Abstract
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.
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 fractionalorder chaos systems, and chaos synchronization [1–33].
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.(i)Lyapunov way: there have been few results on parameter estimation method of fractionalorder 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)NonLyapunov way via artificial intelligence methods, for examples, differential evolution [7] and particle swarm optimization [9], in which the commensurate fractionalorder chaos systems and simplest case with one unknown order for normal fractionalorder chaos systems are discussed: however, to the best of our knowledge, little work in nonLyapunov way has been done to the parameters and orders estimation of noncommensurate and hyper fractionalorder chaos systems. And there is no general mathematical model that has been purposed for all these kinds of identification.
We consider the following fractionalorder chaos system with time delays where denotes the state vector. denotes the original parameters, is the time delay, and (, ) is the fractional derivative orders. Consider
Normally the function is known. And the , , and are unknown; then will be the parameters to be estimated.
Then a correspondent system is constructed as follows where , , , and are the correspondent variables to those in (1), and function is the same. The two systems (1) and (3) have the same initial condition .
Then the objective is obtained as follows:
How could we identify the fractional system, when some fractional chaotic differential equations are unknown? That is,
Now the problem of parameters estimation (4) becomes another much more complicated question, fractionalorder chaos reconstruction problem [36], to find the forms of fractionalorder equations as in (5). In [36] a novel nonLyapunov reconstruction method based on a novel united mathematical model was proposed to reconstruct the unknown equations .
When it comes to the system (1) with neither nor are known, the united model is not effective. For the united mathematical model [36], to be identified is only instead of . That is, of (3) are not included. Actually, if the are taken into consideration in united model (5), then the basic parameters’ setting to be reconstructed in [36] will be basic set with extra and so forth and the input variables with extra and, and so forth. Although for the candidates “programs” in [36] the maximum depth of tree is , 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 timeconsuming 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 in unknown, it is really difficult to generate an individual with , 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 in are integer orders than improper fractional orders.
Therefore, to estimate the of (3) with unknown systematic parameters is still a question to be solved for parameters and orders estimation of noncommensurate and hyperfractionalorder 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 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 2, a general mathematical model not only for fractional chaos parameters identification but also for reconstruction in nonLyapunov way is newly proposed and analyzed in three submodels A, B, and C. And a simple review was given on nonLyapunov parameters estimation methods for fractionalorder 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 fractionalorder 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 NonLyapunov Way
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.
2.1. Gao’s Mathematical Model
Now we consider the general forms of fractionalorder chaos system (1). To make the system (1) more clear, we take its equivalent form as the following system:
And a correspondent system (7) is constructed as follow:
To have simple forms, we take .
Then novel objective function (fitness) (8) in this paper comes into being from (20) and (21) as follows: where is the number of sample points for methods resolving the fractional systems and is the time step.
Now a novel Gao mathematical model for fractional chaos reconstruction comes into being as Figure 1 shows, where functions , , fractional orders , , time delays , , and systematic parameters , , respectively.
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 (8) to be optimized can also be any kind of (9) to be minimized by artificial intelligence methods as follows:
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 , fractional orders , and time delays .
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 [40–45]. And GL, RL, and Caputo fractional derivatives are equivalent under some conditions [46].
The continuous integrodifferential operator [47, 48] is used, and we consider the continuous function . The GL fractional derivatives are defined as follows: where means the integer part of , , and which are the bounds of operation for , .
We take ideas of a numerical solution method [47, 48] obtained by relationship (12) derived from the GL definition to resolve system. That is, where is the memory length, , is the time step of calculation, and are binomial coefficients (,). When, for numerical computation, the following are used:
then in general, for the simplest case (15) of (6) as following.
Let . It can have the approximate value as (16), when it used for calculating
And let ; then (15) can be expressed as where in the above is defined as or for all .
