In this paper, a nonLyapunov 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 fractionalorder chaos’ identification are converted into a multiple modal nonnegative functions’ minimization through a proper translation, which takes fractionalorders and parameters as its particular independent variables. And the objective is to find the best combinations of fractionalorders 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 fractionalorder chaotic systems, with the advantages of high precision and robustness.
The applications of fractional differential equations began to appeal to related scientists recently [
However, there are some systematic parameters and orders that are unknown for the fractionalorder chaos systems in controlling and synchronization. It is difficult to identify the parameters in the fractionalorder chaotic systems with unknown parameters. Hitherto, there have been two main approaches in parameters’ identification for fractionalorder chaos systems.
Lyapunov way: there have been few results on parameter estimation method of fractionalorder chaotic systems based on chaos synchronization [
NonLyapunov way via artificial intelligence methods, for examples, differential evolution [
We consider the following fractionalorder chaos system with time delays
Normally the function
Then a correspondent system is constructed as follows
Then the objective is obtained as follows:
How could we identify the fractional system, when some fractional chaotic differential equations
Now the problem of parameters estimation (
When it comes to the system (
Therefore, to estimate the
And cuckoo search (CS) is a relatively new and robust optimization algorithm [
To the best of authors’ knowledge, there is no method in nonLyapunov way for noncommensurate and hyper fractionalorder 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 fractionalorder chaotic systems in a nonLyapunov way. And the illustrative reconstruction simulations in different chaos systems are discussed, respectively.
The rest is organized as follows. In Section
In this section, a general mathematical model for fractional chaos parameters identification in nonLyapunov 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.
Now we consider the general forms of fractionalorder chaos system (
And a correspondent system (
To have simple forms, we take
Then novel objective function (fitness) (
Now a novel Gao mathematical model for fractional chaos reconstruction comes into being as Figure
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, submodels A, B, and C.
And objective function (
It should be noticed here that the independent variables in function (
And for the submodel A, that is,
it can also be written as follows:
There exist several definitions of fractional derivatives. Among these, the GrünwaldLetnikov (GL), the RiemannLiouville (RL), and the Caputo fractional derivatives are the commonly used [
The continuous integrodifferential operator [
We take ideas of a numerical solution method [
then in general, for the simplest case (
Let
And let
Equation (
With the ideas from iteration of (
And if
To the best of our knowledge, there is no work that has been done to reconstruct the fractional chaos systems under condition that both
In this submodel,
It is should be noticed that there are few methods for reconstruction of fractionalorder chaos systems [
However, there are a few results for normal chaos systems, as the special cases of fractional chaos systems. For reconstruction of
Considering mathematical submodel A, we have to say that it is really difficult to use the ideas in mathematical submodel B. Let the input variables be taken as
Some examples of the tree structures in GP evolutions.
Illegal candidates
However, when it comes to fractionalorder chaos system, the whole fractionalorder differential equations should be taken into accounts; that is,
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 fractionalorder differential equations.
Thus it is not suitable to use the methods based on GP only to reconstruct the fractionalorder chaos system; neither fractional order
In this submodel C,
There are some estimation methods that have been purposed to identify the unknown parameters and orders for commensurate fractionalorder chaotic systems. However, to the best of our knowledge, no such reconstruction methods have been done for noncommensurate and hyper fractionalorder chaos system; it is necessary to resolve the following equation in nonLyapunov way:
And there exist basic hypotheses in traditional nonLyapunov estimation methods for fractionalorder systems [
This is the basic difference between submodel A, B, and C. And for the case when some chaotic differential equations
We take the fractionalorder Lorénz system (
It is noticed that objective function (
Then the problems of estimation of parameters for chaotic system are transformed into those of nonlinear function optimization (
And considering that the fractional system is very complicated, to simplify the problems, it is reported unknown
However, the case
Equation (
It can be concluded that the parameters’ estimation of fractionalorder chaos system [
And further, the parameters estimation cases that all
However, it should be emphasized here that, for reconstruction the novel general mathematical model (
Cuckoo search (CS) is an optimization algorithm [
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
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
A Lévy flight is performed for cuckoo
Based on the above rules and ideas, the basic steps of the CS can be summarised as shown in pseudocode of Algorithm
(1)
(2)
(3)
(4)
(5)
(6)
(7)
(8)
(9)
(10)
(11)
(12)
(13)
It should be noticed that, in each iteration of Algorithm
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 Algorithm
Differential evolution (DE) algorithm was proposed by Storn et al. [
To apply the mutation operator, firstly choose randomly four mutually different individuals in the current population
Following the crossover phase, the crossover operator is applied on
Then it comes to the replacement phase. To maintain the population size, we have to compare the fitness of
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 Algorithm
(1)
(2)
(3)
(4)
(5)
(6)
(7)
(8)
(9)
(10)
In each iteration of Algorithm
And
The task of this section is to find a simple but effective approach for unknown
Now we can propose a novel approach for hyper, proper, and improper fractional chaos systems. The pseudocode of the proposed reconstruction is given in Algorithm
(1)
(2)
(3)
(4)
(5)
(6)
(7)
To test Algorithm
Here we discuss the noncommensurate fractional Lorénz system [
Fractionalorder Arneodo’s system [
Fractionalorder Duffing’s system [
Fractionalorder GenesioTesi’s systems [
Fractionalorder financial systems [
Fractionalorder Lü system [
Improper fractionalorder Chen system [
Fractionalorder Rössler system [
Fractionalorder Chua’s oscillator [
Hyperfractionalorder Lorénz system [
Hyperfractionalorder Lü system [
Hyperfractionalorder Liu system [
Hyperfractionalorder Chen system [
Hyperfractionalorder Rössler system [
A fourwing fractionalorder system [
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 to
Detail parameters setting for different systems.
FO systems  Unknown  Lower boundary  Upper boundary  Step 


