# Identification of Unknown Parameters and Orders via Cuckoo Search Oriented Statistically by Differential Evolution for Noncommensurate Fractional-Order Chaotic Systems

Authors: Fei Gao; Xue-Jing Lee; Heng-qing Tong; Feng-xia Fei; Hua-ling Zhao
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

In this paper, a non-Lyapunov novel approach is proposed to estimate the unknown parameters and orders together for noncommensurate and hyper fractional chaotic systems based on cuckoo search oriented statistically by the differential evolution (CSODE). Firstly, a novel Gaos’ mathematical model is proposed and analyzed in three submodels, not only for the unknown orders and parameters’ identification but also for systems’ reconstruction of fractional chaos systems with time delays or not. Then the problems of fractional-order chaos’ identification are converted into a multiple modal nonnegative functions’ minimization through a proper translation, which takes fractional-orders and parameters as its particular independent variables. And the objective is to find the best combinations of fractional-orders and systematic parameters of fractional order chaotic systems as special independent variables such that the objective function is minimized. Simulations are done to estimate a series of noncommensurate and hyper fractional chaotic systems with the new approaches based on CSODE, the cuckoo search, and Genetic Algorithm, respectively. The experiments’ results show that the proposed identification mechanism based on CSODE for fractional orders and parameters is a successful method for fractional-order chaotic systems, with the advantages of high precision and robustness.


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## Body

## 1. Introduction

The applications of fractional differential equations began to appeal to related scientists recently [1–27] in the following areas: bifurcation, hyperchaos, proper and improper fractional-order chaos systems, and chaos synchronization [1–33].However, there are some systematic parameters and orders that are unknown for the fractional-order chaos systems in controlling and synchronization. It is difficult to identify the parameters in the fractional-order chaotic systems with unknown parameters. Hitherto, there have been two main approaches in parameters’ identification for fractional-order chaos systems.(i)
Lyapunov way: there have been few results on parameter estimation method of fractional-order chaotic systems based on chaos synchronization [34] and methods for parameter estimation of uncertain fractional order complex networks [35].  However, the design of controller and the updating law of parameter identification are still a tough task with technique and sensitively depend on the considered systems.(ii)
Non-Lyapunov way via artificial intelligence methods, for examples, differential evolution [7] and particle swarm optimization [9], in which the commensurate fractional-order chaos systems and simplest case with one unknown order for normal fractional-order chaos systems are discussed: however, to the best of our knowledge, little work in non-Lyapunov way has been done to the parameters and orders estimation of noncommensurate and hyper fractional-order chaos systems. And there is no general mathematical model that has been purposed for all these kinds of identification.We consider the following fractional-order chaos system with time delays(1)α𝒟tqY(t)=f(Y(t),Y0(t),θ,τ),
where Y(t)=(y1(t),y2(t),…,yn(t))T∈ℜn denotes the state vector. θ=(θ1,θ2,…,θn)T denotes the original parameters, τ=(τ1,τ2,…,τn) is the time delay, and q=(q1,q2,…,qn) (0<qi<1, i=1,2,…,n) is the fractional derivative orders. Consider
(2)f|(Y(t),Y0(t),θ,τ)=(f1,f2,…,fn)|(Y(t),Y0(t),θ,τ).Normally the functionf is known. And the θ, q, and τ are unknown; then Θ=(θ1,θ2,…,θn,q1,q2,…,qn,τ1,τ2,…,τn) will be the parameters to be estimated.Then a correspondent system is constructed as follows(3)α𝒟tqY~(t)=f(Y~(t),Y0(t),θ~,τ~),
where Y~(t), θ~, q~, and τ~ are the correspondent variables to those in (1), and function f is the same. The two systems (1) and (3) have the same initial condition Y0(t).Then the objective is obtained as follows:(4)Θ*=argminΘF=argminΘ∑t‍∥Y(t)-Y~(t)∥2.How could we identify the fractional system, when some fractional chaotic differential equationsf=(f1,f2,…,fn) are unknown? That is,
(5)(f1,f2,…,fn)*=argmin(f1,f2,…,fn)F.Now the problem of parameters estimation (4) becomes another much more complicated question, fractional-order chaos reconstruction problem [36], to find the forms of fractional-order equations as in (5). In [36] a novel non-Lyapunov reconstruction method based on a novel united mathematical model was proposed to reconstruct the unknown equations (f1,f2,…,fn).When it comes to the system (1) α𝒟tqY(t)=f(Y(t),Y0(t),θ) with neither q nor θ are known, the united model is not effective. For the united mathematical model [36], to be identified is only (f1,f2,…,fn) instead of q. That is, α𝒟tqY~(t) of (3) are not included. Actually, if the q are taken into consideration in united model (5), then the basic parameters’ setting to be reconstructed in [36] will be basic set {×,÷,+,-} with extra {=,𝒟tq} and so forth and the input variables {x1,x2,…,xn} with {Y} extra and, and so forth. Although for the candidates “programs” in [36] the maximum depth of tree is 6, considering that the maximum number of nodes per tree is infinite, there will be infinite illegal candidates on will be generated. Then, the one hand, the most time-consuming thing for novel united model (5) is to kill these illegal individuals from the legal individuals. However, these defaults are not solved in [36]. On the other hand, as q∈𝒟tq in unknown, it is really difficult to generate an individual with {×,÷,+,-,=,𝒟tq}, in neither illegal nor legal cases. And up till now, there is no existing way to resolve these defaults. And we can conclude from simulations [36] that the proposed method is much more efficient for the systems when coefficients q in 𝒟tq are integer orders than improper fractional orders.Therefore, to estimate theq of (3) with unknown systematic parameters θ is still a question to be solved for parameters and orders estimation of noncommensurate and hyperfractional-order chaos systems.And cuckoo search (CS) is a relatively new and robust optimization algorithm [37, 38], inspired by the obligate brood parasitism of some cuckoo species by laying their eggs in the nests of other host birds (of other species). The searching performance is mainly based on the Lévy flights mathematically [37–39], which essentially provide a random walk while their random steps are drawn from a Lévy distribution for large steps [37–39]. However, in CS evolution, the Lévy flights in each main iteration are used twice. It has two results: the CS searching performance becomes a little strong, but the redundant evaluations for the objective function are generated too. Therefore, some more efforts are needed to improve the performance of CS.To the best of authors’ knowledge, there is no method in non-Lyapunov way for noncommensurate and hyper fractional-order chaotic systems’ parameters and orders estimation so far. The objective of this work is to present a novel simple but effective approach to estimate the noncommensurate and hyper fractional-order chaotic systems in a non-Lyapunov way. And the illustrative reconstruction simulations in different chaos systems are discussed, respectively.The rest is organized as follows. In Section2, a general mathematical model not only for fractional chaos parameters identification but also for reconstruction in non-Lyapunov way is newly proposed and analyzed in three submodels A, B, and C. And a simple review was given on non-Lyapunov parameters estimation methods for fractional-order and normal chaos systems. In Section 3, a novel method with proposed united model based on cuckoo search oriented statistically by differential evolution (CSODE) is proposed. And simulations are done to a series of different noncommensurate and hyper fractional-order chaotic systems by a novel method based on CSODE, single cuckoo search, and Genetic Algorithm, respectively, in Section 4. Conclusions are summarized briefly in Section 5.


## 2. Gao’s Mathematical Model for Fractional Chaos Reconstruction and Orders Estimation in Non-Lyapunov Way

In this section, a general mathematical model for fractional chaos parameters identification in non-Lyapunov way is proposed. A detail explanation for the general mathematical model will be given in the following subsections in three aspects, submodel A, B, and C.


### 2.1. Gao’s Mathematical Model

Now we consider the general forms of fractional-order chaos system (1). To make the system (1) more clear, we take its equivalent form as the following system:
(6)α𝒟tq1y1(t)=f1(y2(t-τ),…,yn(t),yn(t-τ)t,y1(t),y1(t-τ),y2(t),y2(t-τ),…,yn(t),yn(t-τ)),α𝒟tq2y2(t)=f2(y2(t-τ),…,yn(t),yn(t-τ)t,y1(t),y1(t-τ),y2(t),y2(t-τ),…,yn(t),yn(t-τ))⋮α𝒟tqnyn(t)=fn(y2(t-τ),…,yn(t),yn(t-τ)t,y1(t),y1(t-τ),y2(t),y2(t-τ),…,yn(t),yn(t-τ)),L=(y1,y2,…,yn).And a correspondent system (7) is constructed as follow:
(7)α𝒟tq~1y~1(t)=f~1(y2(t-τ),…,yn(t),yn(t-τ)t,y~1(t),y~1(t-τ~),y~2(t),y~2(t-τ~),…,y~n(t),y~n(t-τ~)),α𝒟tq~2y~2(t)=f~2(y2(t-τ),…,yn(t),yn(t-τ)t,y~1(t),y~1(t-τ~),y~2(t),y~2(t-τ~),…,y~n(t),y~n(t-τ~))⋮α𝒟tq~ny~n(t)=f~n(y2(t-τ),…,yn(t),yn(t-τ)t,y~1(t),y~1(t-τ~),y~2(t),y~2(t-τ~),…,y~n(t),y~n(t-τ~)),L~=(y~1,y~2,…,y~n).To have simple forms, we takeα=0.Then novel objective function (fitness) (8) in this paper comes into being from (20) and (21) as follows:
(8)F=∑t=0·hT·h‍∥L~-L∥2,
where T is the number of sample points for methods resolving the fractional systems and h is the time step.Now a novel Gao mathematical model for fractional chaos reconstruction comes into being as Figure1 shows, where functions f=(f1,f2,…,fn), f~=(f~1,f~2,…,f~n), fractional orders q=(q1,q2,…,qn), q~=(q~1,q~2,…,q~n), time delays τ=(τ1,τ2,…,τn), τ~=(τ~1,τ~2,…,τ~n), and systematic parameters θ=(θ1,θ2,…,θn), θ~=(θ~1,θ~2,…,θ~n), respectively.Figure 1
Gao’s mathematical model for fractional chaos reconstruction.A detail explanation for the general mathematical model will be given in the following subsections in three aspects, sub-models A, B, and C.And objective function (8) to be optimized can also be any kind of (9) to be minimized by artificial intelligence methods as follows:
(9)F=∑i=1N‍∥Li-L~i∥2,G=1N∑i=1N‍∥Li-L~i∥2,H=∑i=1N‍∥Li-L~i∥2,W=1N∑i=1N‍∥Li-L~i∥2.


