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

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 problemsoffractional-orderchaos’identificationareconvertedintoamultiplemodalnonnegativefunctions’minimizationthroughapropertranslation,whichtakesfractional-ordersandparametersasitsparticularindependentvariables.Andtheobjectiveistofindthebestcombinationsoffractional-ordersandsystematicparametersoffractionalorderchaoticsystemsasspecialindependentvariablessuchthattheobjectivefunctionisminimized.SimulationsaredonetoestimateaseriesofnoncommensurateandhyperfractionalchaoticsystemswiththenewapproachesbasedonCSODE,thecuckoosearch,andGeneticAlgorithm,respectively.Theexperiments’resultsshowthattheproposedidentificationmechanismbasedonCSODEforfractionalordersandparametersisasuccessfulmethodforfractional-orderchaoticsystems,withtheadvantagesofhighprecisionandrobustness.

put and analysed in three sub-models, 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 fractionalorder chaos' identification are converted into a multiple modal non-negative functions' minimization through a proper translation, which takes fractionalorders and parameters as its particular independent variables. And the objective is to find 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 non-commensurate and hyper fractional chaotic systems with the new approaches based on CSODE, the cuckoo search and differential evolution respectively. The experiments' results show that the proposed identification mechanism based on CSODE for fractional-orders and parameters is a successful methods for fractional-order chaotic systems, with the advantages of high precision and robustness.
Keywords: Non-commensurate and hyper fractional-order chaotic system, unknown fractional orders, unknown systematic parameters, cuckoo search oriented statistically the differential evolution, non-negative special

Introduction
The applications of fractional differential equations began to appeal to related scientists recently  in following areas, bifurcation, hyperchaos, proper and improper fractional-order chaos systems and chaos synchronization .
However, there are some systematic parameters and orders 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 at two main approaches in parameters' identification for fractional-order chaos systems.
• 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 is still a tough task with technique and sensitively depends on the considered systems.
• non-Lyapunov way via artificial intelligence methods For examples, such as 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 non-commensurate and hyper fractional-order chaos systems.
And there are no general mathematical model has been purposed for all these kinds of identification.
We consider the following fractional-order chaos system with time delays.
Then a correspondent system are constructed as following.
whereỸ (t),θ,q,τ are the correspondent variables to those in equation (1), and function f are the same. The two systems (1) (2) have the same initial condition Y 0 (t).
Then the objective is obtained as following, When some the fractional chaotic differential equations f = (f 1 , f 2 , ..., f n ) are unknown, how to identify the fractional system? That is, .., f n ) * = arg min Now the problem of parameters estimation (3) become another much more complicated question, fractional-order chaos reconstruction problem [36], to find the forms of fractional order equations as in (4). In Reference [36] a novel non-Lyapunov reconstruction method based on a novel united mathematical model was proposed to reconstruct the unknown equations (f 1 , f 2 , ..., f n ).
When it comes to the system with neither q nor θ are known, the united model are not effective. For the united mathematical model [36], to be identified is only (f 1 , f 2 , ..., f n ) instead of q. That is α D q tỸ (t) of the equation (2) are not included. Actually, if the q are taken into consideration in the united model (4), then the basic parameters' setting to be reconstructed in Reference [36] will be basic set {×, ÷, +, −} with extra {=, D q t } etc., and the input variables {x 1 , x 2 , ..., x n } with {Y } extra and etc. Although for the candidates "programs" in Reference [36] the maximum depth of tree is 6, considering the maximum number of nodes per tree is infinite, there will be infinite illegal candidates will be generated. Then, in one hand, the most time-consuming thing for the novel united model(4) is to kill these illegal individuals from the legal individuals. However, these defaults are not solved in Reference [36]. In the other hand, as q ∈ D q t in unknown, it is really difficult to generate an individual with {×, ÷, +, −, =, D q t }, 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 are much more efficient for the systems with coefficients in (f 1 , f 2 , ..., f n ) as integers than as improper fractions. Therefore, to estimate the q of the equation (2) with unknown systematic parameters θ is still a question to be solved for parameters and orders estimation of non-commensurate and hyper fractional-order chaos systems.
And Cuckoo search (CS) is an 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][38][39], which essentially provide a random walk while their random steps are drawn from a Lévy distribution for large steps [37][38][39]. However, in of CS evolution, the Lévy flights in each main iteration are used twice. It has two results, the CS' searching performance become a little strong but the redundant evaluations for the objective function are generated too. Therefore, some more efforts are need to improve the performance of CS.
To the best of authors' knowledge, there are no methods in non-Lyapunov way for non-commensurate 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 non-commensurate and hyper fractional order chaotic systems in a non-Lyapunov way. And the illustrative reconstruction simulations in different chaos systems system are discussed respectively.
The rest is organized as follows. In Section 2, a general mathematical model not only for fractional chaos parameters identification but also for reconstruction in non-Lyapunov way are newly proposed and analyzed in three sub-models A, B and C. And a simple review was given on non-Lyapunove parameters estimation methods for fractional-order and normal chaos systems.
In section 3, a novel methods with proposed united model based on Cuckoo search oriented by differential evolution statistically (CSODE) is proposed.
And simulations are done to a series of different non-commensurate and hyper fractional order chaotic systems by a novel method based on CSODE in Section 4. Conclusions are summarized briefly in Section 5.

