Bifurcation and Stability Analysis of Saturated Nonlinear Systems

Saturated nonlinear systems arise in a wide variety of engineering, physical, biological, and control applications in which the system input, actuator output, or state variable is restricted by physical limitations. Saturation introduces an additional nonlinear characteristic into an otherwise nonlinear dynamical system and may significantly influence its equilibrium structure, local stability, oscillatory behavior, and bifurcation characteristics. The present paper investigates the stability and bifurcation behavior of a class of saturated nonlinear dynamical systems using equilibrium analysis, Jacobian linearization, eigenvalue analysis, and parameter-dependent bifurcation theory. A general mathematical model containing a saturation function is formulated, and the system is analyzed separately in the unsaturated and saturated operating regions. The equilibrium conditions are derived, followed by local stability analysis based on the eigenvalues of the Jacobian matrix. Particular attention is given to saddle-node and Hopf bifurcations, as well as the qualitative changes that may occur when an equilibrium crosses a saturation boundary. A representative second-order nonlinear model is considered to demonstrate how actuator saturation modifies the effective feedback gain and consequently changes the stability characteristics of the system. The analysis shows that saturation can alter the location of equilibrium points, stability margins, and critical bifurcation parameters. The study also establishes that conventional smooth bifurcation theory may need to be supplemented by piecewise-smooth analysis when trajectories interact with saturation boundaries. The results provide a useful theoretical framework for analyzing constrained nonlinear systems and have potential applications in nonlinear control, robotics, mechanical systems, electrical systems, aerospace engineering, and other dynamical processes involving bounded control actions.