### 2.2. Mathematical Submodel A

It should be noticed here that the independent variables in function (8) in the general model in Figure 1 are not always the parameters and fractional orders. They can be the special variables, for instance, functions f~=(f~1,f~2,…,f~n), fractional orders q~=(q1,q2,…,qn), and time delays τ~=(τ~1,τ~2,…,τ~n).And for the submodel A, that is,(10)(f~,q~,τ~)*=argmin(f~,q~,τ~)F,it can also be written as follows:(11)((f~1,f~2,…,f~n),(q~1,q~2,…,q~n),(τ~1,τ~2,…,τ~n))*=argmin((f~1,f~2,…,f~n),(q~1,q~2,…,q~n),(τ~1,τ~2,…,τ~n))F.There exist several definitions of fractional derivatives. Among these, the Grünwald-Letnikov (G-L), the Riemann-Liouville (R-L), and the Caputo fractional derivatives are the commonly used [40–45]. And G-L, R-L, and Caputo fractional derivatives are equivalent under some conditions [46].The continuous integrodifferential operator [47, 48] is used, and we consider the continuous function f(t). The G-L fractional derivatives are defined as follows:
(12)α𝒟tqf(t)=limh→01hq∑j=0[(t-α)/h]‍(-1)j(qj)f(t-jh),
where [x] means the integer part of x, α, and t which are the bounds of operation for α𝒟tqf(t), q∈ℜ.We take ideas of a numerical solution method [47, 48] obtained by relationship (12) derived from the G-L definition to resolve system. That is,
(13)(k-Lm/h)𝒟tkqf(t)≈1hq∑j=0k‍(-1)j(qj)f(tk-j),
where Lm is the memory length, tk=kh, h is the time step of calculation, and (-1)j(qj) are binomial coefficients cj(q) (j=0,1,…,). When, for numerical computation, the following are used:
(14)c0(q)=1,cj(q)=(1-1+qj)cj-1(q),then in general, for the simplest case (15) of (6) as following. 								(15)0𝒟tqf(y(t))=f(t,y(t),y(t-τ)).Letykτ=y(tk-τ(tk)). It can have the approximate value as (16), when it used for calculating
(16)ykτ=yk+1+[τ/h]τ/h-[τ/h]h+yk+[τ/h]1+[τ/h]-τ/hh.And letyk=y(tk); then (15) can be expressed as
(17)h-q∑j=0k‍cj(q)y(tk-j)=f(tk,y(tk),y(tk-τ(tk))),y(tk)=f(tk,y(tk),y(tk-τ(tk)))hq-∑j=vk‍cj(q)y(tk-j),yk=f(tk,yk,ykτ)hq-∑j=vk‍cj(q)yk-j,
where v in the above is defined as
(18)v={k-Lmh,k>Lmh,1,k<Lmh
or v=1 for all k.Equation (17) is an implicit nonlinear equation respect to yk. Now we can construct an iteration algorithm to solve yk as following (19). 								(19)yk(l+1)=f(tk,yk(l),ykτ(l))hq-∑j=vk‍cj(q)yk-j,
where l is the iteration number. When |yk(l+1)-yk(l)|<δ (normally the given error δ<10-6), we consider yk=yk(l+1) to be the solution of the simplest equation (15). And if the derivative of f exists |f|≤M (M is constant), and hqM<1, then iteration (19) converges to a constant as long as the calculus step h is smaller enough.With the ideas from iteration of (19), the systems (6) and (7) are solved as follows:
(20)xik(l+1)=fi(tk-1,tk,x1k(l),x1k,τ(l),…,xi-1k(l),xi-1k,τ(l),xik-1(l),xik-1,τ(l),…,xnk-1(l),xnk-1,τ(l))hqi-∑j=vk‍cj(q)xik-j,(21)x~ik(l+1)=f~i(tk-1,tk,x~1k(l),x~1k,τ~(l),…,x~i-1k(l),x~i-1k,τ~(l),x~ik-1(l),x~ik-1,τ~(l),…,x~nk-1(l),x~nk-1,τ~(l))hq~i-∑j=vk‍cj(q~)x~ik-j.And ifτ=(0,0,…,0), then the system (6) can be solved as [36]
(22)xik(l+1)=fi(x1k(l),…,xi-1k(l),xik-1(l),…,xnk-1(l))hqi-∑j=vk‍cj(qi)xik-j.To the best of our knowledge, there is no work that has been done to reconstruct the fractional chaos systems under condition that bothf, q and τ are unknown in submodel A as (10) neither for time-delays free nor with time-delays chaos systems.


### 2.3. Mathematical Submodel B

In this sub-model,f are unknown, but τ and q are definite. Then to be estimated is only the fractional differential equations f; that is,
(23)(f~)*=argmin(f~)F.It is should be noticed that there are few methods for reconstruction of fractional-order chaos systems [36] so far.However, there are a few results for normal chaos systems, as the special cases of fractional chaos systems. For reconstruction off=(f1,f2,…,fn) with the non-Lyapunov methods, they are mainly from symbolic regression through genetic programming (GP) [49–51], and some are from evolutionary algorithms [52–62].Considering mathematical sub-model A, we have to say that it is really difficult to use the ideas in mathematical sub-model B. Let the input variables be taken asx, y, and z and let the basic operators set used be {+,-,×,÷,𝒟tp,=}, where fractional-order p∈[0,1] is uncertain. Now we consider the easiest cases that the fractional order differential equation 𝒟tq1x=f1(x,y,z) is unknown. Then we will see the individuals as following with the ideas similar to methods for the normal chaos of sub-model B. Figures 2(a) and 2(b) are the normal and correct candidate individuals only for the right part f1 of dx/dt=f1(x,y,z) of the normal chaos systems.Some examples of the tree structures in GP evolutions.
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Illegal candidatesHowever, when it comes to fractional-order chaos system, the whole fractional-order differential equations should be taken into accounts; that is,𝒟tq1x=f1(x,y,z) with q=p∈[0,1] uncertain and f1 unknown. Figure 2(c) shows a correct candidate. And when the evolutions (crossover, mutation, and selection) go on, there are some wrong and illegal candidates generated as Figures 2(d), 2(e), and 2(f) show. Figure 2(d) is a wrong candidate with 𝒟tp(x-y). Figure 2(e) is a wrong candidate with 𝒟tp1(x-y) and 𝒟tp2(y). Here it should be noticed that random p1,p2∈[0,1]. Figure 2(f) is a wrong candidate with not only 𝒟tp1(x-y), 𝒟tp2(z), and 𝒟tp3(z) but also extra {=}. Here it should be noticed that random p1,p2,p3∈[0,1].So long as the evolutions (crossover, mutation, and selection) go on, these wrong candidate inevitably exists, although, in the genetic programming, the tree depth is set to be limited with unlimited leaves. And these kinds of wrong individuals will become heavy burden for both the genetic evolution and resolving of the fractional-order differential equations.Thus it is not suitable to use the methods based on GP only to reconstruct the fractional-order chaos system; neither fractional orderq nor equations fi are unknown. However, if it is only considering the unknown equations fi with definite certain fractional order q, these methods will be impressive and efficient as in [36].


### 2.4. Mathematical Submodel C

In this sub-model C,(q1≠q2≠⋯≠qn), systematic parameters and time delays (θ1,θ2,…,θn),τ~=(τ~1,τ~2,…,τ~n)∈(f1,f2,…,fn) are unknown for noncommensurate and hyperfractional-order chaos system.There are some estimation methods that have been purposed to identify the unknown parameters and orders for commensurate fractional-order chaotic systems. However, to the best of our knowledge, no such reconstruction methods have been done for noncommensurate and hyper fractional-order chaos system; it is necessary to resolve the following equation in non-Lyapunov way:(24)(q~,θ~,τ~)*=argmin(q~,θ~,τ~)F.And there exist basic hypotheses in traditional non-Lyapunov estimation methods for fractional-order systems [7, 9, 28]. That is, the parameters and fractional orders are partially known or the known data series coincide with definite forms f=(f1,f2,…,fn) of fractional chaotic differential equations except for some uncertain parameters and fractional orders Θ=(θ1,θ2,…,θn,q1,q2,…,qn).This is the basic difference between submodel A, B, and C. And for the case when some chaotic differential equationsf=(f1,f2,…,fn) are unknown, there are some chaos system reconstruction methods. Then the cases [63–75] can be thought of as special cases of chaos reconstruction, when the exact forms of chaotic differential equations f=(f1,f2,…,fn) are available, but some parameters are unknown.


#### 2.4.1. Parameters Estimation for Fractional-Order Chaos Systems

We take the fractional-order Lorénz system (25) [3, 8, 24] as an example, which is generalized from the first canonical chaotic attractor found in 1963, Lorénz system [76]:
(25)α𝒟tq1x=σ·(y-x),α𝒟tq2y=γ·x-x·z-y,α𝒟tq3z=x·y-b·z,L=(x,y,z),
where q1, q2, and q3 are the fractional orders. When (q1,q2,q3)=(0.993,0.993,0.993), σ=10, γ=28, b=8/3, and α=0, intimal point (0.1,0.1,0.1) system (25) is chaotic. Generally when the dimension
(26)∑‍=q1+q2+q3>2.91
for fractional system (25) is chaotic [3, 8, 24], the form of function (10) can also be as follows:
(27)p2=F(σ,γ,b,q1,q2,q3)=∑t=0·hT·h‍∥L~-L∥2.It is noticed that objective function (27) can be any forms of correspond equations (9).Then the problems of estimation of parameters for chaotic system are transformed into those of nonlinear function optimization (27). And the smaller the p2 is, the better are combinations of parameter (σ,γ,b,q1,q2,q3). The independent variables of these functions are θ=(σ,γ,b,q1,q2,q3).And considering that the fractional system is very complicated, to simplify the problems, it is reported unknownq=q1=q2=q3,σ,γ,b or case of σ,γ,b are known and only one qi are unknown for the similar fractional-order chaos systems, such as fractional order Lü system [16, 77], fractional-order Chen system [27, 78], and fractional Lorénz system [3, 8, 24], discussed in [7, 9]. This is the basic idea for the recently proposed methods for fractional chaos system [7, 9].However, the caseq1≠q2≠q3 is not included in the above non-Lyapunov ideas or not fully discussed either for noncommensurate fractional chaos systems.


