Study On The Chaos Ergodicity Of A Fractional-Order Uneven Hopfield Neural Network

In this thesis, we study the chaotic behavior of a fractional-order (FO) nonlinear dynamic system referred to as uneven Hopfield neural network in the literature. Our research is one of the comprehensive research projects carried out at the Lab 570 (Lab of Modern Circuits and Systems) of UESTC, where almost every renowned autonomous nonlinear system have been investigated recently, such as Chen, Lv, Lorenz 84, Rossler, Jerk equation, etc. In a recent publication[1], W.M. Ahmad and J.C. Sprott addressed a conjecture in which he claim that A third order chaotic autonomous nonlinear system, with the appropriate nonlinearity and control parameter is chaotic for any fractional order 2 +ξ, 1>ξ > 0. For the integer order uneven Hopfield neural networkx|·=tanh(y-a~*x)y|·=tanh(z-b~*x) (1)z|·=c~*x~2-d~*xTransforming (1) into its FO form by set ,,, we will have the corresponding FO uneven Hopfield neural network as the following (2) where α,βand γare all real numbers. Our goal is not only to testify the mentioned above Sprott conjecture, but to study in detail the ergodicity of the chaotic behavior for the uneven Hopfield neural network (2) with respect to differential order (α,β,γ) variation as well. Thus we change these parameters and let the total order of the uneven Hopfield neural network δ=α+β+γbe varied from 2.7 to 3.3, under these designations, through computer simulations we found that the chaotic behavior of the uneven Hopfield neural network can be realized provided the control parameters a, b, and c are properly chosen. Also the renowned Lorenz system is investigated in the same way. Therefore, our study confirms the following Observation The chaotic behavior of uneven Hopfield neural network as well as the Lorenz system exhibits ergodicity with respect to the differentiation order variation in an interval of (3 – ξ >δ >3+ξ?), where ξ∈[0,1]. The issue on generality of this observation is of both academic and technologic significance. The issue might be investigated through experimental study on other nonlinear dynamic systems on the one hand, and by the theoretical proof on the other hand. The later topic seems to be a great challenge to the scientists especially the mathematicians.