Generalized Synchronization of Nonlinear Chaotic Systems through Natural Bioinspired Controlling Strategy

and Applied Analysis 3 Definition 2. If the undisturbedmotion is stable on the partial regionΩ, and there exists a δ > 0, so that on the given partial regionΩ when

In our real world, various kinds of nonlinear chaotic biosystems describing natural phenomenon are found to have some states always positive.It means these states are always in the first quadrant, such as the three species prey-predator systems [15,16,21], double Mackey-Glass systems [17,22,23], energy communication system in biological research [19,24], and virus-immune system [20].For the three species preypredator systems, which consist of two competing preys and one predator can be described by the following set of nonlinear differential equations: where ,   ,   ,   , and   ,  = 1, 2 are the model parameters assuming only positive values, and the functions Φ  (, ),  = 1, 2 represent the densities of the two prey species and  represents the density of the predator species.The predator  consumes the preys ,  according to the response functions [25].The prescribed model characterized by nonlinear response since amount of food consumed by predator per unit time depends upon the available food sources from the two preys  and .And for the virus-immune system, a mathematical model is considered as follows: where , , and  represent the population concentrations of uninfected, infected target cells and virus, respectively.We denote by the  constant supply of target cells from its precursor, where  is the rate at which new T cells are created.These cells have finite life time and  1 represent the average death rates of these cells,  2 and  3 represent the death rate of the infected cells before viral production commences, and  4 is the viral clearance rate constant.These target cells are assumed to grow logistically with specific growth rate  and carrying capacity Γ. Above all, their states are all positive.As a result, a natural bioinspired stability theory is proposed for generalized synchronization in this paper.It means that there exists a given functional relationship between the states of the master and that of the slave.Via using the natural bioinspired stability theory, the new Lyapunov function is a simple linear homogeneous function of states and introduces less simulation error.
In this paper, a new Lyapunov control function based on the bioinspired stability theory is proposed to synchronize fuzzy chaotic systems.This new Lyapunov control function is designed as a simple linear homogeneous function of states and the process of designing controller to synchronize two fuzzy chaotic systems becomes much simpler.Furthermore, we design three different synchronization situations to show the effectiveness of our strategy-(1) CASE I, synchronization of identical systems: synchronization of master fuzzy Lorenz system and slave fuzzy Lorenz system, (2) CASE II, synchronization of two different systems: synchronization of master fuzzy Lorenz system and slave fuzzy Chen system, and (3) CASE III, synchronization of complicated different systems: synchronization of master fuzzy Lorenz system and complicated fuzzy chaotic system (slave fuzzy Chen system and functional fuzzy Rossler's system).
The layout of the rest of the paper is as follows.In Section 2, the natural bioinspired stability theory is introduced.In Section 3, modeling of chaotic systems is proposed.In Section 4, three simulation examples are given.In Section 5, conclusions are given.