### 2.5. The Main Differences between Submodels A, B, and C

Equation (10) is the crucial turning point that changing from the parameters estimation into functions reconstruction and orders estimation, in other words, both fractional-order estimation and fractional chaos systems’ reconstruction.It can be concluded that the parameters’ estimation of fractional-order chaos system [7, 9] is a special case of fractional-order chaos reconstruction here as (10). In their researches, the forms of the fractional-order differential equations (f1,f2,…,fn) are known, but some parameters (θ1,θ2,…,θn) of these equations are unknown, and only one fractional order and some of these systematic parameters (θ1,θ2,…,θn) are estimated [7, 9].And further, the parameters estimation cases that all(f1,f2,…,fn) are known but parameters (θ1,θ2,…,θn) of these equations are unknown and the reconstruction case that some of (f1,f2,…,fn) are unknown, in Section 2.3 for the normal chaos system, are the special cases of fractional-order chaos systems’ reconstruction (10).However, it should be emphasized here that, for reconstruction the novel general mathematical model (10) for fractional chaos parameters identification in non-Lyapunov way, with uncertain different fractional order q; that is, q1≠q2≠⋯≠qn∈{×,÷,+,-,𝒟tq1,𝒟tq2,…,𝒟tqn}; it is really difficulty to generate a proper candidate from this basic set as shown in Figure 2. Then, it is not easy to reconstruct the fractional-order differential equations and identify the fractional orders together. And only the simplest case that with definite q=q1=q2=⋯=qn is discussed [36].


## 2.1. Gao’s Mathematical Model

Now we consider the general forms of fractional-order chaos system (1). To make the system (1) more clear, we take its equivalent form as the following system:
(6)α𝒟tq1y1(t)=f1(y2(t-τ),…,yn(t),yn(t-τ)t,y1(t),y1(t-τ),y2(t),y2(t-τ),…,yn(t),yn(t-τ)),α𝒟tq2y2(t)=f2(y2(t-τ),…,yn(t),yn(t-τ)t,y1(t),y1(t-τ),y2(t),y2(t-τ),…,yn(t),yn(t-τ))⋮α𝒟tqnyn(t)=fn(y2(t-τ),…,yn(t),yn(t-τ)t,y1(t),y1(t-τ),y2(t),y2(t-τ),…,yn(t),yn(t-τ)),L=(y1,y2,…,yn).And a correspondent system (7) is constructed as follow:
(7)α𝒟tq~1y~1(t)=f~1(y2(t-τ),…,yn(t),yn(t-τ)t,y~1(t),y~1(t-τ~),y~2(t),y~2(t-τ~),…,y~n(t),y~n(t-τ~)),α𝒟tq~2y~2(t)=f~2(y2(t-τ),…,yn(t),yn(t-τ)t,y~1(t),y~1(t-τ~),y~2(t),y~2(t-τ~),…,y~n(t),y~n(t-τ~))⋮α𝒟tq~ny~n(t)=f~n(y2(t-τ),…,yn(t),yn(t-τ)t,y~1(t),y~1(t-τ~),y~2(t),y~2(t-τ~),…,y~n(t),y~n(t-τ~)),L~=(y~1,y~2,…,y~n).To have simple forms, we takeα=0.Then novel objective function (fitness) (8) in this paper comes into being from (20) and (21) as follows:
(8)F=∑t=0·hT·h‍∥L~-L∥2,
where T is the number of sample points for methods resolving the fractional systems and h is the time step.Now a novel Gao mathematical model for fractional chaos reconstruction comes into being as Figure1 shows, where functions f=(f1,f2,…,fn), f~=(f~1,f~2,…,f~n), fractional orders q=(q1,q2,…,qn), q~=(q~1,q~2,…,q~n), time delays τ=(τ1,τ2,…,τn), τ~=(τ~1,τ~2,…,τ~n), and systematic parameters θ=(θ1,θ2,…,θn), θ~=(θ~1,θ~2,…,θ~n), respectively.Figure 1
Gao’s mathematical model for fractional chaos reconstruction.A detail explanation for the general mathematical model will be given in the following subsections in three aspects, sub-models A, B, and C.And objective function (8) to be optimized can also be any kind of (9) to be minimized by artificial intelligence methods as follows:
(9)F=∑i=1N‍∥Li-L~i∥2,G=1N∑i=1N‍∥Li-L~i∥2,H=∑i=1N‍∥Li-L~i∥2,W=1N∑i=1N‍∥Li-L~i∥2.


## 2.2. Mathematical Submodel A

It should be noticed here that the independent variables in function (8) in the general model in Figure 1 are not always the parameters and fractional orders. They can be the special variables, for instance, functions f~=(f~1,f~2,…,f~n), fractional orders q~=(q1,q2,…,qn), and time delays τ~=(τ~1,τ~2,…,τ~n).And for the submodel A, that is,(10)(f~,q~,τ~)*=argmin(f~,q~,τ~)F,it can also be written as follows:(11)((f~1,f~2,…,f~n),(q~1,q~2,…,q~n),(τ~1,τ~2,…,τ~n))*=argmin((f~1,f~2,…,f~n),(q~1,q~2,…,q~n),(τ~1,τ~2,…,τ~n))F.There exist several definitions of fractional derivatives. Among these, the Grünwald-Letnikov (G-L), the Riemann-Liouville (R-L), and the Caputo fractional derivatives are the commonly used [40–45]. And G-L, R-L, and Caputo fractional derivatives are equivalent under some conditions [46].The continuous integrodifferential operator [47, 48] is used, and we consider the continuous function f(t). The G-L fractional derivatives are defined as follows:
(12)α𝒟tqf(t)=limh→01hq∑j=0[(t-α)/h]‍(-1)j(qj)f(t-jh),
where [x] means the integer part of x, α, and t which are the bounds of operation for α𝒟tqf(t), q∈ℜ.We take ideas of a numerical solution method [47, 48] obtained by relationship (12) derived from the G-L definition to resolve system. That is,
(13)(k-Lm/h)𝒟tkqf(t)≈1hq∑j=0k‍(-1)j(qj)f(tk-j),
where Lm is the memory length, tk=kh, h is the time step of calculation, and (-1)j(qj) are binomial coefficients cj(q) (j=0,1,…,). When, for numerical computation, the following are used:
(14)c0(q)=1,cj(q)=(1-1+qj)cj-1(q),then in general, for the simplest case (15) of (6) as following. 								(15)0𝒟tqf(y(t))=f(t,y(t),y(t-τ)).Letykτ=y(tk-τ(tk)). It can have the approximate value as (16), when it used for calculating
(16)ykτ=yk+1+[τ/h]τ/h-[τ/h]h+yk+[τ/h]1+[τ/h]-τ/hh.And letyk=y(tk); then (15) can be expressed as
(17)h-q∑j=0k‍cj(q)y(tk-j)=f(tk,y(tk),y(tk-τ(tk))),y(tk)=f(tk,y(tk),y(tk-τ(tk)))hq-∑j=vk‍cj(q)y(tk-j),yk=f(tk,yk,ykτ)hq-∑j=vk‍cj(q)yk-j,
where v in the above is defined as
(18)v={k-Lmh,k>Lmh,1,k<Lmh
or v=1 for all k.Equation (17) is an implicit nonlinear equation respect to yk. Now we can construct an iteration algorithm to solve yk as following (19). 								(19)yk(l+1)=f(tk,yk(l),ykτ(l))hq-∑j=vk‍cj(q)yk-j,
where l is the iteration number. When |yk(l+1)-yk(l)|<δ (normally the given error δ<10-6), we consider yk=yk(l+1) to be the solution of the simplest equation (15). And if the derivative of f exists |f|≤M (M is constant), and hqM<1, then iteration (19) converges to a constant as long as the calculus step h is smaller enough.With the ideas from iteration of (19), the systems (6) and (7) are solved as follows:
(20)xik(l+1)=fi(tk-1,tk,x1k(l),x1k,τ(l),…,xi-1k(l),xi-1k,τ(l),xik-1(l),xik-1,τ(l),…,xnk-1(l),xnk-1,τ(l))hqi-∑j=vk‍cj(q)xik-j,(21)x~ik(l+1)=f~i(tk-1,tk,x~1k(l),x~1k,τ~(l),…,x~i-1k(l),x~i-1k,τ~(l),x~ik-1(l),x~ik-1,τ~(l),…,x~nk-1(l),x~nk-1,τ~(l))hq~i-∑j=vk‍cj(q~)x~ik-j.And ifτ=(0,0,…,0), then the system (6) can be solved as [36]
(22)xik(l+1)=fi(x1k(l),…,xi-1k(l),xik-1(l),…,xnk-1(l))hqi-∑j=vk‍cj(qi)xik-j.To the best of our knowledge, there is no work that has been done to reconstruct the fractional chaos systems under condition that bothf, q and τ are unknown in submodel A as (10) neither for time-delays free nor with time-delays chaos systems.


## 2.3. Mathematical Submodel B

In this sub-model,f are unknown, but τ and q are definite. Then to be estimated is only the fractional differential equations f; that is,
(23)(f~)*=argmin(f~)F.It is should be noticed that there are few methods for reconstruction of fractional-order chaos systems [36] so far.However, there are a few results for normal chaos systems, as the special cases of fractional chaos systems. For reconstruction off=(f1,f2,…,fn) with the non-Lyapunov methods, they are mainly from symbolic regression through genetic programming (GP) [49–51], and some are from evolutionary algorithms [52–62].Considering mathematical sub-model A, we have to say that it is really difficult to use the ideas in mathematical sub-model B. Let the input variables be taken asx, y, and z and let the basic operators set used be {+,-,×,÷,𝒟tp,=}, where fractional-order p∈[0,1] is uncertain. Now we consider the easiest cases that the fractional order differential equation 𝒟tq1x=f1(x,y,z) is unknown. Then we will see the individuals as following with the ideas similar to methods for the normal chaos of sub-model B. Figures 2(a) and 2(b) are the normal and correct candidate individuals only for the right part f1 of dx/dt=f1(x,y,z) of the normal chaos systems.Some examples of the tree structures in GP evolutions.
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Illegal candidatesHowever, when it comes to fractional-order chaos system, the whole fractional-order differential equations should be taken into accounts; that is,𝒟tq1x=f1(x,y,z) with q=p∈[0,1] uncertain and f1 unknown. Figure 2(c) shows a correct candidate. And when the evolutions (crossover, mutation, and selection) go on, there are some wrong and illegal candidates generated as Figures 2(d), 2(e), and 2(f) show. Figure 2(d) is a wrong candidate with 𝒟tp(x-y). Figure 2(e) is a wrong candidate with 𝒟tp1(x-y) and 𝒟tp2(y). Here it should be noticed that random p1,p2∈[0,1]. Figure 2(f) is a wrong candidate with not only 𝒟tp1(x-y), 𝒟tp2(z), and 𝒟tp3(z) but also extra {=}. Here it should be noticed that random p1,p2,p3∈[0,1].So long as the evolutions (crossover, mutation, and selection) go on, these wrong candidate inevitably exists, although, in the genetic programming, the tree depth is set to be limited with unlimited leaves. And these kinds of wrong individuals will become heavy burden for both the genetic evolution and resolving of the fractional-order differential equations.Thus it is not suitable to use the methods based on GP only to reconstruct the fractional-order chaos system; neither fractional orderq nor equations fi are unknown. However, if it is only considering the unknown equations fi with definite certain fractional order q, these methods will be impressive and efficient as in [36].


