Introduction: Nonlinear dynamical systems show complex behaviour sensitive to initial conditions. Controlling and synchronising chaos has applications in engineering, science, secure communications, encryption, biology, chemistry, finance, neural networks, cryptography, and medicine.
Aim: This paper aims to analyze the synchronisation, tracking control, and stability of hyperchaotic 5D Lorenz systems. A linear feedback controller is built to ensure asymptotic stability of the two identical hyperchaotic 5D Lorenz systems developing from distinct initial conditions, based on Lyapunov stability theory and active backstepping nonlinear approaches.
Methods: The three positive Lyapunov exponents and complex dynamical behaviour of the hyperchaotic 5D system are demonstrated. The control functions for the corresponding control and synchronisation of the hyperchaotic 5D Lorenz system are designed using the active backstepping nonlinear technique. The nonlinear controllers of the intended backstepping can stabilise and direct the hyperchaotic 5D Lorenz system at any place to follow any smooth function of time trajectory. The proposed method integrates the selection of a Lyapunov function with the creation of active control, and it is a systematic design technique. To validate the feasibility and effectiveness of the proposed control technique, numerical simulation results are presented.
Conclusion: The use of an active backstepping control approach to control and synchronise the hyperchaotic 5D Lorenz system stabilises chaotic motion, simplifies design without needing eigenvalues, and efficiently controls high-dimensional hyperchaotic systems, outperforming conventional chaos. Thus, numerical simulations confirm effectiveness.