Definition of the Stability on Partial Region.
Consider the differential equations of disturbed motion of a nonautonomous system in the normal form where the function   is defined on the intersection of the partial region Ω (shown in Figure 1) and and  >  0 , where  0 and  are certain positive constants.  , which vanishes when the variables   are all zero, is a real valued function of ,  1 , . . .,   .It is assumed that   is smooth enough to ensure the existence and uniqueness of the solution of the initial value problem.When   does not contain  explicitly, the system is autonomous.Obviously,   = 0 ( = 1, . . ., ) is a solution of (3).We are interested in the asymptotical stability of this zero solution on partial region Ω (including the boundary) of the neighborhood of the origin which in general may consist of several subregions (Figure 1).Definition 1.For any given number  > 0, if there exists a  > 0, such that on the closed given partial region Ω when for all  ≥  0 , the inequality is satisfied for the solutions of (3) on Ω, then the disturbed motion   = 0 ( = 1, . . ., ) is stable on the partial region Ω.
Definition 2. If the undisturbed motion is stable on the partial region Ω, and there exists a   > 0, so that on the given partial region Ω when the equality lim is satisfied for the solutions of (3) on Ω, then the undisturbed motion   = 0 ( = 1, . . ., ) is asymptotically stable on the partial region Ω.
The intersection of Ω and region defined by ( 7) is called the region of attraction.
then, it is a semidefinite function on the partial region whose sense is opposite to that of , or if it becomes zero identically, then the undisturbed motion is stable on the partial region.
Proof.Let us assume for the sake of definiteness that  is a positive definite function.Consequently, there exists a sufficiently large number  0 and a sufficiently small number ℎ < , such that on the intersection Ω 1 of partial region Ω and and  ≥  0 , the following inequality is satisfied  (,  1 , . . .,   ) ≥  ( 1 , . . .,   ) , where  is a certain positive definite function which does not depend on .Besides that, (13) may assume only negative or zero value in this region.
Let  be an arbitrarily small positive number.We will suppose that in any case  < ℎ.Let us consider the aggregation of all possible values of the quantities  1 , . . .,   , which are on the intersection  2 of Ω 1 and and let us designate by  > 0 the precise lower limit of the function  under this condition.By virtue of ( 10), we will have We will now consider the quantities   as functions of time which satisfy the differential equations of disturbed motion.We will assume that the initial values  0 of these functions for  =  0 lie on the intersection Ω 2 of Ω 1 and the region where  is so small that  ( 0 ,  10 , . . .,  0 ) < .
By virtue of the fact that ( 0 , 0, . . ., 0) = 0, such a selection of the number  is obviously possible.We will suppose that in any case the number  is smaller than .Then the inequality being satisfied at the initial instant will be satisfied, in the very least, for a sufficiently small  −  0 , since the functions   () very continuously with time.We will show that these inequalities will be satisfied for all values  >  0 .Indeed, if these inequalities were not satisfied at some time, there would have to exist such an instant  =  for which this inequality would become an equality.In other words, we would have and consequently, on the basis of ( 19) On the other hand, since  < ℎ, the inequality ( 9) is satisfied in the entire interval of time [ 0 , ], and consequently, in this entire time interval / ≤ 0. This yields  (,  1 () , . . .,   ()) ≤  ( 0 ,  10 , . . .,  0 ) , which contradicts (20) on the basis of (19).Thus, the inequality ( 23) must be satisfied for all values of  >  0 .Finally, we must point out that from the view-point of mathematics, the stability on partial region in general does not be related logically to the stability on whole region.If an undisturbed solution is stable on a partial region, it may be either stable or unstable on the whole region and vice versa.In specific practical problems, we do not study the solution starting within Ω 2 and running out of Ω.

Theorem 9. If in satisfying the conditions of Theorem 8 the derivative 𝑑𝑉/𝑑𝑡 is a definite function on the partial region with opposite sign to that of 𝑉 and the function 𝑉 itself permits an infinitesimal upper limit, then the undisturbed motion is asymptotically stable on the partial region.
Proof.Let us suppose that  is a positive definite function on the partial region and that, consequently, / is negative definite.Thus on the intersection Ω 1 of Ω and the region defined by (9) and  ≥  0 there will be satisfied not only the inequality (10), but also the following inequality: where  1 is a positive definite function on the partial region independent of .
Let us consider the quantities   as functions of time which satisfy the differential equations of disturbed motion assuming that the initial values  0 =   ( 0 ) of these quantities satisfy the inequalities (18).Since the undisturbed motion is stable in any case, the magnitude  may be selected so small that for all values of  ≥  0 the quantities   remain within Ω 1 .Then, on the basis of (24) the derivative of function (,  1 (), . . .,   ()) will be negative at all times and, consequently, this function will approach a certain limit, as  increases without limit, remaining larger than this limit at all times.We will show that this limit is equal to some positive quantity different from zero.Then for all values of  ≥  0 the following inequality will be satisfied: where  > 0.
Since  permits an infinitesimal upper limit, it follows from this inequality that where  is a certain sufficiently small positive number.Indeed, if such a number  did not exist, that is, if the quantity ∑    () was smaller than any preassigned number no matter how small, then the magnitude (,  1 (), . . .,   ()), as follows from the definition of an infinitesimal upper limit, would also be arbitrarily small, which contradicts (25).
If for all values of  ≥  0 the inequality (26) is satisfied, then (24) shows that the following inequality will be satisfied at all times: where  1 is positive number different from zero which constitutes the precise lower limit of the function  1 (,  1 (), . . .,   ()) under condition (26).Consequently, for all values of  ≥  0 we will have which is, obviously, in contradiction with (25).The contradiction thus obtained shows that the function (,  1 (), . . .,   ()) approached zero as  increase without limit.Consequently, the same will be true for the function ( 1 (), . . .,   ()) as well, from which it follows directly that lim which proves the theorem.called Partial Region, which is inspired via the biological behavior in nature and is used to control the states of a system which exist in the first quadrant to achieve the goal system with a set of simple controllers through designing a simpler and more convenient Lyapunov function.