## 2.4. Mathematical Submodel C

In this sub-model C,(q1≠q2≠⋯≠qn), systematic parameters and time delays (θ1,θ2,…,θn),τ~=(τ~1,τ~2,…,τ~n)∈(f1,f2,…,fn) are unknown for noncommensurate and hyperfractional-order chaos system.There are some estimation methods that have been purposed to identify the unknown parameters and orders for commensurate fractional-order chaotic systems. However, to the best of our knowledge, no such reconstruction methods have been done for noncommensurate and hyper fractional-order chaos system; it is necessary to resolve the following equation in non-Lyapunov way:(24)(q~,θ~,τ~)*=argmin(q~,θ~,τ~)F.And there exist basic hypotheses in traditional non-Lyapunov estimation methods for fractional-order systems [7, 9, 28]. That is, the parameters and fractional orders are partially known or the known data series coincide with definite forms f=(f1,f2,…,fn) of fractional chaotic differential equations except for some uncertain parameters and fractional orders Θ=(θ1,θ2,…,θn,q1,q2,…,qn).This is the basic difference between submodel A, B, and C. And for the case when some chaotic differential equationsf=(f1,f2,…,fn) are unknown, there are some chaos system reconstruction methods. Then the cases [63–75] can be thought of as special cases of chaos reconstruction, when the exact forms of chaotic differential equations f=(f1,f2,…,fn) are available, but some parameters are unknown.


### 2.4.1. Parameters Estimation for Fractional-Order Chaos Systems

We take the fractional-order Lorénz system (25) [3, 8, 24] as an example, which is generalized from the first canonical chaotic attractor found in 1963, Lorénz system [76]:
(25)α𝒟tq1x=σ·(y-x),α𝒟tq2y=γ·x-x·z-y,α𝒟tq3z=x·y-b·z,L=(x,y,z),
where q1, q2, and q3 are the fractional orders. When (q1,q2,q3)=(0.993,0.993,0.993), σ=10, γ=28, b=8/3, and α=0, intimal point (0.1,0.1,0.1) system (25) is chaotic. Generally when the dimension
(26)∑‍=q1+q2+q3>2.91
for fractional system (25) is chaotic [3, 8, 24], the form of function (10) can also be as follows:
(27)p2=F(σ,γ,b,q1,q2,q3)=∑t=0·hT·h‍∥L~-L∥2.It is noticed that objective function (27) can be any forms of correspond equations (9).Then the problems of estimation of parameters for chaotic system are transformed into those of nonlinear function optimization (27). And the smaller the p2 is, the better are combinations of parameter (σ,γ,b,q1,q2,q3). The independent variables of these functions are θ=(σ,γ,b,q1,q2,q3).And considering that the fractional system is very complicated, to simplify the problems, it is reported unknownq=q1=q2=q3,σ,γ,b or case of σ,γ,b are known and only one qi are unknown for the similar fractional-order chaos systems, such as fractional order Lü system [16, 77], fractional-order Chen system [27, 78], and fractional Lorénz system [3, 8, 24], discussed in [7, 9]. This is the basic idea for the recently proposed methods for fractional chaos system [7, 9].However, the caseq1≠q2≠q3 is not included in the above non-Lyapunov ideas or not fully discussed either for noncommensurate fractional chaos systems.


## 2.4.1. Parameters Estimation for Fractional-Order Chaos Systems

We take the fractional-order Lorénz system (25) [3, 8, 24] as an example, which is generalized from the first canonical chaotic attractor found in 1963, Lorénz system [76]:
(25)α𝒟tq1x=σ·(y-x),α𝒟tq2y=γ·x-x·z-y,α𝒟tq3z=x·y-b·z,L=(x,y,z),
where q1, q2, and q3 are the fractional orders. When (q1,q2,q3)=(0.993,0.993,0.993), σ=10, γ=28, b=8/3, and α=0, intimal point (0.1,0.1,0.1) system (25) is chaotic. Generally when the dimension
(26)∑‍=q1+q2+q3>2.91
for fractional system (25) is chaotic [3, 8, 24], the form of function (10) can also be as follows:
(27)p2=F(σ,γ,b,q1,q2,q3)=∑t=0·hT·h‍∥L~-L∥2.It is noticed that objective function (27) can be any forms of correspond equations (9).Then the problems of estimation of parameters for chaotic system are transformed into those of nonlinear function optimization (27). And the smaller the p2 is, the better are combinations of parameter (σ,γ,b,q1,q2,q3). The independent variables of these functions are θ=(σ,γ,b,q1,q2,q3).And considering that the fractional system is very complicated, to simplify the problems, it is reported unknownq=q1=q2=q3,σ,γ,b or case of σ,γ,b are known and only one qi are unknown for the similar fractional-order chaos systems, such as fractional order Lü system [16, 77], fractional-order Chen system [27, 78], and fractional Lorénz system [3, 8, 24], discussed in [7, 9]. This is the basic idea for the recently proposed methods for fractional chaos system [7, 9].However, the caseq1≠q2≠q3 is not included in the above non-Lyapunov ideas or not fully discussed either for noncommensurate fractional chaos systems.


## 2.5. The Main Differences between Submodels A, B, and C

Equation (10) is the crucial turning point that changing from the parameters estimation into functions reconstruction and orders estimation, in other words, both fractional-order estimation and fractional chaos systems’ reconstruction.It can be concluded that the parameters’ estimation of fractional-order chaos system [7, 9] is a special case of fractional-order chaos reconstruction here as (10). In their researches, the forms of the fractional-order differential equations (f1,f2,…,fn) are known, but some parameters (θ1,θ2,…,θn) of these equations are unknown, and only one fractional order and some of these systematic parameters (θ1,θ2,…,θn) are estimated [7, 9].And further, the parameters estimation cases that all(f1,f2,…,fn) are known but parameters (θ1,θ2,…,θn) of these equations are unknown and the reconstruction case that some of (f1,f2,…,fn) are unknown, in Section 2.3 for the normal chaos system, are the special cases of fractional-order chaos systems’ reconstruction (10).However, it should be emphasized here that, for reconstruction the novel general mathematical model (10) for fractional chaos parameters identification in non-Lyapunov way, with uncertain different fractional order q; that is, q1≠q2≠⋯≠qn∈{×,÷,+,-,𝒟tq1,𝒟tq2,…,𝒟tqn}; it is really difficulty to generate a proper candidate from this basic set as shown in Figure 2. Then, it is not easy to reconstruct the fractional-order differential equations and identify the fractional orders together. And only the simplest case that with definite q=q1=q2=⋯=qn is discussed [36].


## 3. Cuckoo Search Oriented Statistically by Differential Evolution

### 3.1. Cuckoo Search

Cuckoo search (CS) is an optimization algorithm [37, 38], inspired by the obligate brood parasitism of some cuckoo species by laying their eggs in the nests of other host birds (of other species). And some host birds can come into direct conflict with the intruding cuckoos.CS is based on three idealized rules.(i)
Only one egg is laid and is dumped into a randomly chosen nest by each cuckoo at timet.(ii)
The best nests with high quality of eggs (candidate solutions) will be copied to the next generation directly.(iii)
The number of available host nests is fixed, and an alien egg will be discovered by a host bird with probabilitypa∈[0,1]. If so, the host can either throw the egg away or abandon the nest so as to build a completely new nest in a new location.A Lévy flight is performed for cuckooi when a new candidate x(t+1) is generated [37–39]:
(28)xi(t+1)=xi(t)+a⊕Lévy(λ),
where a>0 is the step size which should be related to the scales of the problem of interest. Normally, a=O(1). The product ⊕ means entrywise multiplications. Lévy flights essentially provide a random walk while their random steps are drawn from a Lévy distribution for large steps
(29)Lévy~u=t-λ(1<λ≤3),
which has an infinite variance with an infinite mean, and essentially form a random walk process obeying a power-law step-length distribution with a heavy tail [37–39].Based on the above rules and ideas, the basic steps of the CS can be summarised as shown in pseudocode of Algorithm1.Algorithm 1:The basic steps of the cuckoo search.
(1)Basic  parameters'  settingObjective function f(x),x=(x1,…,xd),
initial population of n host nests xi(i=1,2,…,n), boundaries for each
dimension xi=(xi,1,…,xi,d) and so forth.
(2)WhileTermination condition is not satisfieddo
(3) Get a cuckoo (i,i=1,2,…,n) randomly by Lévy flights (28).
(4)  Evaluate its fitness Fi;
(5)  If  (Fi>Fj)
(6)  Replace j by the new solution
(7)  End
(8)  Abandon a fraction (pa) of worse nests.
(9)  Build new ones at new locations via Lévy flights (28) (i,i=1,2,…,n).
(10)   Keep the best solutions (or nests with quality solutions).
(11)    Rank the solutions and find the current best.
(12)end  while
(13)OutputGlobal optimum QgIt should be noticed that, in each iteration of Algorithm1, there are two rounds of evaluation of the fitness: one is after getting a cuckoo by Lévy flights and the other is after abandon the worse nests with probability pa and building the new nest at the new locations. It is also showed in the original MATLAB code in [38].This might be the reason that CS is efficient. Because CS uses Lévy flights twice and evaluates the candidates twice in one generation. However, there is one evaluation for the whole population in normal swarm intelligent methods. If we consider the number of evaluating the fitness function by these two evaluations, they might not be economic.Thus, we can make some modifications here to accelerate the CS as Algorithm1 by decreasing the evaluation number for the fitness.


