It is an important issue to estimate parameters of fractional-order chaotic systems in nonlinear science, which has received increasing interest in recent years. In this paper, time delay and fractional order as well as system’s parameters are concerned by treating the time delay and fractional order as additional parameters.
The parameter estimation is converted into a multi-dimensional optimization problem. A new scheme based on artificial bee colony (ABC) algorithm is proposed to solve the optimization problem. Numerical experiments are performed on two typical time-delay fractional-order chaotic systems to verify the effectiveness of the proposed method.
Caputo Fractional-order Derivative
In general, three best-known definitions of fractional-order derivatives are widely used: Grunwald-Letnikov, Riemann-Liouville and Caputo definitions. In particular, the main advantage of Caputo fractional-order derivative is that it owns same initial conditions with integer-order derivatives, which have well-understood features of physical situations and more applicable to real world problems. Thus, the Caputo fractional-order derivative is employed in this paper.
Besides, due to multiple variables in the problem and multiple local search optima in the objective functions, it is easily trapped into local optimal solution and the computation amount is great. So it is not easy to search the global optimal solution effectively and accurately using the traditional general methods. Therefore, we aim to solve this problem by the effective artificial bee colony algorithm in this paper. The general principle of parameter estimation by ABC algorithm is shown in Fig. 1.
A NOVEL PARAMETER ESTIMATION SCHEME
In the natural bee swarm, there are three kinds of honey bees to search foods generally, which include the employed bees the onlookers and the scouts (both onlookers and the scouts are also called unemployed bees). The employed bees search the food around the food source in their memory. At the same time, they pass their food information to the onlookers. The onlookers tend to select good food sources from those found by the employed bees, then further search the foods around the selected food source.
In particular, the smaller F is, the better combination of parameters (α1, c, τ) is. The distribution of the objective function value for the time-delay fractional-order financial system is shown in Fig. 2. As viewed in different colors in Fig. 2, it can be found that the objective function values are smaller in the neighborhood of the point (α1, c, τ) = (0.76, 1, 0.08) than those in other places.
In this paper, the parameter estimation of time-delay fractional-order chaotic systems is concerned by converting it into an optimization problem. A method based on artificial bee colony algorithm is proposed to deal with the optimization problem via functional extreme model. In simulations, the proposed method is applied to identify two typical time-delay fractional-order chaotic systems. And the simulation results show that the fractional order, the time delay and the system’s parameter of chaotic system can be successfully estimated with the proposed scheme.
The aim of this paper is to design a scheme based on ABC algorithm to estimate the unknown fractional orders, system’s parameters and time delays. The proposed method can avoid the design of parameter update law in synchronization analysis of the time-delay fractional-order chaotic systems with unknown parameters. Though it is not good enough, we hope this method will contribute to the application of chaos control and synchronization for the time-delay fractional-order chaotic systems.
Source: Erciyes University
Authors: Wenjuan Gu | Yongguang Yu | Wei Hu