Research Article | Open Access
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 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.
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  and methods for parameter estimation of uncertain fractional order complex networks . 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  and particle swarm optimization , 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 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, fractional-order chaos reconstruction problem , to find the forms of fractional-order equations as in (5). In  a novel non-Lyapunov 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 , 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  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  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 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 . 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  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 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 Section 2, 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
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, sub-models A, B, and C.
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 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 .
The continuous integrodifferential operator [47, 48] is used, and we consider the continuous function . The G-L 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 G-L 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:
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.
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 time-delays free nor with time-delays chaos systems.
2.3. Mathematical Submodel B
In this sub-model, 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 fractional-order chaos systems  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 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 as , , and and let the basic operators set used be , where fractional-order 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 sub-model 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.
(f) Illegal candidates
However, when it comes to fractional-order chaos system, the whole fractional-order 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 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 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 .
2.4. Mathematical Submodel C
In this sub-model C, , systematic parameters and time delays 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:
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 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 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 : 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:
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 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 case 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 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 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 ; 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 fractional-order differential equations and identify the fractional orders together. And only the simplest case that with definite is discussed .
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 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 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 .
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 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 on . Then a trial vector is generated by in the current population , where , the index is randomly chosen, and CR is a user-defined 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 Fractional-Order Chaos Systems
The task of this section is to find a simple but effective approach for unknown 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 Algorithm 3.
4.2. Noncommensurate and Hyperfractional-Order Chaos Systems
To test Algorithm 3, 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 .
Example 3. Fractional-order Duffing’s system .
Example 9. Fractional-order Chua’s oscillator .
Example 10. Hyperfractional-order Lorénz system .
Example 11. Hyperfractional-order Lü system .
Example 12. Hyperfractional-order Liu system .
Example 13. Hyperfractional-order Chen system .
Example 14. Hyperfractional-order Rössler 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 , generation is set to 500, and the default values are , , and ; Table 1 gives the detail setting for each system.
|Number of sample points for resolving the fractional chaos systems.|
Table 2 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.
|Success means that the solution is less than in 100 independent simulations.|
No. of average evaluation for objective function (NEOF).
Success means that the solution is less than in 100 independent simulations.
|Success means that the solution is less than in 100 independent simulations.|
Number 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 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.
|Success means that the solution is less than|