### 3.2. Differential Evolution Algorithm

Differential evolution (DE) algorithm was proposed by Storn et al. [79–82]. DE utilizes Mn–dimensional vectors, Xi=(xi1,…,xin)∈S, i=1,…,M, as a population for each iteration, called a generation, of the algorithm. For each vector Xi(G)=(Xi1(G),Xi2(G),…,Xin(G)), i=1,2,…,M, there are three main genetic operators acting [79–82].To apply the mutation operator, firstly choose randomly four mutually different individuals in the current populationXr1(G),Xr2(G),Xr3(G),Xr4(G) (r1≠r2≠r3≠r4≠i) to compose a differential vector D(G)=[Xr1(G)-Xr2(G)]+[Xr3(G)-Xr4(G)]; then combines it with the current best individual Xbest(G) to get a perturbed vector V=(V1,V2,…,Vn) [79, 83] as follows:
(30)V=Xbest(G)+CF×D(G),
where CF>0 is a user-defined real parameter, called mutation constant, which controls the amplification of the difference between two individuals to avoid search stagnation.Following the crossover phase, the crossover operator is applied onXi(G). Then a trial vector U=(U1,U2,…,Un) is generated by
(31)Um={Vm,if(rand(0,1)<CR)or(m=k),Xim(G),if(rand(0,1)≥CR)and(m≠k),
in the current population [79], where m=1,2,…,n, the index k∈{1,2,…,n} is randomly chosen, and CR is a user-defined crossover constant [79, 83] in the range [0,1]. In other words, the trial vector consists of some of the components of the mutant vector and at least one of the components of a randomly selected individual of the population.Then it comes to the replacement phase. To maintain the population size, we have to compare the fitness ofU and Xi(G) and then choose the better:
(32)Xi(G+1)={U,ifF(U)<F(Xi(G)),Xi(G),otherwise.


### 3.3. Cuckoo Search Oriented Statistically by Differential Evolution

Considering the redundant evaluation for the fitness function of CS and the efficiency of DE, we can propose a novel cuckoo search oriented statistically by differential evolution as shown in Algorithm2.Algorithm 2:Cuckoo search oriented statistically by differential evolution (CSODE).
(1)Basic  parameters'  setting Objective function f(x),x=(x1,…,xd),
initial population of n host nests xi(i=1,2,…,n), boundaries for each
dimension xi=(xi,1,…,xi,d) and so forth.
(2)While Termination condition is not satisfied do
(3)   If pDe<0.2, generating candidate cuckoo population (i,i=1,2,…,n)
randomly from current population by (31).
(4)   Updating the current cuckoo swarm and the candidate swarm with (32).
(5)   Abandon a fraction (pa) of worse nests.
(6)   Build new ones at new locations via Lévy flights (28).
(7)   Keep the best solutions (or nests with quality solutions).
(8)   Rank the solutions and find the current best.
(9)end  while
(10)OutputGlobal optimum QgIn each iteration of Algorithm2, Lévy flights (28) are used once for each location. And differential evolution operation is used with a probability pDe less than 0.2. In this way, the evaluations for the fitness function are reduced nearly by 80% compared to original Algorithm 1.AndpDe in Algorithm 2 CSODE should not be too big. Otherwise, it will cause Algorithm 2 to be much more like a DE algorithm rather than a cuckoo searcher algorithm. It will be illustrated in the section of simulations. Actually, our original idea is to let CS be oriented not controlled by DE.


## 3.1. Cuckoo Search

Cuckoo search (CS) is an optimization algorithm [37, 38], inspired by the obligate brood parasitism of some cuckoo species by laying their eggs in the nests of other host birds (of other species). And some host birds can come into direct conflict with the intruding cuckoos.CS is based on three idealized rules.(i)
Only one egg is laid and is dumped into a randomly chosen nest by each cuckoo at timet.(ii)
The best nests with high quality of eggs (candidate solutions) will be copied to the next generation directly.(iii)
The number of available host nests is fixed, and an alien egg will be discovered by a host bird with probabilitypa∈[0,1]. If so, the host can either throw the egg away or abandon the nest so as to build a completely new nest in a new location.A Lévy flight is performed for cuckooi when a new candidate x(t+1) is generated [37–39]:
(28)xi(t+1)=xi(t)+a⊕Lévy(λ),
where a>0 is the step size which should be related to the scales of the problem of interest. Normally, a=O(1). The product ⊕ means entrywise multiplications. Lévy flights essentially provide a random walk while their random steps are drawn from a Lévy distribution for large steps
(29)Lévy~u=t-λ(1<λ≤3),
which has an infinite variance with an infinite mean, and essentially form a random walk process obeying a power-law step-length distribution with a heavy tail [37–39].Based on the above rules and ideas, the basic steps of the CS can be summarised as shown in pseudocode of Algorithm1.Algorithm 1:The basic steps of the cuckoo search.
(1)Basic  parameters'  settingObjective function f(x),x=(x1,…,xd),
initial population of n host nests xi(i=1,2,…,n), boundaries for each
dimension xi=(xi,1,…,xi,d) and so forth.
(2)WhileTermination condition is not satisfieddo
(3) Get a cuckoo (i,i=1,2,…,n) randomly by Lévy flights (28).
(4)  Evaluate its fitness Fi;
(5)  If  (Fi>Fj)
(6)  Replace j by the new solution
(7)  End
(8)  Abandon a fraction (pa) of worse nests.
(9)  Build new ones at new locations via Lévy flights (28) (i,i=1,2,…,n).
(10)   Keep the best solutions (or nests with quality solutions).
(11)    Rank the solutions and find the current best.
(12)end  while
(13)OutputGlobal optimum QgIt should be noticed that, in each iteration of Algorithm1, there are two rounds of evaluation of the fitness: one is after getting a cuckoo by Lévy flights and the other is after abandon the worse nests with probability pa and building the new nest at the new locations. It is also showed in the original MATLAB code in [38].This might be the reason that CS is efficient. Because CS uses Lévy flights twice and evaluates the candidates twice in one generation. However, there is one evaluation for the whole population in normal swarm intelligent methods. If we consider the number of evaluating the fitness function by these two evaluations, they might not be economic.Thus, we can make some modifications here to accelerate the CS as Algorithm1 by decreasing the evaluation number for the fitness.


## 3.2. Differential Evolution Algorithm

Differential evolution (DE) algorithm was proposed by Storn et al. [79–82]. DE utilizes Mn–dimensional vectors, Xi=(xi1,…,xin)∈S, i=1,…,M, as a population for each iteration, called a generation, of the algorithm. For each vector Xi(G)=(Xi1(G),Xi2(G),…,Xin(G)), i=1,2,…,M, there are three main genetic operators acting [79–82].To apply the mutation operator, firstly choose randomly four mutually different individuals in the current populationXr1(G),Xr2(G),Xr3(G),Xr4(G) (r1≠r2≠r3≠r4≠i) to compose a differential vector D(G)=[Xr1(G)-Xr2(G)]+[Xr3(G)-Xr4(G)]; then combines it with the current best individual Xbest(G) to get a perturbed vector V=(V1,V2,…,Vn) [79, 83] as follows:
(30)V=Xbest(G)+CF×D(G),
where CF>0 is a user-defined real parameter, called mutation constant, which controls the amplification of the difference between two individuals to avoid search stagnation.Following the crossover phase, the crossover operator is applied onXi(G). Then a trial vector U=(U1,U2,…,Un) is generated by
(31)Um={Vm,if(rand(0,1)<CR)or(m=k),Xim(G),if(rand(0,1)≥CR)and(m≠k),
in the current population [79], where m=1,2,…,n, the index k∈{1,2,…,n} is randomly chosen, and CR is a user-defined crossover constant [79, 83] in the range [0,1]. In other words, the trial vector consists of some of the components of the mutant vector and at least one of the components of a randomly selected individual of the population.Then it comes to the replacement phase. To maintain the population size, we have to compare the fitness ofU and Xi(G) and then choose the better:
(32)Xi(G+1)={U,ifF(U)<F(Xi(G)),Xi(G),otherwise.


## 3.3. Cuckoo Search Oriented Statistically by Differential Evolution

Considering the redundant evaluation for the fitness function of CS and the efficiency of DE, we can propose a novel cuckoo search oriented statistically by differential evolution as shown in Algorithm2.Algorithm 2:Cuckoo search oriented statistically by differential evolution (CSODE).
(1)Basic  parameters'  setting Objective function f(x),x=(x1,…,xd),
initial population of n host nests xi(i=1,2,…,n), boundaries for each
dimension xi=(xi,1,…,xi,d) and so forth.
(2)While Termination condition is not satisfied do
(3)   If pDe<0.2, generating candidate cuckoo population (i,i=1,2,…,n)
randomly from current population by (31).
(4)   Updating the current cuckoo swarm and the candidate swarm with (32).
(5)   Abandon a fraction (pa) of worse nests.
(6)   Build new ones at new locations via Lévy flights (28).
(7)   Keep the best solutions (or nests with quality solutions).
(8)   Rank the solutions and find the current best.
(9)end  while
(10)OutputGlobal optimum QgIn each iteration of Algorithm2, Lévy flights (28) are used once for each location. And differential evolution operation is used with a probability pDe less than 0.2. In this way, the evaluations for the fitness function are reduced nearly by 80% compared to original Algorithm 1.AndpDe in Algorithm 2 CSODE should not be too big. Otherwise, it will cause Algorithm 2 to be much more like a DE algorithm rather than a cuckoo searcher algorithm. It will be illustrated in the section of simulations. Actually, our original idea is to let CS be oriented not controlled by DE.


## 4. A Novel Unknown Parameters and Orders Identification Method Based on CSODE for Noncommensurate Fractional-Order Chaos Systems

The task of this section is to find a simple but effective approach for unknownq and systematic parameters in (24) for noncommensurate fractional-order chaos based on CSODE in non-Lyapunov way.


