Chaotic System part 10 doc

Error dynamics of the coupled master-slave Chen system when controller is switched on 4.2 Anti-synchronization of two identical coupled chaotic Lee systems The studied chaotic Lee syste

0 5 10 15 -1

0 1

-2 -1 0 1

-0.5 0 0.5 1

Time (s)Fig 5 Error dynamics of the coupled master-slave Chen system

when controller is switched on

4.2 Anti-synchronization of two identical coupled chaotic Lee systems

The studied chaotic Lee system is described by the following differential equations (Juhn et

-20 0 20

0 20 40 60 80 100 120

x1(t) x2(t)

Fig 6 Three-dimensional view of the Lee chaotic attractor

Let us consider the master Lee system ( ) Sm given by (63):

u t is the scalar active control

For the following state error vector components, defined relatively to anti-synchronization

The problem of chaos anti-synchronization between two identical Lee chaotic dynamical

systems is solved here by the design of a state feedback structure ki(.), , ,∀ =i 1 2 3, and the

choice of nonlinear functions fi(.), , , , ∀ = i 1 2 3 such that:

The nonlinear elements (.)f i and (.)k i have to be chosen to make the instantaneous

characteristic matrix of the closed-loop system in the arrow form and the closed-loop error

system asymptotically stable

From the possible solutions, allowing to put the instantaneous characteristic matrix of (68)

under the arrow form, let consider the following:

the overvaluing matrix is in arrow form and has non negative off-diagonal elements and

nonlinearities isolated in either one row or one column

By the use of the proposed theorem 2, stability and anti-synchronization properties are

satisfied for the both following sufficient conditions (72) and (73):

Various choices of the gain vector K(.), (.)K =[k1(.) k2(.) 0 ,] are possible, such as the

following linear one:

chaotically, as shown in Fig 7., and after activating the controller, Fig 8 shows three

parametrically harmonically excited 3D systems evolve in the opposite direction The

trajectories of error system (68) imply that the asymptotical anti-synchronization has been,

successfully, achieved, Fig 9

5 Hybrid synchronization by a nonlinear state feedback controller –

Application to the Chen–Lee chaotic system (Hammami, 2009)

Let consider two coupled chaotic Chen and Lee systems (Juhn et al., 2009) The following

nonlinear differential equations, of the form (1), correspond to a master system (Tam & Si

with:

2

0 0

x x and x m3 are state variables, and a b c, , and d four system parameters

For the following parameters (a b c d =, , , ) (5 10 0 3 3 8,− , , ,− ) and initial condition

1( )0 2( )0 3( )0 T 1 5 32 13 T,

is a chaotic attractor, as shown in Fig 10

-20 0 20

-50 0 50

0 50 100

Time (s)Fig 7 Error dynamics ( eAS1, eAS2, eAS3) of the coupled master-slave Lee system

when the active controller is deactivated

-20 0 20

xm2 xs2

-200 0 200

Time (s)

xm3 xs3

Fig 8 Partial time series of anti-synchronization for Lee chaotic system

when the active controller is switched on

0 5 10 15 -1

0 1 2

-2 0 2 4

-1 0 1 2

Time (s)Fig 9 Error dynamics of the coupled master-slave Lee system when control is activated

-40 -20 0 20 40

-50 0

5005 10 15 20

xm1(t) xm2(t)

Fig 10 The 3-dimensional strange attractor of the chaotic Chen–Lee master system

For the Chen–Lee slave system, described in the state space by:

x and the synchronization state variable x m2 facing to x s2

Then, the hybrid synchronization errors between the master and the slave systems

( ) AS ( ) S ( ) AS ( ) ,T

e t =⎡⎣e t e t e t ⎤⎦ are such as:

1 1 1

( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )

e=⎡⎣e e e ⎤⎦ so that the slave system, characterized by (76) and (77),

synchronizes and anti-synchronizes, simultaneously, to the master one, described by (1) and

(75), by assuring that the synchronization error e S2 and the anti-synchronization errors

1

AS

e and e AS3 decay to zero, within a finite time

Thus, for a state feedback controller of the form (79), K(.), K(.)={ }k ij(.) , , , , ,∀i j=1 2 3 and

by considering the differential systems (1), (75), (76), (77) and (78), we obtain the following

state space description of the error resulting system:

By respect to the stabilisability conditions announced in the above-mentioned theorem 2, the

dynamic error system (80) is first characterized by an instantaneous arrow form matrix

0 0

(.)(.)

