Abstract
In this research work, we describe Halvorsen circulant chaotic systems and its qualitative properties. We show that Halvorsen circulant chaotic system is dissipative and that it has an unstable equilibrium at the origin. The Lyapunov exponents of Halvorsen circulant chaotic system are obtained as L-1 = 0.8109, L-2 = 0 and L-3 = -4.6255. The Kaplan-Yorke dimension of the Halvorsen circulant chaotic system is obtained as D-KY = 2.1753. Next, this work describes the adaptive control of the Halvorsen circulant chaotic system with unknown parameters. Also, this work describes the adaptive synchronization of the identical Halvorsen circulant chaotic systems with unknown parameters. The adaptive feedback control and synchronization results are proved using Lyapunov stability theory. MATLAB simulations are depicted to illustrate all the main results for the Halvorsen circulant chaotic system.