Existence of hidden attractors in nonlinear hydro-turbine governing systems and its stability analysis

Abstract

This work studies the stability and hidden dynamics of the nonlinear hydro-turbine governing system with an output limiting link, and propose a new six-dimensional system, which exhibits some hidden attractors. The parameter switching algorithm is used to numerically study the dynamic behaviors of the system. Moreover, it is investigated that for some parameters the system with a stable equilibrium point can generate strange hidden attractors. A self-excited attractor with the change of its parameters is also recognized. In addition, numerical simulations are carried out to analyze the dynamic behaviors of the proposed system by using the Lyapunov exponent spectra, Lyapunov dimensions, bifurcation diagrams, phase space orbits, and basins of attraction. Consequently, the findings in this work show that the basins of hidden attractors are tiny for which the standard computational procedure for localization is unavailable. These simulation results are conducive to better understanding of hidden chaotic attractors in higher-dimensional dynamical systems, and are also of great significance in revealing chaotic oscillations such as uncontrolled speed adjustment in the operation of hydropower station due to small changes of initial values.