Chapter 8 Pragmatical Hybrid Projective Generalized Synchronization of New
8.3. Numerical results of PHPGS by GYC partial region stability theory
Take new Mathieu-Duffing system with Bessel function parameters (7.3) as master system, where a, b, c, d, e, f are uncertain parameters and following new Mathieu-Duffing system as slave system:
1 2
k =k =k =k =100 , in order that the error dynamics always happens in first quadrant.
In order to lead( , , , )y y y y1 2 3 4 to(g x1 1+k g x1, 2 2+k g x2, 3 3+k g x3, 4 4+k4) , u u u 1, ,2 3 and u4 are added to each equation of Eq.(8.7), respectively:
1 2 1 Fig. 8.1. By GYC partial region asymptotical stability theorem, one can choose a
Lyapunov function in the form of a positive definite function of e , 1 e , 2 e , 3 e , 4 a, b,
c, d, e, f in first quadrant:
1 2 3 4
quadrant.
The Lyapunov asymptotical stability theorem can not be satisfied in this case. The common origin of error dynamics and parameter update dynamics cannot be determined to be asymptotically stable. By GYC pragmatical asymptotical stability theorem (see Appendix ) D is a 10 -manifold, n = 10 and the number of error state variables p = 4.
Whene1 = e2 = e = 3 e4= 0 and
a
, b, c, d, e, f take arbitrary values , V = , 0 so X is of 6 dimensions, m = n – p = 10 – 4 = 6. m + 1≤ n are satisfied. By the GYC pragmatical asymptotical stability theorem, not only error vector e tends to zero but also all estimated parameters approach their uncertain parameters. PHPGS of chaotic systems by GYC partial region stability theory is accomplished. The equilibrium point e1 = e2= e = 3 e4 = a = b = c = d = e = f = 0 is asymptotically stable. The numerical results are shown in Figs 8.3-8.5, The generalized synchronization is accomplished with g = 1.5, 1 − g = 0.5,2 g = 1.5,3 − and g = 24 while
1(0) 105.000001,
e = e2(0) 94.999999,= e3(0) 105.000001= and e4(0) 119.99999= . Four error states versus time are shown in Fig. 8.2. The estimated parameters approach the uncertain parameters of the chaotic system as shown in Figs. 8.3-5. The initial values of estimated parameters are ˆ(0) 15a = , ˆ(0) 1b = , ˆ(0) 0.005c = , ˆ(0)d = −23 ,
ˆ(0) 0.002
e = and ˆ (0) 14f = .
Fig. 8.1 Phase portraits of error dynamics.
Fig. 8.2 The time histories of errors (e1, e2, e , 3 e4).
Fig. 8.3 The time histories of a and b.
Fig. 8.4 The time histories of c and d.
Fig. 8.5 The time histories of e and f .
Chapter 9 Conclusions
Chaotic system features that it has complex dynamical behaviors and sensitive behavior dependence initial conditions. Because of this property, chaotic systems are thought difficult to be synchronized or controlled. In practice, some or all of the system parameters are uncertain. Additionally, these parameters change at every time. A lot of researchers have studied to solve this problem by different control theories. There are many control techniques which are presented to synchronize and control chaotic systems, such as backstepping design method [2], impulsive control method [3], invariant manifold method [4], adaptive control method [5], linear and nonlinear feedback control method [6], and active control approach [7], PC method [8], etc. In this thesis, we have studied the chaos of a new Mathieu – Duffing system by phase portraits, Poincaré maps, power spectrum and Lyapunov exponent diagram in Chapter 2.
In Chapter 3, by the pragmatical asymptotical stability theorem, the estimated parameters approach uncertain parameters can be answered strictly. In the current scheme of adaptive synchronization [12-15], the traditional Lyapunov stability theorem and Babalat lemma are used to prove that the error vector approaches zero, as time approaches infinity. But the question of that why the estimated parameters also approach uncertain parameters remains unanswered. By the pragmatical asymptotical stability theorem, the question can be answered strictly. A new Mathieu – Duffing system and a new Duffing – van der Pol system are used as simulated example. Pragmatical hybrid projective hyper chaotic generalized synchronization of chaotic systems by adaptive backstepping control is accomplished.