### 4.1. A Novel Unknown Parameters and Orders Identification Method

Now we can propose a novel approach for hyper, proper, and improper fractional chaos systems. The pseudocode of the proposed reconstruction is given in Algorithm3.Algorithm 3:A novel unknown parameters and orders identification method based on differential evolution algorithms for noncommensurate and hyperfractional-order chaos systems.
(1)Basic  parameters'  setting  for  Algorithm2.
(2)InitializeGenerate the initial population.
(3)WhileTermination condition is not satisfieddo
(4)Algorithm 2 with fitness with (24).
(5)Boundary  constraintsFor each xik∈Xi,k=1,2,…,D, if xi1 is
beyond the boundary, it is replaced by a random number in the boundary.
(6)end  while
(7)OutputGlobal optimum xBest


### 4.2. Noncommensurate and Hyperfractional-Order Chaos Systems

To test Algorithm3, some different well-known and widely used noncommensurate and hyper fractional order chaos systems are choose as follows. To have a comparative result, these systems are taken from [36].Example 1.
Here we discuss the noncommensurate fractional Lorénz system [8, 36].Example 2.
Fractional-order Arneodo’s system  [36, 47, 84].Example 3.
Fractional-order Duffing’s system [47].Example 4.
Fractional-order Genesio-Tesi’s systems [47, 85].Example 5.
Fractional-order financial systems  [36, 47, 85].Example 6.
Fractional-order Lü system [16, 47].Example 7.
Improper fractional-order Chen system [27, 47, 78].Example 8.
Fractional-order Rössler system [12, 47].Example 9.
Fractional-order Chua’s oscillator [86].Example 10.
Hyperfractional-order Lorénz system [87].Example 11.
Hyperfractional-order Lü system [88].Example 12.
Hyperfractional-order Liu system [89].Example 13.
Hyperfractional-order Chen system [90].Example 14.
Hyperfractional-order Rössler system [12].Example 15.
A four-wing fractional-order system [91, 92] both incommensurate and hyperchaotic
(33)𝒟tq1x1=ax1-x2x3+x4,𝒟tq2x2=-bx2+x1x3,𝒟tq3x3=x1x2-cx3+x1x4,𝒟tq4x4=-x2.
When (a,b,c)=(8,40,49), (q1,q2,q3,q4)=(1,0.95,0.9,0.85), and initial point is (1,-2,3,1) [91], system (33) is chaotic.


### 4.3. Simulations

For systems to be identified, the parameters of the proposed method are set as follows. The parameters of the simulations are fixed: the size of the population was set equal toM=40, generation is set to 500, and the default values are CF=1, CR=0.85, and pDE=0.2; Table 1 gives the detail setting for each system.Table 1
Detail parameters setting for different systems.
F-O systems
Unknown
Lower boundary
Upper boundary
Step
NOSP
a
Example1
(σ, γ, b, q1, q2, q3)
5, 20, 0.1, 0.1, 0.1, 0.1
15, 30, 10, 1, 1, 1
0.01
100
Example2
(β1, β2, β3, β4, q1, q2, q3)
−6, 2, 0.1, −1.5, 0.1, 0.1, 0.1
−5, 5, 1, −0.5, 1, 1, 1
0.005
200
Example3
(a, b, c, q1, q2)
0.1, 0.1, 0.1, 0.1, 0.5
1, 1, 2, 1, 1.5
0.0005
500
Example4
(β1, β2, β3, β4, q1, q2, q3)
1, 1, 0.1, 0.1, 0.5, 0.5, 0.1
2, 2, 1, 1.5, 1.5, 1.5, 1
0.005
200
Example5
(a, b, c, q1, q2, q3)
0.5, 0.01, 0.5, 0.5, 0.1, 0.1
1.5, 1, 1.5, 1.5, 1, 1
0.005
200
Example6
(a, b, c, q1, q2, q3)
30, 0.1, 15, 0.1, 0.1, 0.1
40, 10, 25, 1, 1, 1
0.01
100
Example7
(a, b, c, d, q1, q2, q3)
30, 0.1, 20, −10, 0.5, 1, 1
40, 10, 30, −0.1, 2, 2, 2
0.01
100
Example8
(a, b, c, q1, q2, q3)
0.1, 0.1, 5, 0.1, 0.1, 0.1
1, 1, 15, 1, 1, 1
0.01
100
Example9
(α, β, γ, ζ, a, b, q1, q2, q3, q4)
5, 10, 0.1, 0.1, 0.3, 0.1, 0.1, 0.1, 0.1, 0.1
10, 20, 1, 2, 0.3, 1, 1, 1, 1, 1
0.01
100
Example10
(a, b, c, d, q1, q2, q3, q4)
5, 0.1, 20, −2, 0.1, 0.1, 0.1, 0.1
15, 5, 30, −0.1, 1, 1, 1, 1
0.01
100
Example11
(a, b, c, d, q1, q2, q3, q4)
30, 0.1, 15, 0.1, 0.1, 0.1, 0.1, 0.1
40, 5, 25, 5, 1, 1, 1, 1
0.005
200
Example12
(a, b, c, d, q1, q2, q3, q4)
5, 0.5, 1, 0.1, 0.1, 0.1, 0.1, 0.1
15, 1.5, 10, 1, 1, 1, 1, 1
0.005
100
Example13
(a, b, c, d, q1, q2, q3, q4)
30, 0.1, 10, 0.1, 0.1, 0.1, 0.1, 0.1, 0.1
40, 5, 20, 10, 1, 1, 1, 1, 1
0.005
200
Example14
(a, b, c, d, q1, q2, q3, q4)
0.1, 0.1, 0.1, 0.01, 0.1, 0.1, 0.1, 0.1
1, 5, 1, 1, 1, 1, 1, 1
0.005
200
Example15
(a, b, c, q1, q2, q3, q4)
5, 38, 45, 0.5, 0.5, 0.5, 0.5
10, 45, 50, 1, 1, 1, 1
0.005
200
aNumber of sample points for resolving the fractional chaos systems.Table2 shows the simulation results of the above fractional-order chaotic systems. And some simulations are done by single cuckoo search (CS) methods. In these cases, all the other parameters for the algorithms are the same as for CSODE. The simulation results are listed in Table 3.Table 2
Simulation results for different fractional-order chaos systems by CSODE.
F-O system
StD
Mean
Min.
Max.
Success ratea
NEOFb
Example1
2.2025
e
-
04
6.6862
e
-
04
3.1179
e
-
06
5.5594
e
-
03
100%
24013
Example2
4.7346
e
-
06
1.5061
e
-
05
3.2217
e
-
07
1.4355
e
-
04
100%
24073
Example3
5.8073
e
-
09
1.4890
e
-
08
1.3035
e
-
10
1.3548
e
-
07
100%
24016
Example4
1.3724
e
-
04
5.1929
e
-
04
7.6119
e
-
07
3.8262
e
-
03
100%
24073
Example5
2.7967
e
-
08
2.9039
e
-
08
1.5084
e
-
09
1.4728
e
-
07
100%
24013
Example6
2.7950
e
-
04
6.5471
e
-
04
6.8653
e
-
06
5.9367
e
-
03
100%
24038
Example7
6.5650
e
-
03
8.3188
e
-
03
1.6981
e
-
04
4.3289
e
-
02
83%
24074
Example8
1.8666
e
-
08
2.1368
e
-
08
1.2494
e
-
09
1.2324
e
-
07
100%
24013
Example9
1.4857
e
-
04
1.4529
e
-
04
1.9011
e
-
05
1.0265
e
-
03
100%
24073
Example10
5.1741
e
-
03
4.4624
e
-
03
5.0835
e
-
04
2.4994
e
-
02
87%
24041
Example11
1.2904
e
-
02
1.2573
e
-
02
5.8260
e
-
04
5.5329
e
-
02
100%c
24041
Example12
6.2490
e
-
06
4.1206
e
-
06
3.3623
e
-
07
2.3320
e
-
05
100%
24101
Example13
5.4440
e
-
02
4.5710
e
-
02
6.0972
e
-
03
2.7476
e
-
01
87%c
24022
Example14
2.0859
e
-
04
1.5394
e
-
04
3.1938
e
-
05
7.9808
e
-
04
100%
24041
Example15
4.5612
e
-
01
1.8007
e
+
00
3.1952
e
-
03
1.6640
e
+
01
56%c
24024
aSuccess means that the solution is less than 1e-2 in 100 independent simulations.
bNo. of average evaluation for objective function (NEOF).
cSuccess means that the solution is less than 1e-1 in 100 independent simulations.Table 3
Simulation for fractional-order systems by single cuckoo search.
System
StD
Mean
Min.
Max.
Success ratea
NEOFb
Example1
3.8566
e
-
01
6.8548
e
-
01
1.1083
e
-
01
2.2250
0%
40040
Example2
5.2993
e
-
04
1.1322
e
-
03
2.9726
e
-
04
2.7731
e
-
03
100%
40040
Example3
2.2560
e
-
04
3.8504
e
-
04
7.2187
e
-
05
1.4730
e
-
03
100%
40040
Example4
4.0392
e
-
03
1.8645
e
-
03
1.0365
e
-
03
1.0422
e
-
02
100%
40040
Example6
9.2611
e
-
01
1.8303
6.0910
e
-
01
6.4697
0%
40040
Example9
8.1825
e
-
02
4.0495
e
-
02
1.5058
e
-
02
2.3472
e
-
01
0%
40040
Example10
9.5917
4.4177
3.1812
2.5456
e
+
01
0%
40040
Example11
3.8130
7.1730
2.5631
2.7080
0%
40040
Example13
8.8389
3.5136
2.6274
2.4436
e
+
01
0%
40040
Example15
9.5876
e
-
02
1.4075
e
-
01
3.3331
e
-
02
5.7142
e
-
01
43%
40040
aSuccess means that the solution is less than 1e-1 in 100 independent simulations.
bNumber of evaluation for objective function.And comparisons of CSODE with evolutionary algorithms such as Genetic Algorithms are done. Here we choose the GA toolbox from MATLAB 2013a; most of the parameters are chosen as default in Matlab, except that population size is80, generation number is 600, exiting the GA’s evolution with average fitness value changes less than 10-16, and “Vectorized” is “on,” “UseParallel” is “always.” And simulation results are showed in Table 4.Table 4
Simulation for fractional-order systems by Genetic Algorithm.
System
StD
Mean
Min.
Max.
Success ratea
NEOFb
Example1
8.8774
7.6047
1.4454
17.78
0%
48080
Example2
0.098989
0.076223
0.0057602
0.1894
37%
48080
Example5
0.0074472
0.042908
0.035629
0.050513
100%
48080
Example6
0.52621
5.6488
5.0984
6.1469
0%
48080
Example8
0.025535
0.036555
0.007422
0.055056
40%
48080
Example10
5.5344
36.469
30.181
40.602
0%
48080
Example11
14.492
17.303
5.2875
33.397
0%
48080
Example13
4.8245
10.34
4.773
13.299
0%
48080
Example15
6.3815
721.09
713.72
724.77
0%
34294
aSuccess means that the solution is less than 1e-1 in 100 independent simulations.
bNo. of evaluation for objective function.The following figures give an illustration of how the self growing evolution process works by CSODE (Algorithm3). In which, Figures 3, 4, 5, 6, 7, 8, 9, 10, and 11 show the simulation evolution results of the above fractional order chaotic systems with optimization process of objective function’s evolution and the parameters and orders uncertain of the above fractional order chaotic systems.Evolution process for fractional-order Lorénz system.
(a)
(b)Evolution process for fractional-order Arneodo system.
(a)
(b)Evolution process for fractional-order financial system.
(a)
(b)Evolution process for fractional-order Lü system.
(a)
(b)Evolution process for fractional-order improper Chen system.
(a)
(b)Evolution process for hyper fractional-order Lü system.
(a)
(b)Evolution process for hyper fractional-order Liu system.
(a)
(b)Evolution process for hyper fractional-order Rössler system.
(a)
(b)Evolution process for fractional-order four-wing system (33).
(a)
(b)From the simulations results of the above fractional-order chaos system, it can be concluded that the proposed method is efficient. And from above figures, it can be concluded that the estimated systems are self-growing under the genetic operations of the proposed methods.To test the performance of the proposed method (Algorithm3), some more simulations are done to the four-wing incommensurate hyperfractional-order chaotic system (33) in the following cases A, B, C, and D. In these cases, each with only one condition is changed according to the original setting for system (33). The other parameters for the algorithms are the same as those for CSODE. The simulation results are listed in Table 5.(i)
Case A: enhancing the defined intervals of the unknown parameters and orders to[0,10]×[30,45]×[40,50]×[0.1,1]×[0.1,1]×[0.1,1]×[0.1,1].(ii)
Case B: reducing the number of samples for computing system (33) from 200 to 100.(iii)
Case C: increasing the iteration numbers of Algorithm3 from 500 to 800.(iv)
Case D: changing the population size of Algorithm3 from 40 to 80.Table 5
Simulation results for system (33).
System (33)
StD
Mean
Min.
Max.
Success rate
NEOFb
Case A
2.3271
e
+
02
2.2944
e
+
02
3.4062
e
-
02
5.2168
e
+
02
31%a
24000
Case Ac
3.5361
e
+
02
4.3992
e
+
02
2.9705
e
-
04
7.2477
e
+
02
39%a
24040
Case B
3.3699
e
-
02
1.6530
e
-
01
1.5989
e
-
05
8.3946
e
-
01
96%d
23800
Case C
6.2779
e
-
03
2.4339
e
-
02
1.4415
e
-
05
1.7882
e
-
01
98%d
39360
Case D
7.5365
e
-
01
2.3014
e
+
00
1.1463
e
-
02
1.6874
e
+
01
41%d
48320
aSuccess means that the solution is less than 1 in 100 independent simulations.
bNumber of evaluation for objective function.
cBy single DE with 600 generations.
dSuccess means that the solution is less than 1e-1 in 100 independent simulations.Figure12 shows the correspondent simulation results for system (33).Simulation results for system (33).
(a)
(b)From results of Tables2, 3, and 5 and Figure 12, we can conclude that minimizing the number of samples for computing the system (33) as case B, enhancing the iteration numbers as case C, and changing the population size of Algorithm 3 as case D, will make Algorithm 3 much efficient and achieve a much more higher precision. However if the defined intervals of the unknown parameters of system (33) are enhanced, then the results will go to the opposite way. That is, the success rate is from 90% to 20% as case A.If the number of evaluation for objective function is considered, it is that minimizing the number of samples for computing the system (33) as case B is the best way to achieve higher efficiency and precision.And according to Tables2 and 3, Algorithm 3 based on CSODE is much better than single cuckoo search.


