Active and Backstepping Control for Stabilization and Synchronization of a Four-Dimensional Hyperchaotic Finance System

This paper addresses the stabilization and drive–response synchronization of the four-dimensional hyperchaotic finance model using active backstepping (ABS) and active control (AC). The contribution is not the introduction of a new control paradigm but a unified implementation of AC and ABS for the Yu finance model, together with explicit Lyapunov convergence estimates, reproducible numerical benchmarking, and robustness-oriented performance assessment. For the ideal full-state-feedback setting, Lyapunov arguments establish exponential stabilization for the ABS-controlled system and exponential synchronization for both AC and ABS. The numerical protocol quantifies settling time, norm-relative overshoot, envelope-based decay rate, integrated control energy, CPU time, and actuator peak/RMS values. The results show that AC provides smooth and energy-efficient synchronization, whereas ABS gives fast convergence and a nonlinear Lyapunov-based stabilization framework but requires higher actuation effort. Robustness tests under parameter mismatch and additive measurement noise indicate bounded trajectories and decaying synchronization errors under the tested perturbation levels. The results also clarify the trade-off between convergence speed, control energy, implementation complexity, and actuator feasibility.