Uncorrected Author Proof Journal of Intelligent & Fuzzy Systems xx (20xx) x–xx DOI:10.3233/JIFS-169816 IOS Press 1 Modeling of fractional order chaotic systems using artificial bee colony optimization and ant colony optimization 1 2 3 Sangeeta Gupta ∗ , Varun Upadhyaya, Ayush Singh, Pragya Varshney and Smriti Srivastava 4 Department of Instrumentation and Control Engineering, NSIT, New Delhi, India 5 Abstract. In this paper, a modified Artificial Bee Colony algorithm is proposed. Then estimation of the parameters of fractional order chaotic systems is performed using the proposed Artificial Bee Colony algorithm and Ant Colony algorithm. For the purpose of modeling, four fractional order chaotic systems viz. Financial System, Chen System, Lorenz’s system and 3 Cell Net system have been considered. Each chaotic system is defined by a set of fractional-order differential equations. These equations comprise of several variables – model parameters, derivative orders, and initial conditions. For the system’s entire state and future values to be known, the values of all the parameters have to be estimated to a reasonable degree of accuracy. It is a general practice to use modern evolutionary algorithms to solve such problems. Simulations on both nature inspired optimization algorithms are performed and estimated values of parameters determined. Comparisons with existing scheme of Artificial Bee Colony based parameter estimation are also performed. Observations reveal that the results of the modified ABC algorithm outperform those of other techniques for all the four cases. 6 7 8 9 10 11 12 13 14 15 Keywords: Artificial bee colony optimization (ABC), ant colony optimization (ACO), fractional order chaotic systems 16 1. Introduction 17 A non-linear system, having deterministic, irreg- 18 ular and complex behavior and which is sensitive 19 to initial conditions can be referred to as a chaotic 20 system [1]. Researchers are nowadays working on 21 methods of chaos control, by virtue of its potential 22 applications in various fields of sciences and engi- 23 neering [2–6]. Fractional systems are known to be 24 expressed by fractional differential equations, which 25 when applied to control problems, provide better con- 26 trol due to the increased number of tuning parameters. 27 Emphasis has now been diverted to study of 28 fractional-order chaotic systems. It is observed that 29 several fractional-order systems having order less 30 ∗ Corresponding author. Sangeeta Gupta, Instrumentation and Control Engineering Department, NSIT, New Delhi, India. E-mail: sangeeta22011@gmail.com. than three exhibit chaos. Some of such systems are: 31 the fractional order (f-o) Chua’s circuit [6], f-o Chen 32 system [5], f-o Financial system [7], f-o Lorenz sys- 33 tem [8], and many more [9–11]. Chaotic dynamics are 34 expressed as systems which are no-linear and possess 35 positive energy. 36 For control and synchronization of f-o chaotic 37 systems, parameter estimation is essential. Dynamic 38 estimation of all parameters of chaotic and hyper- 39 chaotic systems are performed using a variational 40 calculus based method [12]. The method of sym- 41 bolic time series analysis is utilized for parameter 42 estimation of higher dimension systems exhibiting 43 chaos [13]. Nature inspired optimization algorithms 44 have also been used for estimation of parameters 45 of f-o systems showing chaotic behaviour, such as 46 the Locust Search Algorithm in [14], differential 47 evolution in [15], Quantum Parallel Particle Swarm 48 Optimization Algorithm in [16], and Artificial Bee 49 1064-1246/18/$35.00 © 2018 – IOS Press and the authors. All rights reserved