In Chapter 4, a new kind of symplectic synchronization and a new control Lyapunov
function are proposed. A new kind of symplectic synchronization plays a ‘‘interwined’’
role, so we call the“master’’ system partner A, the ‘‘slave’’ system partner B. Using the new control Lyapunov function
( ) exp( T ) 1
V e = ke e − (9.1) , the error tolerance introduced by using this new control Lyapunov function can be reduced marvelously to 10−17 of that using traditional control Lyapnov function
( ) T
V e =e e (9.2) In Chapter 5, by the GYC partial region stability theory, chaos control is achieved.
In GYC partial region stability theory, Lyapunov function is simpler a traditional Lyapunov function of error states, which is a linear homogenous function of error states.
The simulation error can be reduced by using the GYC partial region stability and simple controllers. A new Mathieu – Duffing system in the first quadrant is used as simulated examples which effectively confirm the scheme.
In Chapter 6, by using the GYC partial region stability theory, the Lyapunov function is a simple linear homogeneous function of error states and the controllers are simpler than traditional controllers and so reduce the simulation error. Two new Mathieu – Duffing systems are in chaos generalized synchronization successfully.
In Chapter 7, the chaotic behaviors of a new Mathieu – Duffing systems Bessel function parameters are first proposed. The chaotic behaviors of a new Mathieu – Duffing systems with Bessel function parameters is studied numerically by phase portraits, Poincaré maps, bifurcation diagram and Lyapunov exponent diagram.
In Chapter 8, by the GYC pragmatical asymptotical stability theorem and GYC partial region stability theory, the error vector tends to zero and the estimated parameters approach uncertain values is guaranteed and the controllers of are simpler than traditional controllers and so reduce the simulation error. Pragmatical hybrid projective generalized
synchronization of new Mathieu-Duffing systems with Bessel function parameters by adaptive control is achieved. The GYC pragmatical asymptotical stability theorem and GYC partial region stability theory are powerful to synchronize and control chaotic systems. The security of communication is greatly increased.
Appendix A
GYC Pragmatical asymptotical theorem
The stability for many problems in real dynamical systems is actual asymptotical stability, although it may not be mathematical asymptotical stability. The mathematical asymptotical stability demands that trajectories from all initial states in the neighborhood of zero solution must approach the origin as t→∞. If there are only a small part or even one of the initial states from which the trajectories or trajectory do not approach the origin as t→∞, the zero solution is not mathematically asymptotically stable. However, when the probability of occurrence of an event is zero, it means the event does not occur actually. If the probability of occurrence of the event that the trajectries from the initial states are that they do not approach zero when t→∞, is zero, the stability of zero solution is actual asymptotical stability though it is not mathematical asymptotical stability. In order to analyze the asymptotical stability of the equilibrium point of such systems, the pragmatical asymptotical stability theorem is used.
Let X and Y be two manifolds of dimensions m and n(m<n), respectively, and ϕ be a differentiable map from X to Y; then ϕ(X) is a subset of Lebesque measure 0 of Y [74] . For an autonomous system
)
x be an equilibrium point for the system (A.1), then
0
Definition : The equilibrium point for the dynamic system is pragmatically asymptotically stable provided that with initial points on C which is a subset of Lebesque measure 0 of D, the behaviors of the corresponding trajectories cannot be determined, while with initial points on D−C, the corresponding trajectories behave as that agree with traditional asymptotical stability [17, 18].
Theorem: Let V =[x1,x2, xn]T :D→R+ positive definite, analytic on D, where
1, ,2 n
x x x are all space coordinates such that the derivative of V through differential equation, V , is negative semi-definite.
Let X be the m-manifold consisting of point set for which ∀x≠0, 0V(x)= and D is an n-manifold. If m+1<n, then the equilibrium point of the system is pragmatically asymptotically stable.