Example 
( 
5, 20, 0.1, 0.1, 0.1, 0.1  15, 30, 10, 1, 1, 1  0.01  100 
Example 
( 
−6, 2, 0.1, −1.5, 0.1, 0.1, 0.1  −5, 5, 1, −0.5, 1, 1, 1  0.005  200 
Example 
( 
0.1, 0.1, 0.1, 0.1, 0.5  1, 1, 2, 1, 1.5  0.0005  500 
Example 
( 
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 
Example 
( 
0.5, 0.01, 0.5, 0.5, 0.1, 0.1  1.5, 1, 1.5, 1.5, 1, 1  0.005  200 
Example 
( 
30, 0.1, 15, 0.1, 0.1, 0.1  40, 10, 25, 1, 1, 1  0.01  100 
Example 
( 
30, 0.1, 20, −10, 0.5, 1, 1  40, 10, 30, −0.1, 2, 2, 2  0.01  100 
Example 
( 
0.1, 0.1, 5, 0.1, 0.1, 0.1  1, 1, 15, 1, 1, 1  0.01  100 
Example 
( 
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 
Example 
( 
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 
Example 
( 
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 
Example 
( 
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 
Example 
( 
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 
Example 
( 
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 
Example 
( 
5, 38, 45, 0.5, 0.5, 0.5, 0.5  10, 45, 50, 1, 1, 1, 1  0.005  200 
Table
Simulation results for different fractionalorder chaos systems by CSODE.
FO system  StD  Mean  Min.  Max.  Success rate^{a}  NEOF^{b} 

Example 




100%  24013 
Example 




100%  24073 
Example 




100%  24016 
Example 




100%  24073 
Example 




100%  24013 
Example 




100%  24038 
Example 




83%  24074 
Example 




100%  24013 
Example 




100%  24073 
Example 




87%  24041 
Example 




100%^{c}  24041 
Example 




100%  24101 
Example 




87%^{c}  24022 
Example 




100%  24041 
Example 




56%^{c}  24024 
Simulation for fractionalorder systems by single cuckoo search.
System  StD  Mean  Min.  Max.  Success rate^{a}  NEOF^{b} 