Equation (17) is an implicit nonlinear equation respect to . Now we can construct an iteration algorithm to solve as following (19). where is the iteration number. When (normally the given error ), we consider to be the solution of the simplest equation (15). And if the derivative of exists ( is constant), and , then iteration (19) converges to a constant as long as the calculus step is smaller enough.
With the ideas from iteration of (19), the systems (6) and (7) are solved as follows:
And if , then the system (6) can be solved as [36]
To the best of our knowledge, there is no work that has been done to reconstruct the fractional chaos systems under condition that both , and are unknown in submodel A as (10) neither for timedelays free nor with timedelays chaos systems.
2.3. Mathematical Submodel B
In this submodel, are unknown, but and are definite. Then to be estimated is only the fractional differential equations ; that is,
It is should be noticed that there are few methods for reconstruction of fractionalorder 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 of with the nonLyapunov methods, they are mainly from symbolic regression through genetic programming (GP) [49–51], and some are from evolutionary algorithms [52–62].
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 , , and and let the basic operators set used be , where fractionalorder is uncertain. Now we consider the easiest cases that the fractional order differential equation is unknown. Then we will see the individuals as following with the ideas similar to methods for the normal chaos of submodel B. Figures 2(a) and 2(b) are the normal and correct candidate individuals only for the right part of of the normal chaos systems.
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(f) Illegal candidates
However, when it comes to fractionalorder chaos system, the whole fractionalorder differential equations should be taken into accounts; that is, with uncertain and 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 . Figure 2(e) is a wrong candidate with and . Here it should be noticed that random . Figure 2(f) is a wrong candidate with not only , , and but also extra . Here it should be noticed that random .
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 nor equations are unknown. However, if it is only considering the unknown equations with definite certain fractional order , these methods will be impressive and efficient as in [36].
2.4. Mathematical Submodel C
In this submodel C, , systematic parameters and time delays are unknown for noncommensurate and hyperfractionalorder chaos system.
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 [7, 9, 28]. That is, the parameters and fractional orders are partially known or the known data series coincide with definite forms of fractional chaotic differential equations except for some uncertain parameters and fractional orders .
This is the basic difference between submodel A, B, and C. And for the case when some chaotic differential equations 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 are available, but some parameters are unknown.
2.4.1. Parameters Estimation for FractionalOrder Chaos Systems
We take the fractionalorder Lorénz system (25) [3, 8, 24] as an example, which is generalized from the first canonical chaotic attractor found in 1963, Lorénz system [76]: where , , and are the fractional orders. When , , , , and , intimal point system (25) is chaotic. Generally when the dimension for fractional system (25) is chaotic [3, 8, 24], the form of function (10) can also be as follows:
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 is, the better are combinations of parameter . The independent variables of these functions are .
And considering that the fractional system is very complicated, to simplify the problems, it is reported unknown or case of are known and only one are unknown for the similar fractionalorder chaos systems, such as fractional order Lü system [16, 77], fractionalorder 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 case is not included in the above nonLyapunov 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 fractionalorder estimation and fractional chaos systems’ reconstruction.
It can be concluded that the parameters’ estimation of fractionalorder chaos system [7, 9] is a special case of fractionalorder chaos reconstruction here as (10). In their researches, the forms of the fractionalorder differential equations are known, but some parameters of these equations are unknown, and only one fractional order and some of these systematic parameters are estimated [7, 9].
And further, the parameters estimation cases that all are known but parameters of these equations are unknown and the reconstruction case that some of are unknown, in Section 2.3 for the normal chaos system, are the special cases of fractionalorder 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 nonLyapunov way, with uncertain different fractional order ; that is, ; 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 fractionalorder differential equations and identify the fractional orders together. And only the simplest case that with definite 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 time .(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 probability . 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 when a new candidate is generated [37–39]: where is the step size which should be related to the scales of the problem of interest. Normally, . 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 which has an infinite variance with an infinite mean, and essentially form a random walk process obeying a powerlaw steplength 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 Algorithm 1.