Modeling of Chaotic Systems
In this section, the well-famous Takagi-Sugeno fuzzy model is applied to model the classical Lorenz system, Chen system, and Rossler's system for further designing the natural bioinspired controller.

Fuzzy Modeling of Master Lorenz
System.For master Lorenz system where , ,  are the parameters.When  = 10,  = 8/3,  = 28, and initial states are (0.5, 1, 5), the dynamic behavior is chaotic.Assume that  1 ∈ [− 1 ,  1 ] and  1 > 0; then Lorenz system can be exactly represented by T-S fuzzy model as follows: Rule 1: Rule 2: where and  1 = 30. 1 and  2 are fuzzy set of Lorenz system.Here, we call (31) the first liner subsystem under the fuzzy rule and (32) the second liner subsystem under the fuzzy rule.The final output of the fuzzy Lorenz system is inferred as follows and the chaotic behavior is shown in Figure 2 Ẋ  where We further summarize all the parameters in Section 3.1 into Table 1.

Fuzzy Modeling of Slave Lorenz
System.For slave Lorenz system where , ,  are the parameters.When  = 10,  = 8/3,  = 28, and initial states are (0.6, 3, 10), the dynamic behavior is chaotic.Assume that  1 ∈ [− 2 ,  2 ] and  2 > 0; then Lorenz system can be exactly represented by T-S fuzzy model as follows: where and  2 = 20.M1 and M2 are fuzzy set of Lorenz system.Here, we call (37) the first linear subsystem under the fuzzy rule and (38) the second linear subsystem under the fuzzy rule.As a result,  1 and  2 are controllers of the first and second subsystems.The final output of the fuzzy Lorenz system is inferred as follows and the chaotic behavior without using controllers is shown in Figure 3 Ẏ where We further summarize all the parameters in Section 3.2 into Table 2.

Fuzzy Modeling of Chen
System as Slave System.For Chen system as slave system where , ,  are the parameters.When  = 35,  = 3,  = 28, and initial states are (0.5, 1, 5), the dynamic behavior is chaotic.Assume that  1 ∈ [− 3 ,  3 ] and  3 > 0; then Chen system can be exactly represented by T-S fuzzy model as follows: Rule 1: Rule 2: Here and  3 = 30. 1 and  2 are fuzzy set of Lorenz system.
Here, we call (43) the first liner subsystem under the fuzzy rule and (44) the second liner subsystem under the fuzzy rule.As a result,  1 and  2 are controllers of the first and second subsystems.The final output of the fuzzy Chen system is inferred as follows and the chaotic behavior without using controllers is shown in Figure 4 Ż where (47)  We further summarize all the parameters in Section 3.3 into Table 3.

Fuzzy Modeling of Rossler's System as Functional System.
For Rossler's system as functional system where , ,  are the parameters.When ℎ = 0.2,  = 0.2,  = 5.7, and initial states are (10,15,11), the dynamic behavior is chaotic.Assume that  1 ∈ [− 4 ,  4 ] and  4 > 0; then Chen system can be exactly represented by T-S fuzzy model as follows: and  4 = 10. 1 and  2 are fuzzy set of Lorenz system.Here, we call (49) the first liner subsystem under the fuzzy rule and (50) the second liner subsystem under the fuzzy rule.The final output of the fuzzy Rossler's system is inferred as follows and the chaotic behavior is shown in Figure 5 Ż where We further summarize all the parameters in Section 3.4 into Table 4.