Proof :Since every point of X can be passed by a trajectory of Eq.(A.1), which is one dimensional, the collection of these trajectories, C, is a (m+1)-manifold [17, 18]. If
) 1
(m+ ＜n, then the collection C is a subset of Lebesque measure 0 of D. By the above definition, the equilibrium point of the system is pragmatically asymptotically stable. □
If an initial point is ergodicly chosen in D, the probability of that the initial point falls on the collection C is zero. Here, equal probability is assumed for every point chosen as an initial point in the neighborhood of the equilibrium point. Hence, the event
that the initial point is chosen from collection C does not occur actually.
Therefore, under the equal probability assumption, pragmatical asymptotical stability becomes actual asymptotical stability. When the initial point falls on D−C, 0V(x)< , the corresponding trajectories behave as if they agree with traditional asymptotical stability because by the existence and uniqueness of the solution of initial-value problem, these trajectories never meet C.
For Eq.(3.39), Eq.(4.51) and Eq.(8.13), the Lyapunov function is a positive definite function of n variables, i.e. p error state variables and n− p=m differences between unknown and estimated parameters, while V =eTCe is a negative semi-definite function of n variables. Since the number of error state variables is always more than one,
>1
p , (m+ )1 <n is always satisfied; by pragmatical asymptotical stability theorem we have and the estimated parameters approach the uncertain parameters. The pargmatical generalized synchronizations is obtained. Therefore, the equilibrium point of the system is pragmatically asymptotically stable. Under the equal probability assumption, it is actually asymptotically stable for both error state variables and parameter variables.
Appendix B
GYC Partial Region Stability Theory
Consider the differential equations of disturbed motion of a nonautonomous system in the normal form
where the function X is defined on the intersection of the partial region s Ω (shown in Fig. 1) and is smooth enough to ensure the existence, uniqueness of the solution of the initial value problem. When X does not contain t explicitly, the system is autonomous. s
Obviously, 0 (xs = s=1, n) is a solution of Eq.( B.1). We are interested to 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 (Fig.
B.1).
2 , ( 1, , )
s s
x <ε s= n
∑
(B.4)is satisfied for the solutions of Eq.(B.27) on Ω , then the disturbed motion
0 ( 1, )
xs = s= n 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
is satisfied for the solutions of Eq.(B.1) on Ω , then the undisturbed motion
0 ( 1, )
xs = s= n is asymptotically stable on the partial region Ω .
The intersection of Ω and region defined by Eq.(B.2) is called the region of attraction. single-valued and have continuous partial derivatives and become zero when
1 n 0
x = =x = . Definition 3:
If there exists t0 > and a sufficiently small 0 h>0, so that on partial region Ω 1 and t≥ , t0 V ≥0 (or ≤0), then V is a positive (or negative) semidefinite, in general
semidefinite, function on the Ω and 1 t≥ . t0
Definition 7: Function with infinitesimal upper bound
If V is bounded, and for any λ>0, there exists μ > , so that on 0 Ω when 1 then V admits an infinitesimal upper bound on Ω . 1
Theorem 1 [33, 34]
If there can be found for the differential equations of the disturbed motion (Eq.(6.1))
a definite function V t x( , , , )1 … xn on the partial region, and for which the derivative with respect to time based on these equations as given by the following :
1
is a semidefinite function on the paritial region whose sense is opposite to that of V, 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 V is a positive definite function.
Consequently, there exists a sufficiently large number t and a sufficiently small 0 number h < H, such that on the intersection Ω of partial region 1 Ω and and t≥ , the following inequality is satisfied t0
1 1
( , , , )n ( , , )n
V t x … x ≥W x … x (B.13) where W is a certain positive definite function which does not depend on t. Besides that, Eq. (B.7) may assume only negative or zero value in this region.
