Adaptive synchronization of fractional order systems of the Lorenz type

Abstract

This paper presents the synchronization analysis of two fractional order systems of the Lorenz type, with unknown parameters. The study was done through simulations in order to understand the mechanisms involved in this synchronization process for later use in the derivation of analytical techniques. The study includes adaptive synchronization between two fractional order systems of the Lorenz type using integer and fractional adaptive laws. Simulation results are shown for the synchronization of integer systems with fractional adaptive laws, and for the synchronization of fractional order systems with integer and fractional adaptive laws.

Adaptive synchronization of fractional Lorenz systems using a reduced number of control signals and parameters

“…5 , the systems exhibit an stable behavior, and when α = 0 . 99 the systems exhibit a chaotic behavior, as it was found in [30] .…”

“…The parameter values for the master system (8) and slave system (9) , which are assumed to be unknown, are σ = 10 , γ = 28 and β = 8 / 3 . It was reported in [30] that for this values, the Lorenz system exhibits a chaotic behavior when the derivation order lies in the interval 0.99 ≤ α ≤ 1.18, when α < 0.99 the Lorenz system tends asymptotically to one of the two attractors and when α > 1. 18 the system exhibits and unstable behavior.…”

“…It is also assumed that an upper bound B γ σ on the sum (γ + σ ) 2 4is known ( ( γ + σ ) 2 B γ σ ). If the following control signals are used in(9) U 1 = −z m e 2 + B γ σ e x m e 2 + y s e 1 − x s e then the controlled system(12) ,(30) is asymptotically stable.Proof. Using the control signals(30) in(12) , the evolution of the synchronization errors results…”