7 6

k k

23 1

(.) ( )(.) ( )

m m

Finally, by considering the fixed values of k11, (.), , , (.)k12 k21 k22 k23 and k31(.), it is

relevant to denote that to satisfy the sufficient condition (30) of theorem 2, for any arbitrary

chosen parameters of correction k13(.) and k32(.), it is necessary to tune the remaining

design parameter k33(.), guaranteeing the hybrid synchronization of the coupled chaotic

studied system such that:

33(.) 0

Then, for the following instantaneous gain matrix K(.), easily obtained:

3 1 2

the studied dynamic error system (80) is asymptotically stable

For the following initial master and slave systems conditions, x m( )0 =⎡⎣1 5 −52 13⎤⎦T,

s

x = −⎡⎣ − ⎤⎦ and without activation of the designed controller, the numerical

simulation results of the above master-slave system are shown in Fig 11

It is obvious, from Fig 12., that the error states grow with time chaotically

Therefore, by designing an adequate nonlinear controlled slave system and under mild

conditions, the hybrid synchronization is achieved within a shorter time, as it is shown in

Fig 13., with an exponentially decaying error, Fig 14

The obtained phase trajectories of the Fig 15., show that the Chen–Lee slave chaotic

attractor is synchronized in a hybrid manner with the master one

-50 0 50

xm2 xs2

-40 -20 0 20

Fig 11 Error dynamics between the master Chen–Lee system

and its corresponding slave system before their hybrid synchronization

0 5 10 15 -100

0 100

-100 0 100

-60 -40 -20 0 20

Time (s)

Fig 12 Evolutions of the hybrid synchronization errors versus time

when the proposed controller is turned off

-50 0 50

xm1 xs1

-50 0 50

-50 0 50

-20 0 20

Time (s)

xm3 xs3

Fig 13 Hybrid synchronization of the master-slave Chen–lee chaotic system

0 5 10 15 -1

0 1

-100 -50 0 50

-2 -1 0 1

Time (s)Fig 14 Exponential convergence of the error dynamics

-30 -20 -10 0 10 20 30 0

5 10 15 20

-15 -10 -5 0

xs2(t)

xs2 vs xs3

Fig 15 2-D projection of the hybrid synchronized attractors

associated to the Chen–Lee chaotic system

6 Conclusion

Appropriate feedback controllers are designed, in this chapter, for the chosen slave system

states to be synchronized, anti-synchronized as well as synchronized in a hybrid manner

with the target master system states It is shown that by applying a proposed control

scheme, the variance of both synchronization and anti-synchronization errors can converge

to zero The synchronisation of two identical Chen chaotic systems, the anti-synchronization

of two identical Lee chaotic systems and, finally, the coexistence of both synchronization

and anti-synchronization for two identical Chen–Lee chaotic systems, considered as a

coupled master-slave systems, are guaranteed by using the practical stability criterion of

Borne and Gentina, associated to the specific matrix description, namely the arrow form

matrix

7 References

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Applications

Design and Applications of Continuous-Time

Chaos Generators

1UAT, CSIC and University of Sevilla

2Universidad Politécnica de Puebla

it must be transitive and it has to be sensitive to initial conditions (Strogatz, 2001) Chaossystems have bounded trajectories in the phase space, and they have at least one positivemaximum Lyapunov exponent (Dieci, 2002; Lu et al., 2005; Muñoz-Pacheco & Tlelo-Cuautle,2010; Ramasubramanian & Sriram, 2000)