Example 



2.2250  0%  40040 
Example 




100%  40040 
Example 




100%  40040 
Example 




100%  40040 
Example 



6.4697  0%  40040 
Example 




0%  40040 
Example 




0%  40040 
Example 



2.7080  0%  40040 
Example 




0%  40040 
Example 




43%  40040 
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 is
Simulation for fractionalorder systems by Genetic Algorithm.
System 
StD  Mean  Min.  Max.  Success rate^{a}  NEOF^{b} 

Example 
8.8774 
7.6047  1.4454  17.78  0%  48080 
Example 
0.098989 
0.076223  0.0057602  0.1894  37%  48080 
Example 
0.0074472  0.042908  0.035629  0.050513  100%  48080 
Example 
0.52621 
5.6488  5.0984  6.1469  0%  48080 
Example 
0.025535 
0.036555  0.007422  0.055056  40%  48080 
Example 
5.5344 
36.469  30.181  40.602  0%  48080 
Example 
14.492  17.303  5.2875  33.397  0%  48080 
Example 
4.8245 
10.34  4.773  13.299  0%  48080 
Example 
6.3815  721.09  713.72  724.77  0%  34294 
The following figures give an illustration of how the self growing evolution process works by CSODE (Algorithm
Evolution process for fractionalorder Lorénz system.
Evolution process for fractionalorder Arneodo system.
Evolution process for fractionalorder financial system.
Evolution process for fractionalorder Lü system.
Evolution process for fractionalorder improper Chen system.
Evolution process for hyper fractionalorder Lü system.
Evolution process for hyper fractionalorder Liu system.
Evolution process for hyper fractionalorder Rössler system.
Evolution process for fractionalorder fourwing system (
From the simulations results of the above fractionalorder 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 selfgrowing under the genetic operations of the proposed methods.
To test the performance of the proposed method (Algorithm
Case A: enhancing the defined intervals of the unknown parameters and orders to
Case B: reducing the number of samples for computing system (
Case C: increasing the iteration numbers of Algorithm
Case D: changing the population size of Algorithm
Simulation results for system (
System ( 
StD  Mean  Min.  Max.  Success rate  NEOF^{b} 

Case A 




31%^{a}  24000 
Case A^{c} 




39%^{a}  24040 
Case B 




96%^{d}  23800 
Case C 




98%^{d}  39360 
Case D 




41%^{d}  48320 
Figure
Simulation results for system (
From results of Tables
If the number of evaluation for objective function is considered, it is that minimizing the number of samples for computing the system (
And according to Tables
The novel Gao mathematical model in Section
The put method based on CSODE consists of numerical optimization problem with unknown fractionalorder differential equations to identify the chaotic systems. Simulation results demonstrate the effectiveness and efficiency of the proposed methods with the Gao mathematical model in Section
Here we have to say that this work is only about the estimation of unknown parameters and orders with the objective function (
In the future, there are three interesting problems to be studied.
Neither the fractionalorders nor some fractional order equations are unknown. That is, the objective function is chosen as (
Time delays and systematic parameters are unknown for fractional timedelay chaos systems. The objective function will be selected as the objective function (
Cases with noises. Normally, the white noise will be added to the Gao three submodels. The similar ideas discussed in Section
In conclusion, it has to be stated that Algorithm
The work is partially supported by the NSFC Projects nos. 91324201, 81271513 of China, the Fundamental Research Funds for the Central Universities of China, the SelfDetermined and Innovative Research Funds of WUT No. 2012Ia035, 2012Ia041, 2013Ia040, and 2010Ia004, Scientific Research Foundation for Returned Scholars from Ministry of Education of China (no. 20111j0032), the HOME Program no. 11044 (Help Our Motherland through Elite Intellectual Resources from Overseas) funded by China Association for Science and Technology, the Natural Science Foundation no. 2009CBD213 of Hubei Province of China, The National Soft Science Research Program 2009GXS1D012 of China, and the National Science Foundation for Postdoctoral Scientists of China no. 20080431004. The work was partially carried out during the tenure of the ERCIM Alain Bensoussan Fellowship Programme, which is supported by the Marie Curie Cofunding of Regional, National, and International Programmes (COFUND) of the European Commission.