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 given as following subsections in three aspects, sub-model A, B, and C.

Gao's mathematical model
Now we consider the general forms of fractional order chaos systems (1). To make the system (1) more clear, we take its equivalent form as following system (5).
To have simple forms, we take α = 0.
Then the novel objective function (fitness) equation (7) in this paper come into being from equations (18) and (19) as below.

Mathematical sub-model A.
It should be noticed here that the independent variables in function (7) in the general model in Figure 1 are not always the parameters and fractional orders. They can be the special variables, for instance, as func- ..,f n ), fractional ordersq = (q 1 , q 2 , ..., q n ) and time delays τ = (τ 1 ,τ 2 , ...,τ n ).
And for the sub-model A, that is It can also be written as following.
The continuous integro-differential operator [47,48] is used, and we consider the continuous function f (t). The G-L fractional derivatives are defined as following.
where [x] means the integer part of x, α, t are the bounds of operation for We take ideas of a numerical solution method [47,48] obtained by the relationship (13) derived from the G-L definition to resolve system. That is, where L m is the memory length, t k = kh, h is the time step of calculation and j , (j = 0, 1, ...,). When for numerical computation, the following are used, Then in general, for the simplest case (14) of equation (5) as following.
Let y kτ = y(t k − τ (t k )) .It can have the approximate value as equation (15), when it used for calculating.
And let y k = y(t k ) , then equation (14) can be expressed as Equation (16) is a implicit nonlinear equation respect to y k . Now we can construct an iteration algorithm to solve y k as following (17).
To our best of knowledge, there is no work have been done to reconstruct the fractional chaos systems under condition that both f, q and τ are unknown in sub-model A as equation (12) neither for time-delays free nor with time-delays chaos systems .

Mathematical sub-model 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 It is should be noticed that there is few method for reconstruction for 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.
Considering mathematical sub-model A, we have to say it is really difficult to use the ideas in mathematical sub-model B. Let the input variables are taken as x, y, z and the basic operators set used be {+, −, ×, ÷, D p t , =}, where fractional order p ∈ [0, 1] is uncertain. Now we consider easiest cases that the is unknown. Then we will see the individuals as following with the ideas similar to methods for the normal chaos of sub-model B. Figure 2 However, when it comes to fractional order chaos system, the whole fractional order differential equations should be taken into accounts, that is Here it should be noticed that random p 1 , p 2 , p 3 ∈ [0, 1]. So long as the evolutions (crossover, mutation and selection) go on, these wrong candidate are inevitably existing, although in the genetic program-ming, the tree depth is set to be limited with unlimited leaves. And these kind of wrong individuals will become heavy burden for both the genetic evolution and resolving 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 order q nor equations f i are unknown. However, if it is only considering the unknown equations f i with definite certain fractional order q , these methods will be impressive and efficient as in Reference [36].

Mathematical sub-model C.
In this sub-model C, (q 1 = q 2 = ... = q n ) , systematic parameters and .., f n ) are unknown for non-commensurate and hyper fractional order chaos system.
There some estimation methods have been purposed to identify the unknown parameters and orders for commensurate fractional order chaotic systems. However, to our best of knowledge, no such reconstruction methods have been done for non-commensurate and hyper fractional order chaos system, it is necessary to resolve the following (22) in non-Lyapuno+v way.

