Modeling of fractional order chaotic systems using artificial bee colony optimization and ant colony optimization

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.

Keywords

Artificial bee colony optimization (ABC)ant colony optimization (ACO)fractional order chaotic systems

1 Introduction

A non-linear system, having deterministic, irregular and complex behavior and which is sensitive to initial conditions can be referred to as a chaotic system [1]. Researchers are nowadays working on methods of chaos control, by virtue of its potential applications in various fields of sciences and engineering [2 –6]. Fractional systems are known to be expressed by fractional differential equations, which when applied to control problems, provide better control due to the increased number of tuning parameters.

Emphasis has now been diverted to study of fractional-order chaotic systems. It is observed that several fractional-order systems having order less than three exhibit chaos. Some of such systems are: the fractional order (f-o) Chua’s circuit [6], f-o Chen system [5], f-o Financial system [20], f-o Lorenz system [8], and many more [9 –11]. Chaotic dynamics are expressed as systems which are non-linear and possess positive energy.

For control and synchronization of f-o chaotic systems, parameter estimation is essential. Dynamic estimation of all parameters of chaotic and hyperchaotic systems are performed using a variational calculus based method [12]. The method of symbolic time series analysis is utilized for parameter estimation of higher dimension systems exhibiting chaos [13]. Nature inspired optimization algorithms have also been used for estimation of parameters of f-o systems showing chaotic behaviour, such as the Locust Search Algorithm in [14], differential evolution in [15], Quantum Parallel Particle Swarm Optimization Algorithm in [16], and Artificial Bee Colony Algorithm in [7]. The authors in [15] have estimated parameters of a fractional-order memristor-based chaotic system using a hybrid artificial bee colony algorithm along with differential evolution. Cuckoo Search algorithm and biogeography based optimization techniques [17, 18] have also been used by researchers.

Artificial Bee Colony (ABC) optimization technique is a heuristic algorithm, which analyses the behaviour of intelligent honey bees. The method was suggested by D. Karaboga [7, 19, 20]. In this method, three types of bees exist: namely employed, scout and onlooker bees. The colony is subdivided into two sub-categories. The primary group of the colony of artificial bees attempts to search for food within the vicinity of food source in their memory. And the second group, which comprises of onlooker bees, first takes the information about the food source from the employed bees within the hive and then intelligently selects the best source.

Another heuristic algorithm commonly being used nowadays is the Ant Colony Optimization (ACO) [21 –23]. It is based on the movement of artificial ants in a pattern so as to minimize the length of the path. These ants deposit pheromones on the nodes of the best path chosen. Depending on the quality of solution (good or bad), the pheromone values are updated (increased or decreased).

A modified ABC Optimization algorithm has been used for estimation of parameters for four f-o chaotic systems in this paper. The differential equations of the systems are known and estimation for some parameters is performed. The four f-o chaotic systems used are: Financial System, Chen System, Lorenz’s system and 3 Cell Net (CNN) system [24]. Simulations are performed and results compared with those obtained using Ant Colony Algorithm. Observations show that the modified ABC algorithm results outperform those of ACO and also the results of [20]. In the paper, in section II, the various f-o chaotic systems used are briefly explained, whereas section III presents the results along with comparison with the work of [20]. Section IV concludes the paper.

2 F-O chaotic systems

2.1 F-O financial system

The financial system used in economic systems [25, 26] has the following fractional order differential equations:- $\begin{matrix} D_{t}^{r_{1}} p (t) = s (t) + (q (t) - l) p (t) \\ D_{t}^{r_{2}} q (t) = 1 - mq (t) - p {(t)}^{2} \\ D_{t}^{r_{3}} s (t) = - p (t) - ns (t) \end{matrix}$ (1)

The system model parameters are l; saving amount; m; cost per investment; and n; elasticity of demand of commercial market ∀ [l, m, n ∈ R³],

The state variables are defined as [p (t), q (t), s (t)] where, interest rate is given by p(t), investment demand by q(t) and price index by s(t).

The fractional orders are [r₁, r₂, r₃] with initial conditions as [q₁ (0), q₂ (0), q₃ (0)].