## 4.1. A Novel Unknown Parameters and Orders Identification Method

Now we can propose a novel approach for hyper, proper, and improper fractional chaos systems. The pseudocode of the proposed reconstruction is given in Algorithm3.Algorithm 3:A novel unknown parameters and orders identification method based on differential evolution algorithms for noncommensurate and hyperfractional-order chaos systems.
(1)Basic  parameters'  setting  for  Algorithm2.
(2)InitializeGenerate the initial population.
(3)WhileTermination condition is not satisfieddo
(4)Algorithm 2 with fitness with (24).
(5)Boundary  constraintsFor each xik∈Xi,k=1,2,…,D, if xi1 is
beyond the boundary, it is replaced by a random number in the boundary.
(6)end  while
(7)OutputGlobal optimum xBest


## 4.2. Noncommensurate and Hyperfractional-Order Chaos Systems

To test Algorithm3, some different well-known and widely used noncommensurate and hyper fractional order chaos systems are choose as follows. To have a comparative result, these systems are taken from [36].Example 1.
Here we discuss the noncommensurate fractional Lorénz system [8, 36].Example 2.
Fractional-order Arneodo’s system  [36, 47, 84].Example 3.
Fractional-order Duffing’s system [47].Example 4.
Fractional-order Genesio-Tesi’s systems [47, 85].Example 5.
Fractional-order financial systems  [36, 47, 85].Example 6.
Fractional-order Lü system [16, 47].Example 7.
Improper fractional-order Chen system [27, 47, 78].Example 8.
Fractional-order Rössler system [12, 47].Example 9.
Fractional-order Chua’s oscillator [86].Example 10.
Hyperfractional-order Lorénz system [87].Example 11.
Hyperfractional-order Lü system [88].Example 12.
Hyperfractional-order Liu system [89].Example 13.
Hyperfractional-order Chen system [90].Example 14.
Hyperfractional-order Rössler system [12].Example 15.
A four-wing fractional-order system [91, 92] both incommensurate and hyperchaotic
(33)𝒟tq1x1=ax1-x2x3+x4,𝒟tq2x2=-bx2+x1x3,𝒟tq3x3=x1x2-cx3+x1x4,𝒟tq4x4=-x2.
When (a,b,c)=(8,40,49), (q1,q2,q3,q4)=(1,0.95,0.9,0.85), and initial point is (1,-2,3,1) [91], system (33) is chaotic.


