Implementation of Liu Chaotic System with Linear Feedback Synchronization Control

This article presents the implementation of the Liu chaotic system alongside its linear feedback synchronization control program. The Liu chaotic system represents a nonlinear dynamical system characterized by chaotic behavior on its attractor. Research has demonstrated its broad applications in communication systems, encryption technologies, and random number generation. Consequently, investigating control methodologies for the Liu chaotic system has gained significant importance. The linear feedback synchronization control method operates by regulating feedback signals within the control system to achieve synchronization among multiple chaotic systems. This approach typically involves calculating error signals between drive and response systems, then applying proportional feedback gains to minimize synchronization errors. In our implementation, the Liu chaotic system is mathematically described by a set of three coupled differential equations: dx/dt = a(y - x) dy/dt = bx - kxz dz/dt = -cz + hx² The synchronization control algorithm employs a linear feedback controller that computes the state differences between master and slave systems, then applies corrective signals through carefully tuned gain matrices. The synchronization stability is ensured through Lyapunov stability analysis, where appropriate feedback gains are selected to guarantee asymptotic synchronization. We will examine how to implement the linear feedback synchronization control program for regulating the Liu chaotic system, including numerical integration methods for solving the differential equations and parameter tuning strategies for optimal synchronization performance. The practical applications and potential value of this controlled chaotic system in secure communications and cryptographic systems will also be discussed.