Using modified ABC and ANT optimization algorithm, the values of [r₁, m, n] were estimated using the Euclidian Norm of the error in the outputs as the cost function. $J (\bar{r_{1}}, \bar{l}, \bar{m}) = \sum_{k = 1}^{N} \underset{2}{∥ Q_{k} - \bar{Q_{k}} ∥}$ (2)

For the system [r₁, r₂, r₃] = [0.76, 1, 1]; [l, m, n] = [3, 0.1, 1], [q₁ (0), q₂ (0), q₃ (0)] = [0, 1, 0.3].

For the values to be estimated, the initial span chosen are r₁ ∈ [0.4, 1.4], m ∈ [0, 1] and n ∈ [0.5, 1.5].

2.2 F-O chen system

Chen discovered a simple 3-dimensional autonomous system in 1999 [5], which shows chaotic phenomena and is defined as:- $\begin{matrix} D_{t}^{r_{1}} p (t) = l (q (t) - p (t)) \\ D_{t}^{r_{2}} q (t) = (n - l) p (t) - p (t) s (t) + nq (t) \\ D_{t}^{r_{3}} s (t) = p (t) q (t) - ms (t) \end{matrix}$ (3)

where the model has system parameters [l, m, n ∈ R³], fractional orders [r₁, r₂, r₃] and initial conditions [q₁ (0), q₂ (0), q₃ (0)].

For this system, choice of fractional orders is [r₁, r₂, r₃] = [0.94, 0.94, 0.94], model parameters as [l, m, n] = [35, 3, 27]; and initial conditions is [q₁ (0), q₂ (0), q₃ (0)] = [0.2, 0, 0.5].

For estimation, the ranges chosen are r₁ ∈ [0.44, 1.44], m ∈ [2.5, 3.5] and n ∈ [26.7, 30.3].

2.3 F-O lorenz’s system

The f-o Lorenz system [8] differential equations are: $\begin{matrix} D_{t}^{r_{1}} p (t) = σ (q (t) - p (t)) \\ D_{t}^{r_{2}} q (t) = p (t) (ρ - s (t)) - q (t) \\ D_{t}^{r_{3}} s (t) = p (t) s (t) - β q (t) \end{matrix}$ (4)

In the equations, the system model parameters are given by the matrix [σ, ρ, β > 0], σ being Prandtl number and ρ Rayleigh number. All symbols have their usual meanings.

For the system, $\begin{matrix} [r_{1}, r_{2}, r_{3}] = [0.993, 0.993, 0.993] \\ [σ, ρ, β] = [10, 28, 2.67] \\ [q_{1} (0), q_{2} (0), q_{3} (0)] = [0.1, 0.1, 0.1] \end{matrix}$

The initial estimates are chosen as q₁ ∈ [0.493, 1.493] ρ ∈ [25, 31] β ∈ [2.17, 3.17].

2.4 F-O 3 cells net (CNN) system

The CNN system introduced in [24, 27 –30] is a combination of resistors, capacitors, controlled and independent sources connected in parallel. Out of these, resistors and capacitors are linear, whereas the sources are both linear and non-linear. Each CNN cell is connected only to its adjacent cells. The fractional order differential equations which define the CNN system are:- $\begin{matrix} D_{t}^{r_{1}} p (t) = - p (t) + lf (p (t)) - jf (q (t)) - jf (s (t)) \\ D_{t}^{r_{2}} q (t) = - q (t) - jf (p (t)) + mf (q (t)) - kf (s (t)) \\ D_{t}^{r_{3}} s (t) = - s (t) - jf (p (t)) + kf (q (t)) + nf (s (t)) \end{matrix}$ (5)

[l, m, n, j, k] are the system model parameters, [r₁, r₂, r₃] the fractional orders and [q₁ (0), q₂ (0), q₃ (0)] the initial conditions. Using modified ABC optimization and ANT optimization algorithms, the values of [r₁, m, n] were estimated using the Euclidian Norm of the error in the outputs as the cost function.

For the system, the choices done are $\begin{matrix} [r_{1}, r_{2}, r_{3}] = [0.99, 0.99, 0.99], \\ [l, m, n, j, k] = [1.24, 1.1, 1.0, 4.4, 3.21] \\ [q_{1} (0), q_{2} (0), q_{3} (0)] = [0.1, 0.1, 0.1] \end{matrix}$

For the values to be estimated, r₁ ∈ [0.49, 1.49], m ∈ [0.6, 1.6] and n ∈ [0.5, 1.5].

