Displays of high-grade hyperchaos and bi-stability in an erbium-doped fiber three-ring mutual-coupled laser

A novel erbium-doped fiber three-ring mutual-coupled laser system is innovated to produce quasi-periodic, hyperchaotic, and other dynamical behaviors. Its optics three-body laser physics model of the ring-coupledring-coupled- ring-coupled-ring construction is developed to present. Bi-stability is foretold to exist in three laser rings of the system in mathematics, and is illustrated to prove in our numerical simulation. The hyperchaotic and quasi-periodic behaviors, as well as the coexistent scenario are shown routes from the perioddoubling to chaos. The dynamics distributions, such as the stable region, one-periodic region, quasi-perodic region, chaotic region, are displayed to inspect by the bifurcation diagram. We report our study of Lyapunov exponents (LEs) of the system to describe and indicate its dynamical behaviors, such as quasi-period, and hyperchaos possessing multiple positive LEs, such as three positive LEs, or four positive LEs, or five positive LEs, or six positive LEs. Then, a classification of a hyperchaotic grade is introduced and defined by numbers of positive LEs. And a hyperchaotic distribution and a fine structure of hyperchaotic region are discovered via a few LE distributions while a diagnosis of hyperchaos is indicated, which points to a precision route to hyperchaos to produce hyperchaos and multi-grade hyperchaos. The work shows an important in our study of hyperchaotic emitters, hyperchaotic fiber laser technologies, and applications.