Nowadays, several chaos generators have been implemented with electronic devices andcircuits in order to have a major impact on many novel applications, as the ones reported in(Cruz-Hernández et al., 2005; Ergun & Ozoguz, 2010; Gámez-Guzmán et al., 2008; Lin & Wang,2010; Strogatz, 2001; Trejo-Guerra et al., 2009) Furthermore, this chapter is mainly devoted

to highlight the design automation of continuous-time multi-scroll chaos generators, theirimplementations by using behavioral models of commercially available electronic devices,their experimental realizations and applications to secure communications A review ofthe double-scroll Chua’s circuit is also presented along with the generation of hyperchaos

by Coupling Two Chua’s circuits Basically, we present the generation of multi-scrollattractors by using saturated nonlinear function series We show their implementation

by using traditional operational amplifiers (opamps) and current-feedback operationalamplifiers (CFOAs) (Senani & Gupta, 1998) Besides, we summarize some performances

of multi-scroll chaos generators by using opamps (Muñoz-Pacheco & Tlelo-Cuautle, 2010),CFOAs (Trejo-Guerra, Sánchez-López, Tlelo-Cuautle, Cruz-Hernández & Muñoz-Pacheco,2010), current conveyors (CCs) (Sánchez-López et al., 2010) and unity-gain cells (UGCs)(Sánchez-López et al., 2008) However, not only the CC and the UGC can be taken fromthe commercially available CFOA AD844, but also they can be designed with standarintegrated complementary metal-oxide-semiconductor (CMOS) technology (Trejo-Guerra,Tlelo-Cuautle, Muñoz-Pacheco, Cruz-Hernández & Sánchez-López, 2010)

The usefulness of the chaos generators is highlighted through the physical realization of

a secure communication system by applying Hamiltonian forms and observer approach(Cruz-Hernández et al., 2005) This chapter finishes by listing several trends on theimplementation of chaos generators by using integrated CMOS technology, which may opennew lines for research covering the behavioral modeling, synthesis, design and simulation ofintegrated chaotic oscillators

1.1 Description of a chaos system

In order to make any quantitative progress in understanding a system, a mathematical model

is required The model may be formulated in many ways, but their essential feature allows

us to predict the behavior of the system, sometimes by given its initial conditions and aknowledge of the external forces which affect it In electronics, the mathematical descriptionfor a dynamical system, most naturally adopted for behavioral modeling, is done by using theso-called state-space representation, which basically consists of a set of differential equationsdescribing the evolution of the variables whose values at any given instant determine thecurrent state of the system These are known as the state variables and their values at anyparticular time are supposed to contain sufficient information for the future evolution of thesystem to be predicted, given that the external influences (or input variables) which act upon

it are known

In the state-space approach, the differential equations are of first order in the time-derivative,

so that the initial values of the variables will suffice to determine the solution In general, thestate-space description is given in the form:

˙x=f(x, u, t)

where the dot denotes differentiation with respect to time (t) and the functions f and h are

in general nonlinear In (1), the variety of possible nonlinearities is infinite, but it maynevertheless be worthwhile to classify them into some general categories For example:There are simple analytic functions such as powers, sinusoids and exponentials of a singlevariable, or products of different variables A significant feature of these functions is that theyare smooth enough to possess convergent Taylor expansions at all points and consequentlycan be linearized (Strogatz, 2001) A type of nonlinear function frequently used in systemmodeling is the piecewise-linear (PWL) approximation, which consists of a set of linearrelations valid in different regions (Elhadj & Sprott, 2010; Lin & Wang, 2010; Lü et al.,2004; Muñoz-Pacheco & Tlelo-Cuautle, 2009; Sánchez-López et al., 2010; Suykens et al., 1997;Trejo-Guerra, Sánchez-López, Tlelo-Cuautle, Cruz-Hernández & Muñoz-Pacheco, 2010; Yalçin

et al., 2002) The use of PWL approximations have the advantage that the dynamical equationsbecome linear or linearized in any particular region, and hence the solutions for differentregions can be joined together at the boundaries

When applying PWL approximation to a system described by (1), the resulting linearizedsystem has finite dimensional state-space representation, as a result the equations describing

a linear behavioral model become:

˙x=Ax+Bu

where A, B, C, and D are matrices (possibly time-dependent) of appropriate dimensions The

great advantage of linearity is that, even in the time dependent case, a formal solution can