## 4.3. Simulations

For systems to be identified, the parameters of the proposed method are set as follows. The parameters of the simulations are fixed: the size of the population was set equal toM=40, generation is set to 500, and the default values are CF=1, CR=0.85, and pDE=0.2; Table 1 gives the detail setting for each system.Table 1
Detail parameters setting for different systems.
F-O systems
Unknown
Lower boundary
Upper boundary
Step
NOSP
a
Example1
(σ, γ, b, q1, q2, q3)
5, 20, 0.1, 0.1, 0.1, 0.1
15, 30, 10, 1, 1, 1
0.01
100
Example2
(β1, β2, β3, β4, q1, q2, q3)
−6, 2, 0.1, −1.5, 0.1, 0.1, 0.1
−5, 5, 1, −0.5, 1, 1, 1
0.005
200
Example3
(a, b, c, q1, q2)
0.1, 0.1, 0.1, 0.1, 0.5
1, 1, 2, 1, 1.5
0.0005
500
Example4
(β1, β2, β3, β4, q1, q2, q3)
1, 1, 0.1, 0.1, 0.5, 0.5, 0.1
2, 2, 1, 1.5, 1.5, 1.5, 1
0.005
200
Example5
(a, b, c, q1, q2, q3)
0.5, 0.01, 0.5, 0.5, 0.1, 0.1
1.5, 1, 1.5, 1.5, 1, 1
0.005
200
Example6
(a, b, c, q1, q2, q3)
30, 0.1, 15, 0.1, 0.1, 0.1
40, 10, 25, 1, 1, 1
0.01
100
Example7
(a, b, c, d, q1, q2, q3)
30, 0.1, 20, −10, 0.5, 1, 1
40, 10, 30, −0.1, 2, 2, 2
0.01
100
Example8
(a, b, c, q1, q2, q3)
0.1, 0.1, 5, 0.1, 0.1, 0.1
1, 1, 15, 1, 1, 1
0.01
100
Example9
(α, β, γ, ζ, a, b, q1, q2, q3, q4)
5, 10, 0.1, 0.1, 0.3, 0.1, 0.1, 0.1, 0.1, 0.1
10, 20, 1, 2, 0.3, 1, 1, 1, 1, 1
0.01
100
Example10
(a, b, c, d, q1, q2, q3, q4)
5, 0.1, 20, −2, 0.1, 0.1, 0.1, 0.1
15, 5, 30, −0.1, 1, 1, 1, 1
0.01
100
Example11
(a, b, c, d, q1, q2, q3, q4)
30, 0.1, 15, 0.1, 0.1, 0.1, 0.1, 0.1
40, 5, 25, 5, 1, 1, 1, 1
0.005
200
Example12
(a, b, c, d, q1, q2, q3, q4)
5, 0.5, 1, 0.1, 0.1, 0.1, 0.1, 0.1
15, 1.5, 10, 1, 1, 1, 1, 1
0.005
100
Example13
(a, b, c, d, q1, q2, q3, q4)
30, 0.1, 10, 0.1, 0.1, 0.1, 0.1, 0.1, 0.1
40, 5, 20, 10, 1, 1, 1, 1, 1
0.005
200
Example14
(a, b, c, d, q1, q2, q3, q4)
0.1, 0.1, 0.1, 0.01, 0.1, 0.1, 0.1, 0.1
1, 5, 1, 1, 1, 1, 1, 1
0.005
200
Example15
(a, b, c, q1, q2, q3, q4)
5, 38, 45, 0.5, 0.5, 0.5, 0.5
10, 45, 50, 1, 1, 1, 1
0.005
200
aNumber of sample points for resolving the fractional chaos systems.Table2 shows the simulation results of the above fractional-order chaotic systems. And some simulations are done by single cuckoo search (CS) methods. In these cases, all the other parameters for the algorithms are the same as for CSODE. The simulation results are listed in Table 3.Table 2
Simulation results for different fractional-order chaos systems by CSODE.
F-O system
StD
Mean
Min.
Max.
Success ratea
NEOFb
Example1
2.2025
e
-
04
6.6862
e
-
04
3.1179
e
-
06
5.5594
e
-
03
100%
24013
Example2
4.7346
e
-
06
1.5061
e
-
05
3.2217
e
-
07
1.4355
e
-
04
100%
24073
Example3
5.8073
e
-
09
1.4890
e
-
08
1.3035
e
-
10
1.3548
e
-
07
100%
24016
Example4
1.3724
e
-
04
5.1929
e
-
04
7.6119
e
-
07
3.8262
e
-
03
100%
24073
Example5
2.7967
e
-
08
2.9039
e
-
08
1.5084
e
-
09
1.4728
e
-
07
100%
24013
Example6
2.7950
e
-
04
6.5471
e
-
04
6.8653
e
-
06
5.9367
e
-
03
100%
24038
Example7
6.5650
e
-
03
8.3188
e
-
03
1.6981
e
-
04
4.3289
e
-
02
83%
24074
Example8
1.8666
e
-
08
2.1368
e
-
08
1.2494
e
-
09
1.2324
e
-
07
100%
24013
Example9
1.4857
e
-
04
1.4529
e
-
04
1.9011
e
-
05
1.0265
e
-
03
100%
24073
Example10
5.1741
e
-
03
4.4624
e
-
03
5.0835
e
-
04
2.4994
e
-
02
87%
24041
Example11
1.2904
e
-
02
1.2573
e
-
02
5.8260
e
-
04
5.5329
e
-
02
100%c
24041
Example12
6.2490
e
-
06
4.1206
e
-
06
3.3623
e
-
07
2.3320
e
-
05
100%
24101
Example13
5.4440
e
-
02
4.5710
e
-
02
6.0972
e
-
03
2.7476
e
-
01
87%c
24022
Example14
2.0859
e
-
04
1.5394
e
-
04
3.1938
e
-
05
7.9808
e
-
04
100%
24041
Example15
4.5612
e
-
01
1.8007
e
+
00
3.1952
e
-
03
1.6640
e
+
01
56%c
24024
aSuccess means that the solution is less than 1e-2 in 100 independent simulations.
bNo. of average evaluation for objective function (NEOF).
cSuccess means that the solution is less than 1e-1 in 100 independent simulations.Table 3
Simulation for fractional-order systems by single cuckoo search.
System
StD
Mean
Min.
Max.
Success ratea
NEOFb
Example1
3.8566
e
-
01
6.8548
e
-
01
1.1083
e
-
01
2.2250
0%
40040
Example2
5.2993
e
-
04
1.1322
e
-
03
2.9726
e
-
04
2.7731
e
-
03
100%
40040
Example3
2.2560
e
-
04
3.8504
e
-
04
7.2187
e
-
05
1.4730
e
-
03
100%
40040
Example4
4.0392
e
-
03
1.8645
e
-
03
1.0365
e
-
03
1.0422
e
-
02
100%
40040
Example6
9.2611
e
-
01
1.8303
6.0910
e
-
01
6.4697
0%
40040
Example9
8.1825
e
-
02
4.0495
e
-
02
1.5058
e
-
02
2.3472
e
-
01
0%
40040
Example10
9.5917
4.4177
3.1812
2.5456
e
+
01
0%
40040
Example11
3.8130
7.1730
2.5631
2.7080
0%
40040
Example13
8.8389
3.5136
2.6274
2.4436
e
+
01
0%
40040
Example15
9.5876
e
-
02
1.4075
e
-
01
3.3331
e
-
02
5.7142
e
-
01
43%
40040
aSuccess means that the solution is less than 1e-1 in 100 independent simulations.
bNumber of evaluation for objective function.And comparisons of CSODE with evolutionary algorithms such as Genetic Algorithms are done. Here we choose the GA toolbox from MATLAB 2013a; most of the parameters are chosen as default in Matlab, except that population size is80, generation number is 600, exiting the GA’s evolution with average fitness value changes less than 10-16, and “Vectorized” is “on,” “UseParallel” is “always.” And simulation results are showed in Table 4.Table 4
Simulation for fractional-order systems by Genetic Algorithm.
System
StD
Mean
Min.
Max.
Success ratea
NEOFb
Example1
8.8774
7.6047
1.4454
17.78
0%
48080
Example2
0.098989
0.076223
0.0057602
0.1894
37%
48080
Example5
0.0074472
0.042908
0.035629
0.050513
100%
48080
Example6
0.52621
5.6488
5.0984
6.1469
0%
48080
Example8
0.025535
0.036555
0.007422
0.055056
40%
48080
Example10
5.5344
36.469
30.181
40.602
0%
48080
Example11
14.492
17.303
5.2875
33.397
0%
48080
Example13
4.8245
10.34
4.773
13.299
0%
48080
Example15
6.3815
721.09
713.72
724.77
0%
34294
aSuccess means that the solution is less than 1e-1 in 100 independent simulations.
bNo. of evaluation for objective function.The following figures give an illustration of how the self growing evolution process works by CSODE (Algorithm3). In which, Figures 3, 4, 5, 6, 7, 8, 9, 10, and 11 show the simulation evolution results of the above fractional order chaotic systems with optimization process of objective function’s evolution and the parameters and orders uncertain of the above fractional order chaotic systems.Evolution process for fractional-order Lorénz system.
(a)
(b)Evolution process for fractional-order Arneodo system.
(a)
(b)Evolution process for fractional-order financial system.
(a)
(b)Evolution process for fractional-order Lü system.
(a)
(b)Evolution process for fractional-order improper Chen system.
(a)
(b)Evolution process for hyper fractional-order Lü system.
(a)
(b)Evolution process for hyper fractional-order Liu system.
(a)
(b)Evolution process for hyper fractional-order Rössler system.
(a)
(b)Evolution process for fractional-order four-wing system (33).
(a)
(b)From the simulations results of the above fractional-order chaos system, it can be concluded that the proposed method is efficient. And from above figures, it can be concluded that the estimated systems are self-growing under the genetic operations of the proposed methods.To test the performance of the proposed method (Algorithm3), some more simulations are done to the four-wing incommensurate hyperfractional-order chaotic system (33) in the following cases A, B, C, and D. In these cases, each with only one condition is changed according to the original setting for system (33). The other parameters for the algorithms are the same as those for CSODE. The simulation results are listed in Table 5.(i)
Case A: enhancing the defined intervals of the unknown parameters and orders to[0,10]×[30,45]×[40,50]×[0.1,1]×[0.1,1]×[0.1,1]×[0.1,1].(ii)
Case B: reducing the number of samples for computing system (33) from 200 to 100.(iii)
Case C: increasing the iteration numbers of Algorithm3 from 500 to 800.(iv)
Case D: changing the population size of Algorithm3 from 40 to 80.Table 5
Simulation results for system (33).
System (33)
StD
Mean
Min.
Max.
Success rate
NEOFb
Case A
2.3271
e
+
02
2.2944
e
+
02
3.4062
e
-
02
5.2168
e
+
02
31%a
24000
Case Ac
3.5361
e
+
02
4.3992
e
+
02
2.9705
e
-
04
7.2477
e
+
02
39%a
24040
Case B
3.3699
e
-
02
1.6530
e
-
01
1.5989
e
-
05
8.3946
e
-
01
96%d
23800
Case C
6.2779
e
-
03
2.4339
e
-
02
1.4415
e
-
05
1.7882
e
-
01
98%d
39360
Case D
7.5365
e
-
01
2.3014
e
+
00
1.1463
e
-
02
1.6874
e
+
01
41%d
48320
aSuccess means that the solution is less than 1 in 100 independent simulations.
bNumber of evaluation for objective function.
cBy single DE with 600 generations.
dSuccess means that the solution is less than 1e-1 in 100 independent simulations.Figure12 shows the correspondent simulation results for system (33).Simulation results for system (33).
(a)
(b)From results of Tables2, 3, and 5 and Figure 12, we can conclude that minimizing the number of samples for computing the system (33) as case B, enhancing the iteration numbers as case C, and changing the population size of Algorithm 3 as case D, will make Algorithm 3 much efficient and achieve a much more higher precision. However if the defined intervals of the unknown parameters of system (33) are enhanced, then the results will go to the opposite way. That is, the success rate is from 90% to 20% as case A.If the number of evaluation for objective function is considered, it is that minimizing the number of samples for computing the system (33) as case B is the best way to achieve higher efficiency and precision.And according to Tables2 and 3, Algorithm 3 based on CSODE is much better than single cuckoo search.


## 5. Conclusions

The novel Gao mathematical model in Section2 is not only for fractional chaos parameters identification but also for reconstruction in non-Lyapunov way with three submodels A, B, and C.The put method based on CSODE consists of numerical optimization problem with unknown fractional-order differential equations to identify the chaotic systems. Simulation results demonstrate the effectiveness and efficiency of the proposed methods with the Gao mathematical model in Section2. This is a novel non-Lyapunov way for fractional order chaos’ unknown parameters and orders. The proposed methods solve the question that the unknown fractional-order q are not resolved in [36].Here we have to say that this work is only about the estimation of unknown parameters and orders with the objective function (24) for noncommensurate and hyper fractional-order chaos systems in non-Lyapunov way. It can be concluded that CSODE in Algorithm 3 can be changed to other artificial intelligence methods easily. And from Tables 2, 3, 4, and 5, we can conclude that Algorithm 3 with CSODE is better than CS and GA. And using the system (33), CSODE is better than DE for the bigger scale of the unknown parameters and orders.In the future, there are three interesting problems to be studied.(i)
Neither the fractional-orders nor some fractional order equations are unknown. That is, the objective function is chosen as (10) in the novel mathematical model in Section 2. A simple way for this might be the approaches combining the fractional orders and fractional-order equations together that might be both the estimation methods as artificial intelligent methods and the reconstruction methods as in [36] together in some degree.(ii)
Time delays and systematic parameters are unknown for fractional time-delay chaos systems. The objective function will be selected as the objective function (24) for the time-delay fractional chaotic systems as in Gao’s submodel C, which have special characteristics.(iii)
Cases with noises. Normally, the white noise will be added to the Gao three sub-models. The similar ideas discussed in Section2 will be used.In conclusion, it has to be stated that Algorithm3 for fractional-order chaos systems’ identification in a non-Lyapunov way is a promising direction.



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*Source: 382834-2013-12-10.xml*