Parameters estimation for fractional order chaos systems
We take the fractional order Lorénz system (23) [3,8,24] for example, which is generalized from the first canonical chaotic attractor found in 1963, Lorénz system [76].
The form of function (12) can also be as following: It is noticed that the objective function (24) can be any forms of correspond equations (8),(9),(10), (11).
Then the problems of estimation of parameters for chaotic system are transformed into that of nonlinear function optimization (24). And the smaller p 2 is, the better combinations of parameter (σ, γ, b, q 1 , q 2 , q 3 ) is. The independent variables of these functions are θ = (σ, γ, b, q 1 , q 2 , q 3 ).
And considering the fractional system is very complicated, to simplify the problems, it is reported unknown q = q 1 = q 2 = q 3 , σ, γ, b or case of σ, γ, b are known and only one q i are unknown for the similar fractional order chaos systems, such as fractional order Lü system [16,77] fractional order Chen system [27,78] fractional Lorénz system [3,8,24], discussed in Ref. [7,9].
The above is the basic idea for the recently proposed methods for fractional chaos system [7,9].
However, the case q 1 = q 2 = q 3 are not included in the above non-Lyapunov ideas or not fully discussed either for non-commensurate fractional chaos systems.

The main differences between sub-models A, B and C
Equation (12) 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 (12). In their researches, the forms of the fractional order differential equations (f 1 , f 2 , ..., f n ) are known but some parameters (θ 1 , θ 2 , ..., θ n ) of these equations are unknown, and only one the fractional order and some of these systematic parameters (θ 1 , θ 2 , ..., θ n ) are estimated [7,9].
And further, the parameters estimation cases that all (f 1 , f 2 , ..., f n ) are known but parameters (θ 1 , θ 2 , ..., θ n ) of these equations are unknown, and the reconstruction case that some of (f 1 , f 2 , ..., f n ) are unknown, in Section 2.3 for the normal chaos system, are the special cases of fractional order chaos systems' reconstruction (12).
However, it should be emphasized here that , for reconstruction the novel general mathematical model (12) for fractional chaos parameters identification in non-Lyapunov way, with uncertain different fractional order q, that is q 1 = q 2 = ... = q n ∈ {×, ÷, +, −, D q 1 t , D q 2 t , ..., D qn t }, 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 = q 1 = q 2 = ... = q n are discussed [36].
3. Cuckoo search oriented statistically by differential evolution

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 engage direct conflict with the intruding cuckoos.
CS is based on three idealized rules: • Only one egg is laid, and is dumped into a randomly chosen nest by each cuckoo at time t; • The best nests with high quality of eggs (candidate solutions) will be copied to the next generation directly; • The number of available host nests is fixed, and an alien egg will be discovered by a host bird with probability p a ∈ [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 cuckoo i when a new candidates x (t+1) is generated [37][38][39], 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 entry-wise multiplications. Lévy flights essentially provide a random walk while their random steps are drawn from a Lévy distribution for large steps which has an infinite variance with an infinite mean, essentially form a random walk process obeying a power-law step-length distribution with a heavy tail [37][38][39].
Based on above rules and ideas, the basic steps of the CS can be summarised as following pseudo code Algorithm 1. Get a cuckoo (i, i = 1, 2, ..., n) randomly by Lévy flights (25).

4:
Evaluate its fitness F i ;

5:
If (F i > F j )

6:
Replace j by the new solution 7: End 8: Abandon a fraction (p a ) of worse nests.

10:
Keep the best solutions (or nests with quality solutions).

11:
Rank the solutions and find the current best.
12: end while 13: Output Global optimum Q g there are 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 Algorithm 1 by decreasing the evaluation number for the fitness.
To apply the mutation operator, firstly random choose four mutually different individual in the current population X (G) r 4 ], then combines it with the current best individual X (G) best to get a perturbed vector V = (V 1 , V 2 , · · · , V n ) [79,83] as below: 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.

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 following Algorithm 2.
In each iteration of Algorithm 2, Lévy flights (25) is used once for each location. And differential evolution operation are used with a probability p De less than 0.2. In this way, the evaluations for the fitness function are reduced nearly 80% compared to the original Algorithm 1.
And p De in Algorithm 2 CSODE should not be too big. Otherwise, it will cause the algorithm 2 be much more like a DE algorithm rather than a cuckoo searcher algorithm. It will be illustrated in the section simulations.
Actually, our original idea is let CS oriented not controlled by DE. If p De < 0.2, generating candidate cuckoo population (i, i = 1, 2, ..., n) randomly from current population by equation (27).

4:
Updating the current cuckoo swarm and the candidate swarm with equation (28).