Numerical Simulation Results of the Bioinspired Strategy
In this section, there are three main CASEs for illustrations, The addition of 100 makes the error dynamics always happen in first quadrant.Our goal is   =   + 100; that is, lim The error and error dynamics are Before control action, the error dynamics always happens in first quadrant as shown in Figure 6.By the natural bioinspired stability theory, one can choose a Lyapunov function in the form of a positive definite function in first quadrant: Its time derivative through error dynamics (55) is Choose We obtain which is negative definite function in the first quadrant.
The numerical results are shown in Figures 7 and 8.In Figures 7 and 8, the master and slave systems are two identical Lorenz systems with different initial conditions, where the parameters  = 10,  = 8/3,  = 28,  1 = 30,  2 = 20, initial states are ( 1 ,  2 ,  3 ) = (0.5, 1, 5) for master system, and ( 1 ,  2 ,  3 ) = (0.6, 3, 10) for slave system.The control procedure is arranged that, in the initial 30 sec, the system is in a nature situation without any control.The designed controllers operate after 30 sec to show the high efficiency of our bioinspired controllers.In Figure 7, it is clear that the error states  1 ,  2 ,  3 almost immediately smoothly approach zero after the controllers put in at 30 sec.Also, in Figure 8, the states of the slave system are all synchronized to the states of the target system with high performance.
The addition of 100 makes the error dynamics always happen in first quadrant.Our goal is   =   + 100; that is, lim The error and error dynamics are Before control action, the error dynamics always happens in first quadrant as shown in Figure 9.By the natural bioinspired stability theory, one can choose a Lyapunov function in the form of a positive definite function in first quadrant: Its time derivative through error dynamics (61) is Choose (64) We obtain the error states  1 ,  2 ,  3 almost immediately smoothly approach zero after the controllers put in at 30 sec.Also, in Figure 11, the states of the slave system are all synchronized to the states of the target system with high performance.Case 3. The generalized synchronization error function is   =   −  +  +100, ( = 1, 2, 3), where the states ( 1 ,  2 ,  3 ) of the slave Chen system in (46) are going to achieve generalized synchronization of a complicated designed goal system-the states ( 1 ,  2 ,  3 ) of Lorenz system in (32) with the states ( 1 ,  2 ,  3 ) of the functional system in (52).
The addition of 100 makes the error dynamics always happen in first quadrant.Our goal is   =   +   + 100; that is, lim The error and error dynamics are Before control action, the error dynamics always happens in first quadrant as shown in Figure 12.By the natural bioinspired stability theory, one can choose a Lyapunov function in the form of a positive definite function in first quadrant: Its time derivative through error dynamics (67) is Choose +  (70) We obtain which is negative definite function in the first quadrant.The numerical results are shown in Figures 13 and 14.In Figures 13 and 14, a complicated goal system-Lorenz system with initial state ( 1 ,  2 ,  3 ) = (0.5, 1, 5) in (34) combined with the functional system with initial state ( 1 ,  2 ,  3 ) = (10,15,11) in (52), is considered as the master system and Chen system with initial state ( 1 ,  2 ,  3 ) = (0.5, 1, 5) in ( 46) is referring to the slave system, where the parameters  = 10,  = 8/3,  = 28,  = 35,  = 3,  = 28, ℎ = 0.2,  = 0.2,  = 5.7,  1 = 30,  3 = 30,  4 = 10.The control procedure is arranged that, in the initial 30 sec, the system is in a nature situation without any control.The designed controllers operate after 30 sec to show the high efficiency of our bioinspired controllers.In Figure 12, it is clear that the error states  1 ,  2 ,  3 almost immediately smoothly approach zero after the controllers put in at 30 sec.Also, in Figure 13, the states of the slave system are all synchronized to the states of the target system with high performance.

Conclusions
In our daily life, there are lots of natural systems whose certain states are always positive, that is, exist in the first quadrant.Consequently, a novel control strategy by using bioinspired stability theory is proposed to achieve generalized fuzzy chaos synchronization.In this paper, a new Lyapunov function based on bioinspired control strategy is proposed to directly achieve chaos synchronization of chaotic systems.Via using this strategy, the new Lyapunov function used is a simple linear homogeneous function of states and the process of synchronization of two fuzzy chaotic systems becomes more simple and the controllers operate after 30 sec; the error states  1 ,  2 ,  3 almost immediately smoothly approach zero.Classical Lorenz, Chen, and Rossler's systems are used in numerical examples and are given to demonstrate the effectiveness of the proposed new strategy.Also, this paper provides a new sight of view for researchers to reference and we are going to further apply the novel bioinspired controlling strategy to other real-world applications in the following stage, such as energy communication system in biological research, virus-immune system, and HIV virusimmune system.
Discussion and Statement.Theorems 8 and 9 in Bioinspired GYC Theorem of Stability of Asymptotical Stability on Partial Region provides a new way to investigate the stability problem-to solve the stability problem in the first quadrant,

Figure 4 :
Figure 4: Chaotic behavior of slave fuzzy Chen system.

3 Figure 6 :
Figure 6: Phase portraits of error dynamics for Case 1.

3 Figure 7 :Figure 8 :
Figure 7: Time histories of error dynamics for Case 1-the error states  1 ,  2 ,  3 almost immediately smoothly approach zero after the controllers put in at 30 sec.

3 Figure 9 :
Figure 9: Phase portraits of error dynamics for Case 2.

3 Figure 10 :Figure 11 :
Figure 10: Time histories of error dynamics for Case 2-the error states  1 ,  2 ,  3 almost immediately smoothly approach zero after the controllers put in at 30 sec.

Figure 12 :
Figure 12: Phase portraits of error dynamics for Case 3.