, exhibiting numerous frequency peaks that were randomly distributed with varying amplitudes. The assembled tri-ring laser system demonstrated extensive application potential and significant research value in the fields of fiber laser technology, laser chaos, optical secure communication, optical random signal generator, and laser radar.