3 Evolutionary algorithms

3.1 Ant colony optimization (ACO)

ACO is a Meta Heuristic Search Algorithm. It mimics the technique used by ants to find the optimal path from their colony to the food source. Communication between ants occurs based on the pheromones produced by them through their glands. This pheromone is deposited while walking between the food source and nest, establishing a pheromone trail for the other ants to follow. Ants choose to follow the trail with higher pheromone levels with a higher probability hence increasing the pheromone strength of the trail further. As the pheromone deposited evaporates, the trail is abandoned. The path from the colony to the food which followed by a larger number of ants gets locked.

3.1.1 Initialization

A node matrix consisting of uniformly distributed values of the parameters (Kp, Ki, Kd,λ and μ) within fixed limits, a probability matrix of the same size (initially all nodes have the same probability) and a pheromone matrix are formed. The pheromone matrix, keeps a track of the pheromones strength in the path chosen. In each iteration, the ant matrix is developed which defines the shortest path.

3.1.2 Updating the probability matrix

The probability for an ant to move from one node (a) to another node (a) is calculated as: $P_{ab}^{A} (t) = \frac{{[τ_{ab} (t)]}^{ρ} {[η_{ab}]}^{δ}}{\sum_{a, b \in T^{A}} {[τ_{ab} (t)]}^{ρ} {[η_{ab}]}^{δ}}$ (6)

where, τ_ab is the Pheromone factor, η_ab is the Heuristic factor, ρ, δ are model parameters. The ant traverses a path T^A in a specified time. While traversing, the amount of pheromone released on the path is: $Δ τ_{ab}^{A} = {\begin{matrix} \frac{L^{min}}{L^{A}} & a, b \in T^{A} \\ 0 & otherwise \end{matrix}$ (7)

where L^A, L^min define the objective function and shortest path found in the current iteration.

The amount of pheromone released in a path is updated with each iteration depending upon the route (ab) chosen by the ants. The change in Pheromone concentration and evaporation are given by the formulae: $Δ τ^{k} = Δ τ^{k = 1} = \frac{ξ f_{good}}{f_{bad}}$ (8) $τ_{ab} (t) = σ τ_{ab} (t - 1) + \sum_{A = 1}^{M} Δ τ_{ab}^{A} (t)$ (9)

In (8, 9), M, σ (0 < σ < 1) are the total number of ants and the evaporation rate of pheromone respectively. The most probable path to be taken gives us the best values of the controllerparameters.

3.2 The artificial bee colony algorithm

The artificial bee colony (ABC) algorithm is a metahuristic algorithm that mimics the foraging behavior of honey bees (Pham et al., 2006). In a typical bee colony, one part of the population known as scout bees, continuously search the area surrounding the hive for food sources. They return to the hive and convey their findings through a “waggle dance” (Seeley 1996). This waggle dance contains information about the quality of the food source that each scout bee visited. The best food sources are identified and class of bees known as “forager bees” are sent to explore the neighborhood of these locations. When these bees return to the hive, they too communicate their findings and the best locations are identified again and forager bees are sent to explore these locations again. In this way the optimal food source is eventuallyfound.

While writing the ABC algorithm for minimization of a function (cost function), a food source is analogous to a candidate solution and the waggle dance corresponds to the value of the cost function.

The following variables are used:

Number of scout bees

Number of sites selected as best sites

Number of forager bees selected to explore each “best site”

i^th dimension of the hyper-box centered around a best site

Parameter for neighbourhood shrinking in the t^th iteration

3.2.1 Initialisation

The distribution of Scout bees in an n-dimentional search space is random and the cost function is evaluated for each location. The scout bee population is a fixed system parameter denoted by S_p. The best locations are identified by ranking the cost functions for each location in ascending order.

3.2.2 Forager bees

The best N_b sites are selected and F_p forager bees are sent to explore an area around each location. They are randomly spread inside a fixed n-dimension hyper-box centered around a selected location. This mechanism assigns a large number of bees to explore the area around the best locations. If a forager bee finds a location in the neighborhood of the selected location, which has a lower cost as compared to the location, then it replaces the scout bee that found the location.