Let ε be an arbitrarily small positive number. We shall suppose that in any case ε <h. Let us consider the aggregation of all possible values of the quantities x1, ,… xn, and let us designate by l>0 the precise lower limit of the function W under this condition. by virtue of Eq. (B.5), we shall have
( , , , )1 n
V t x … x ≥l for ( , , )x1 … xn on ω2. (B.15) We shall now consider the quantities x as functions of time which satisfy the s differential equations of disturbed motion. We shall assume that the initial values x of s0
these functions for t= lie on the intersection t0 Ω of 2 Ω and the region 1 obviously possible. We shall suppose that in any case the number δ is smaller than
ε .Then the inequality being satisfied at the initial instant will be satisfied, in the very least, for a sufficiently small t− , since the functions ( )t0 x t very continuously with time. We shall show that s these inequalities will be satisfied for all values t> . Indeed, if these inequalities were t0 not satisfied at some time, there would have to exist such an instant t=T for which this inequality would become an equality. In other words, we would have
2( ) ,
s s
x T =ε
∑
(B.19) and consequently, on the basis of Eq. (B.9)( , ( ), , ( ))1 n
V T x T … x T ≥l (B.20) On the other hand, since ε <h, the inequality (Eq.(B.4)) is satisfied in the entire interval of time [t0, T], and consequently, in this entire time interval dV 0
dt ≤ . This yields
1 0 10 0
( , ( ), , ( ))n ( , , , n ),
V T x T … x T ≤V t x … x (B.21) which contradicts Eq. (B.12) on the basis of Eq. (B.11). Thus, the inequality (Eq.(B.1)) must be satisfied for all values of t> , hence follows that the motion is stable. t0
Finally, we must point out that from the view-point of mathenatics, 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. From the viewpoint of dynamics, we wre not interesting to the solution starting from Ω and going out of 2 Ω .
Theorem 2 [33, 34]
If in satisfying the conditions of theorem 1, 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 V is a positive definite function on the partial region and that consequently, dV
dt is negative definite. Thus on the intersection Ω of Ω and the 1 region defined by Eq. (B.4) and t≥ there will be satisfied not only the inequality t0 (Eq.(B.5)), but the following inequality as will:
1( ,1 n),
dV W x x
dt ≤ − … (B.22) where W is a positive definite function on the partial region independent of t. 1
Let us consider the quantities x as functions of time which satisfy the differential s equations of disturbed motion assuming that the initial values xs0 =x ts( )0 of these quantities satisfy the inequalities (Eq. (B.10)). Since the undisturbed motion is stable in any case, the magnitude δ may be selected so small that for all values of t≥ the t0 quantities x remain within s Ω . Then, on the basis of Eq. (B.13) the derivative of 1 function V t x t( , ( ), , ( ))1 … x tn will be negative at all times and, consequently, this function will approach a certain limit, as t increases without limit, remaining larger than this limit at all times. We shall show that this limit is equal to some positive quantity
different from zero. Then for all values of t≥ the following inequality will be t0 satisfied:
( , ( ), , ( ))1 n
V t x t … x t >α (B.23) where α >0.
Since V permits an infinitesimal upper limit, it follows from this inequality that
2( ) , ( 1, , ),
s s
x t ≥λ s= n
∑
… (B.24)where λ is a certain sufficiently small positive number. Indeed, if such a number λ did not exist, that is , if the quantity s( )
s
∑
x t were smaller than any preassigned number no matter how small, then the magnitude V t x t( , ( ), , ( ))1 … x tn , as follows from the definition of an infinitesimal upper limit, would also be arbitrarily small, which contradicts (B.14).If for all values of t≥ the inequality (Eq. (B.15)) is satisfied, then Eq. (B.13) t0 shows that the following inequality will be satisfied at all times:
1,
dV l
dt ≤ − (B.25) where l is positive number different from zero which constitutes the precise lower limit 1 of the function W t x t1( , ( ), , ( ))1 … x tn under condition (Eq. (B.15)). Consequently, for all
which is, obviously, in contradiction with Eq.(B.14). The contradiction thus obtained shows that the function V t x t( , ( ), , ( ))1 … x tn approached zero as t increase without limit.
Consequently, the same will be true for the function W x t( ( ), , ( ))1 … x tn as well, from which it follows directly that
lim ( ) 0, (s 1, , ),
t x t s n
→∞ = = … (B.26) which proves the theorem.
subregion 2
subregion 3 subregion 1
Ω Ω
Ω
X1 O
X2
Ω1
Ω1
Ω1
Fig. B.1 Partial regions Ω and Ω . 1
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