5:
Abandon a fraction (p a ) of worse nests.

7:
Keep the best solutions (or nests with quality solutions).

8:
Rank the solutions and find the current best.
9: end while 10: Output Global optimum Q g

A novel unknown parameters and orders identification method
based on CSODE for non-commensurate fractional order chaos systems The task of this section is to find a simple but effective approach for unknown q and systematic parameters in equation (22) of for non-commensurate fractional-order chaos based on CSODE in non-Lyapunov way.

A novel unknown parameters and orders identification method
Now we can propose a novel approach for hyper, proper and improper fractional chaos systems. The pseudo-code of the proposed reconstruction is given below.

Algorithm 3 A novel unknown parameters and orders identification method
based on differential evolution algorithms for non-commensurate and hyper fractional order chaos systems 1: Basic parameters' setting for Algorithm 2 .
2: Initialize Generate the initial population.
3: while Termination condition is not satisfied do

5:
Boundary constraints For each x ik ∈ X i , k = 1, 2, ..., D, if x i1 is beyond the boundary, it is replaced by a random number in the boundary.

Non-commensurate and hyper fractional order chaos systems
To test the Algorithm 3, some different well known and widely used noncommensurate and hyper fractional order chaos systems are choose as follow-ing. To have a comparative results, these systems are taken from reference [36].
Example. 15. A four-wing fractional order system [92,93] both incommensurate and hyper chaotic.  Table 1 give the detail setting for each system. Table 2 shows the simulation results of 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.
The following figures give a illustration how the self growing evolution process works by DE Algorithm 3. In which, Figures 3,4 ,5, 6 7 , 8 , 9,10, 11 show the simulation evolution results of above fractional order chaotic systems with optimization process of objective function's evolution and the parameters and orders uncertain of above fractional order chaotic systems.
From the simulations results of above fractional order chaos system, it can be concluded that the proposed method is efficient. And from above figures,   c Success means the the solution is less than 1e − 1 in 100 independent simulations. a Success means the the solution is less than 1e − 1 in 100 independent simulations.
485 490 495 500 10 −5 Figure 11: Evolution process for fractional order four wing system (42) 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 Algorithms 3 , some more simulations are done to the four-wing incommensurate hyper fractional order chaotic system (42) in following cases A,B,C,D. In these cases, each with only one condition is changed according to the original setting for system (42). The other parameters for the algorithms are the same as for CSODE.
The simulation results are listed in Table 4. .
• Case B. Reducing the number of samples for computing system (42) from 200 to 100.
• Case C. Increasing the iteration numbers of Algorithms 3 from 500 to 800.
• Case D. Changing the population size of Algorithms 3 from 40 to 80. Figure 12 show the coresspondent simulation results for system (42).
From results of the Table 2,3, 4 and Figure 12, we can conclude that minimizing the number of samples for computing the system (42) as case B, enhancing the iteration numbers as case C, the population size of Algorithms 3 as case D, will make the Algorithms 3 much more efficient and achieve a much more higher precision. However if the defined intervals of the    Figure 12: Simulation results for system (42) unknown parameters of system (42) 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 No. of evaluation for objective function is considered, it is that minimizing the number of samples for computing the system (42) as case B is the best way to achieve higher efficiency and precision.
And according to Table 2,3, the Algorithms 3 based on CSODE is much more better than single cuckoo search.

Conclusions
The novel Gao's mathematical model in Section 2 is not only for fractional chaos parameters identification but also for reconstruction in non-Lyapunov way with three sub-models 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's mathematical model in Section 2. This is a novel Non-Lyaponov way for fractional order chaos' unknown parameters and orders. The proposed the method solve a the question that the unknown fractional order q are not resolved in reference [36].
Here we have to say that this work is only about the estimation of unknown parameters and orders with the objective function (22)  In the future, there are three interesting problems to be studied.
• Neither the fractional orders nor some fractional order equations are unknown. That is, the objective function is chosen as (12) in the novel mathematic 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 Reference [36] together in some degree.
• Time-delays and systematic parameters are unknown for fractional time-delay chaos systems. The objective function will be selected as the objective function (22) for the time-delay fractional chaotic systems as in Gao's sub-model C , which have special characteristics.
• Cases with noises. Normally, the white noise will be added to the Gao's three sub-models. The similar ideas discussed in Section 2 will be used.
In conclusion, it has to be stated that proposed Algorithms 3 for fractional order chaos systems' identification in a non-Lyapunov way is a promising direction.