The Chaos and Stability of Firefly Algorithm Adjacent Individual

In this paper, in order to overcome the defect of the firefly algorithm, for example, the slow convergence rate, low accuracy and easily falling into the local optima in the global optimization search, we propose a dynamic population firefly algorithm based on chaos. The stability between the fireflies is proved, and the similar chaotic phenomenon in firefly algorithm can be simulated.


Introduction
The optimization of high dimensional functions is the process of selecting the optimum solution from the set of alternative ones [1]. We have to either maximize or minimize the objective functions by calculating the value of function using several input values from the given range of values. The evolutionary algorithms are widely used in optimization problems. They are used in such algorithms, which are reproduction, mutation, crossover, recombination and so on. Firefly algorithm (FA) is a new member of this family, which has been an active focus of research recently, and several modifications and improvements were recorded within the past few years [2][3][4]. The other several methods have been used to improve the performance of firefly algorithm, including chaotic maps. Generally speaking, firefly algorithm can be combined with chaos in two ways. On the one hand, the chaotic map replaces some randomly distributed [5][6]. On the other hand, the firefly intrinsic structure is applied for tuning algorithm parameters using chaotic map FA parameter in order to improve the performance [7][8].
The research focused by scholars is the combination of firefly algorithm with certain chaos. However, the chaotic phenomenon in firefly algorithm may be ignored. In this paper, we have analyzed the stability of adjacent individual in firefly algorithm with method of mathematical method. What's more. The Quasi-chaotic phenomenon that adjacent individual contracts each other is simulated with MATLAB in this paper. The paper's structure is as followed: In Section II, the main idea is the brief overview of the firefly algorithm; In Section III, some theorems of stability and Quasi-chaos of adjacent individual are given. In section IV, some experiments and simulations have been carried out to the performance of chaos in firefly algorithm.

The Firefly Algorithm and Chaos
In this section, some involved theories will be introduced in brief.

Firefly algorithm (FA)
Firefly algorithm (FA), proposed by scholar Yang, is inspired by the phenomenon that the fireflies flash and attract each other in nature [9]. It is one kind of colony searching technology, which simulates the flashing and communication behavior of fireflies. The (1) Where o I is the biggest fluorescence brightness of firefly.  is light intensity absorbed coefficient and it is constant. It is characteristics of the weak that under the influence of the fireflies emit light in the distance and medium by light intensity absorbed coefficient. For two fireflies i and j , the distance is calculated: Equation (2) is the distance between any two fireflies and at and, respectively. Definition 2. 2: The attractiveness of firefly is: Where o  is the attractiveness at 0 r  . Definition 2.3: In every iteration, the fireflies could move to nearby ones with more brightness as determined by Equation (4): Where   i xt and   Step 1: Initialize the parameters of the algorithm. Define the number of fireflies , the biggest attraction o  , the light intensity absorbed coefficient  , the step length factor , the maximum number of generations max T and the search accuracy , respectively. Step 2: Calculate the fitness value of every firefly regarded as the respective maximum fluorescence brightness.
Step 4: Update the locations of fireflies according to (2.4) and random disturb the firefly in the optimum location.
Step 5: Update the fluorescence intensity and attraction after position.
Step 6: If the results meet the requirements, break to step 7; else, the number of iterations plus one, back to step 3.
Step 7: Output the global extreme point and the optimal individual values.

() Om
is time complexity of algorithm .

Chaotic Map (CP)
Chaotic map in natural sciences denotes a deterministic system that behaves unpredictable [10]. In mathematics, chaos does not indicate the system with a complete absence of order, but perfectly ordered system that includes some flavor of randomness. This term was first used by Li and Yorke in [11] at 1975. This phenomenon is usually considered as a part of dynamical systems that changed over the time. In this paper, there are some well-known chaotic maps introduced briefly. a. Sinusodial map This chaotic map was formally defined by the following Equations [12]: The Tinkerbell map was a discrete-time dynamical system given by [14]: Some commonly used values of a, b, c, and d are: The Henon map is a discrete- .In this paper, the former set of parameter has been used to do it, which is showed in Figure 1(c). Like all chaotic maps, the Tinkerbell Map has also been shown to have periods. After a certain number of mapping iterations, every given point showed in the map to the right will find itself once again at its starting location. The chaotic curve of Henon map with 10000 iterations is showed in Figure 1(d), and the initial parameters are: .The map is introduced by Michel Henon as a simplified model of the Poincare section of the Lorenz model. For the classical map, an initial point of the plane will either approach a set of points known as the Henon strange attractor, or diverge to infinity.

Chaotic-FA (CFA)
In this section, we have used four chaotic maps mentioned above to tune the FA parameters and improve the performance, which will lead to a set of Firefly algorithm, or different variants of the chaotic Firefly algorithm. The flowchart of a schematic chaotic-FA (CFA) [5] is presented in Table 1. The following method describes how parameters can be tuned.

The stability and Quasi-chaos of Firefly Algorithm 3.1. The Stability of Firefly Algorithm
In this section, the stability and chaos of firefly algorithm will be proved in details. In the process of evolution, brightness is the matter to attract each other, and other fireflies will move towards the brightest one, it will lead to a stabilization of individuals. So we get the following theorem. Theorem 3.1 If the dynamic behavior of fireflies are stable, it must meet the following conditions: where   0.5 rand

 
Proof: The expressions for updating the position of fireflies are:  (14) According to the theory of stability, the necessary and sufficient condition for the dynamic behavior of fireflies is 1,2 1   , Therefore, the optimal stability condition is

Quasi-chaos of Firefly Algorithm
In the process of iterations, distances will be convergent and fireflies will tend to be stable after a number of iterations or evolutions. Under a certain circumstances, a kind of be similar to chaotic phenomenon will become true, we define it as a Quasi-chaos phenomenon in the paper.

So for firefly
After iterations, we get the propinquity Quasi-chaotic attractor of this map as shown in Figure 2. The simulation is under of circumstance that:For Figure 2(a) the initial locations of adjacent fireflies are 0.01 and 0.09, the attractiveness of them are 0.001 and 0.08, respectively.
However, when these parameters change, the graphics will change greatly. For example, As Figure 2

Experiment and Results
In this section, three complicated functions will be tested using CFA, which the global optimal point can be found simultaneously. And the space locations of adjacent firefly will be simulated to prove the chaotic Algorithm.

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The minimum solution of the Sphere function, Rastrigin function, and Griewank function are located at point x = (0, 0, 0, . . . , 0, 0, 0) with an objective function valued equal to F(x) = 0.0. When applied the CFA algorithm to the function, the algorithm finds the optimal solution after approximately 38, 15, 9 iterations, respectively (Figure 3(a-c)). By testing the four typical chaotic maps in these three complex functions, we are supposed to find that CFA can overcome the defect of the firefly algorithm which slow convergence rate and easily falling into the local optima in the global optimization search,but it still exist the low precision.

Conclusion
This paper has focused on the chaos and stability of adjacent fireflies in firefly algorithm. The main contribution is that the chaotic performance and behavior are proved and tested by using theory of stability in firefly algorithm. Through the above experiments and results, CFA can overcome the defect of the firefly algorithm which slow convergence rate and easily falling into the local optima in the global optimization search.