3.2.3 Global search

In every iteration, while the forager bees are exploring the areas around the best sites, (S_p-N_b) scout bees are scattered with uniform probability globally, i.e in the the entire search space. If a scout bee finds a site yielding a lower cost function than the best sites selected, then it becomes one of the N_b selected sites in the next iteration.

3.2.4 Neighbourhood shrinking

The dimensions of the hyper-box centered around a selected location are initially set as follows: $a_{i} = d_{t} * (max_{i} - min_{i})$ (10)

where i is an integer from 1 to n representing the i^th dimension of the hyper-box $d_{0} = 1$ (11)

For every iteration in which the local search performed by the forager bees does not yield a better site in the neighbourhood of a particular selected site, the neighbourhood size of that selected site is reduced by: $d_{t + 1} = 0.8 * d_{t}$ (12)

In this way the same number of forager bees explore a smaller area around the selected site making their search more extensive.

3.2.5 The ABC algorithm

The algorithm can be summarized as follows:

− Initialize flower patch

− Initialize ‘n’ flower patches of size ‘a’

− Place ‘n’ scout bees randomly within the set limits

− do while stopping condition = TRUE

for i = 1 to n

site_necter(i) = waggle_dance (scout_bee(i))

sort_decending (site_necter);

select top ten sites

for i = 1 to 10

elite_bee(i,x) = scout_bee(i) +- random(0, a/2)

for i = 11 to n

global_search(scout_bee(i));

for i = 1 to 10

for j = 1 to x

if (waggle_dance(elite_bee(i,x)<site_necter(i))

scout_bee(i) = elite_bee(i)

else

shrink_site(i)

end

Subroutines:

− waggle_dance(bee): Compute cost of patch visited by bee

− global_search(bee): assign a random value within the set limits to bee

− shrink_site(i): reduce patch size

Variables:

− n = number of scout bees

− x = number of elite bees per patch

4 Simulations and results

4.1 F-O financial system

Simulations have been performed using artificial bee colony and ACO optimization. While simulating the f-o Financial system using ABC, the authors have chosen the number of scout bees as 50. The algorithm is run for 100 iterations as against 250 iterations in [20]. The authors have tuned the parameters r₁, m and n, whereas in [20], the parameters tuned are α₁, c and τ, the time delay. From the plot of Fig. 1, it is observed that the cost function settles after 75 iterations. For the ACO optimization, the number of ants chosen are 50, and the constants chosen apriori α, β and ρ are set to 0.3426, 0.6595 and 0.0248 respectively. Figure 2 shows the plot of fitness function using ACO.

Fig.1

Fitness plot for Fractional order Financial system using modified Artificial Bee Colony Optimization Algorithm.

Fig.2

Fitness plot for Fractional order Financial system using Ant Colony Optimization Algorithm.

In Table 1, the parameters of the f-o financial system obtained using the two algorithms are presented and compared with those of [20]. Both codes were run for 100 iterations and a simulation time (Tsim) of 10 s.

Table 1

Values of Estimated parameters of f-o financial system for the modified ABC, ACO and ABC of [20] algorithms

	Using modified ABC algorithm	Using ANT colony Optimization algorithm	Using ABC algorithm of [20]
Parameters	No. of scout bees = 50	No. of Ants = 50	No. of scout bees (SN) = 50
	No. of elite bees = 10/site	α= 0.3426
		β= 0.6595
		ρ= 0.0248
Iterations	100	100	250
r₁	0.7599988888	1.2480	0.7599999995
m	0.0992160325	0	−
n	0.988990302	0.5301	0.9999999791
Elapsed Time	192.824285 s	66.080494 s	−
Best Cost	0.1580	22.4458	−
Simulation time(Tsim)		10 s	10 s

4.2 F-O Chen system

Similarly, both ABC and ACO are applied to estimate the parameters of the f-o Chen system. Figure 3 shows that the values of r₁, m and n obtained using the proposed modified ABC algorithm match closely to the results of [20] even when the number of iterations are less. This proves the effectiveness of the proposed algorithm (Table 2). Both codes were run for 100 iterations and a simulation time (Tsim) of 10 s. Only the ABC results are shown for this method, as the latter did not give promising results.

Fig.3

Fitness plot for Fractional order Chen system using modified Artificial Bee Colony Optimization Algorithm.