It should be noticed that, in each iteration of Algorithm 1, 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 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 Algorithm 1 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 M –dimensional vectors, , , as a population for each iteration, called a generation, of the algorithm. For each vector , , there are three main genetic operators acting [79–82].
To apply the mutation operator, firstly choose randomly four mutually different individuals in the current population () to compose a differential vector ; then combines it with the current best individual to get a perturbed vector [79, 83] as follows: where is a userdefined 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 on . Then a trial vector is generated by in the current population [79], where , the index is randomly chosen, and CR is a userdefined crossover constant [79, 83] in the range . 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 of and and then choose the better:
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 Algorithm 2.

In each iteration of Algorithm 2, Lévy flights (28) are used once for each location. And differential evolution operation is used with a probability less than . In this way, the evaluations for the fitness function are reduced nearly by compared to original Algorithm 1.
And 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 FractionalOrder Chaos Systems
The task of this section is to find a simple but effective approach for unknown and systematic parameters in (24) for noncommensurate fractionalorder chaos based on CSODE in nonLyapunov 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 Algorithm 3.

4.2. Noncommensurate and HyperfractionalOrder Chaos Systems
To test Algorithm 3, some different wellknown 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. Fractionalorder Arneodo’s system [36, 47, 84].
Example 3. Fractionalorder Duffing’s system [47].
Example 4. Fractionalorder GenesioTesi’s systems [47, 85].
Example 5. Fractionalorder financial systems [36, 47, 85].
Example 6. Fractionalorder Lü system [16, 47].
Example 7. Improper fractionalorder Chen system [27, 47, 78].
Example 8. Fractionalorder Rössler system [12, 47].
Example 9. Fractionalorder Chua’s oscillator [86].
Example 10. Hyperfractionalorder Lorénz system [87].
Example 11. Hyperfractionalorder Lü system [88].
Example 12. Hyperfractionalorder Liu system [89].
Example 13. Hyperfractionalorder Chen system [90].
Example 14. Hyperfractionalorder Rössler system [12].
Example 15. A fourwing fractionalorder system [91, 92] both incommensurate and hyperchaotic When , , and initial point is [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 to , generation is set to 500, and the default values are , , and ; Table 1 gives the detail setting for each system.
Table 2 shows the simulation results of the above fractionalorder 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.
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 , generation number is , exiting the GA’s evolution with average fitness value changes less than , and “Vectorized” is “on,” “UseParallel” is “always.” And simulation results are showed in Table 4.
The following figures give an illustration of how the self growing evolution process works by CSODE (Algorithm 3). 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.
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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 3), some more simulations are done to the fourwing incommensurate hyperfractionalorder 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 .(ii)Case B: reducing the number of samples for computing system (33) from 200 to 100.(iii)Case C: increasing the iteration numbers of Algorithm 3 from 500 to 800.(iv)Case D: changing the population size of Algorithm 3 from 40 to 80.
Figure 12 shows the correspondent simulation results for system (33).
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From results of Tables 2, 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 to 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 Tables 2 and 3, Algorithm 3 based on CSODE is much better than single cuckoo search.
5. Conclusions
The novel Gao mathematical model in Section 2 is not only for fractional chaos parameters identification but also for reconstruction in nonLyapunov way with three submodels A, B, and C.
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 2. This is a novel nonLyapunov way for fractional order chaos’ unknown parameters and orders. The proposed methods solve the question that the unknown fractionalorder 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 fractionalorder chaos systems in nonLyapunov 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 fractionalorders 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 fractionalorder 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 timedelay chaos systems. The objective function will be selected as the objective function (24) for the timedelay 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 submodels. The similar ideas discussed in Section 2 will be used.
In conclusion, it has to be stated that Algorithm 3 for fractionalorder chaos systems’ identification in a nonLyapunov way is a promising direction.
Acknowledgments
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.