Table 2

Values of Estimated parameters of f-o Chen system for the modified ABC, ACO and ABC of [20] algorithms

	Using modified ABC algorithm	Using ANT colony Optimization algorithm	Using ABC algorithm of [20]
Parameters	No. of scout bees = 50	No. of Ants = 50	No. of scout bees (SN) = 50
	No. of elite bees = 10	α= 0.3426
		β= 0.6595
		ρ= 0.0248
Iterations	100	100	250
r₁	0.9438405367	0.2400	0.9400000256
m	2.50543120	2.5000	−
n	27.0059621153	26.7000	2.9999503453
Elapsed Time	198.877101 s	69.608509 s	−
Best Cost	289.4135	NaN	−
Simulation time(Tsim)	−	10 s	10 s

4.3 F-O Lorenz’s system

The f-o Lorenz’s system is simulated using both modified ABC optimization and Ant Colony optimization. The parameters estimated are r₁, ρ and β. The values of the estimated parameters for same number of iterations are almost same (Table 3). This is further validated with the fitness value plots of Figs. 4 and 5.

Fig.4

Fitness plot for Fractional order Lorenz’s system using modified Artificial Bee Colony Optimization Algorithm.

Fig.5

Fitness plot for Fractional order Lorenz’s system using Ant Colony Optimization Algorithm.

Table 3

Values of Estimated parameters of f-o Lorenz’s system for the modified ABC, ACO algorithms

	Using modified ABC algorithm	Using ANT colony Optimization algorithm
Parameters	No. of scout bees = 50	No. of Ants = 50
	No. of elite bees = 10	α= 0.3426
		β= 0.6595
		ρ= 0.0248
Iterations	100	100
r₁	0.993166311549	1.0261
ρ	28.0480486981	28.0114
β	2.66028460572	2.6076
Elapsed Time	197.476904 s	67.049729 s
Best Cost	2.8743	34.6627
Simulation time (Tsim)	10 s	10 s

4.4 F-O 3 cells net (CNN) system

Table 4 shows the results of parameters estimated using fractional order 3 cells net system. We observe that the estimated value of r₁ using both the methods is somewhat near to the expected value. This result may be improved further by either increasing the number of scout bees or by increasing the number of iterations. The values of m and n are also comparable with the actual values. The fitness plots of Figs. 6 and 7, justify the results. Both codes were run for 100 iterations and a simulation time (Tsim) of 10 s. The results are as follows.

Table 4
Values of Estimated parameters of f-o 3 Cells Net (CNN) system for the modified ABC, ACO algorithms

Using ABC Optimization Using ANT colony Optimization

Parameters No. of scout bees = 50 No. of Ants = 50

No. of elite bees = 10 α= 0.8

β= 0.2

ρ= 0.8

Iterations 100 100

r₁ 1.00902809447585 1.0151

m 1.021893229626 0.9647

n 1.1541210174327 1.2395

Elapsed Time 216.924411 s 68.807140 s

Best Cost 3.1941 5.0920

	Using ABC Optimization	Using ANT colony Optimization
Parameters	No. of scout bees = 50	No. of Ants = 50
	No. of elite bees = 10	α= 0.8
		β= 0.2
		ρ= 0.8
Iterations	100	100
r₁	1.00902809447585	1.0151
m	1.021893229626	0.9647
n	1.1541210174327	1.2395
Elapsed Time	216.924411 s	68.807140 s
Best Cost	3.1941	5.0920

Fig.6

Fitness plot for Fractional order 3 Cells Net (CNN) system using modified Artificial Bee Colony Optimization Algorithm.

Fig.7

Fitness plot for Fractional order 3 Cells Net (CNN) system using Ant Colony Optimization Algorithm.

5 Conclusion

The modified Artificial Bee Colony Optimization algorithm clearly delivers superior results compared to the Ant Colony Optimization algorithm in all cases. The time complexity of the modified ABC algorithm is greater than the ACO because the cost function is called more number of times in the ABC algorithm. Another advantage of the modified ABC algorithm is that its performance does not depend on any tuning of parameters like the ACO algorithm and also, it is not very sensitive to its control parameters like, colony size and works effectively for a wider range of values. The modified ABC algorithm used in this paper incorporates global search and patch shrinking making is more robust and having a higher convergence rate.

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