The first step to analyze a system is to find its equilibrium points, so we need to solve the nonlinear equations as follows

− 10 x 1 + 10 x 2 = 0

296.5 x 1 − 40 x 1 x 3 = 0

10 x 1 x 2 − x 3 = 0

We get the following equilibrium points:

E 0 = ( 0 , 0 , 0 ) , E 1 = ( 593 2 20 , 593 2 20 , 593 80 ) , E 2 = ( − 593 2 20 , − 593 2 20 , 593 80 )

Let f 1 = d x 1 d t , f 2 = d x 2 d t and f 3 = d x 3 d t in the system (1).

Then we get for the vector field

( x ˙ 1 , x ˙ 2 , x ˙ 3 ) T = ( f 1 , f 2 , f 3 ) T

thus the divergence of the vector field V on R 3 yields to:

∇ ⋅ ( x ˙ 1 , x ˙ 2 , x ˙ 3 ) T = ∂ f 1 ∂ x 1 + ∂ f 2 ∂ x 2 + ∂ f 3 ∂ x 3 = − ( ρ + 1 ) = f

Note that f = − ( ρ + 1 ) = − 11 , so the systеm (1) is dissipative for all positive values of ρ , and an exponential rate is:

d V d t = f V ⇒ V ( t ) = V 0 e f t = V 0 e − 11 t

From above equation, the volume element V 0 is contracted by the flow into a volume element V 0 e − 11 t at the time t.

For the numerical solution, we use Runge-Kutta method of order 5^th to solve system (1). With initial states x | x 1 ( 0 ) , x 2 ( 0 ) , x 3 ( 0 ) = [ − 2 , 7 , 12 ]

“As a rule the Lapiynov exponents refer to the average exponential rates of divergence or convergence of nearby trajectories in the phase space. The system is chaotic if there is at least one Lapiynov exponent greater than zero”. The values of lapiynov exponents are: ( L 1 = 2.509426 , L 2 = 0.132019 and L 3 = − 11.818787 ). Therefore, the Lapiynov dimension ‘‘Kaplan-Yοrke dimension’’ is:

D L = 2 + L 1 + L 2 | L 3 | = 2.22349544

So the system (1) is Highly Chaotic System, as shown in Figure 4.

To stabilize highly chaotic system (1), an adaptive control law is designed with unknown parameter α.

As follows:

x ˙ 1 = 10 ( x 2 − x 1 ) + α 1 x ˙ 2 = a x 1 − 40 x 1 x 3 + α 2 x ˙ 3 = 10 x 1 x 2 − x 3 + α 3 (5)

when α 1 , α 2 , α 3 are the feedback controllers.

The adaptive control functions are:

α 1 = − 10 ( x 2 − x 1 ) − μ 1 x 1 α 2 = − a ^ x 1 + 40 x 1 x 3 − μ 2 x 2 α 3 = − 10 x 1 x 2 + x 3 − μ 3 x 3 (6)

where the constants μ i , (i = 1, 2,3) are positive , a ^ is the parameter estimate of α.

Substituting (6) into (5), we get

x ˙ 1 = − μ 1 x 1 x ˙ 2 = ( a − a ^ ) x 1 − μ 2 x 2 x ˙ 3 = − μ 3 x 3 (7)

Let the parameter estimation error

e a = a − a ^ (8)

Using (8), the dynamics (7) can be written compactly as

x ˙ 1 = − μ 1 x 1 x ˙ 2 = e a x 1 − μ 2 x 2 x ˙ 3 = − μ 3 x 3 (9)

The Lapiynov approach is used for derivation of update law for adjusting the parameter estimate a ^ .

Consider the lapiynov function

V ( x 1 , x 2 , x 3 ) = 1 2 ( x 1 2 + x 2 2 + x 3 2 + e a 2 ) (10)

Notice V is positive-definite on R 4 .

Also

e ˙ a = − a ^ ˙ (11)

Differentiating V with substituting (9) and (11), we get:

V ˙ = − μ 1 x 1 2 − μ 2 x 2 2 − μ 3 x 3 2 + e a [ x 1 x 2 − a ^ ˙ ] (12)

In Equation (12), we update estimated parameter by:

a ^ ˙ = x 1 x 2 + μ 4 e a (13)

where the constant μ 4 is a positive.

Now, we substitute (13) into (12), we obtain

V ˙ = − μ 1 x 1 2 − μ 2 x 2 2 − μ 3 x 3 2 − μ 4 e a 2 (14)

Notice V ˙ is negative definite on R 4 .

Thus, by lapiynov stability, Routh-array criteria, Eigenvalues and Hurwitz stability criteria we get the below result.

Proposition 1. The chaotic system (5) with unknown parameter is stabilized for every initial value by adaptive control (6), where the estimated parameter is obtained by (13) and μ 1 , μ 2 , μ 3 , μ 4 are greater than zero.

To simulate the controlled highly chaotic system (7) we take the initial values x | x 1 ( 0 ) , x 2 ( 0 ) , x 3 ( 0 ) = [ 12 , 9 , 17 ] and [ μ 1 , μ 2 , μ 3 ] = [ 30 , 50 , 30 ] .

Figure 5 shows Controlled trajectories of system (1).

We apply adaptive synchronization technique of highly chaotic system with unknown parameter α.

The drive system is

where, (i = 1, 2, 3) are state variables.

As the response system, the controlled highly chaotic dynamics given by

where are nonlinear controllers to be designed, and the state variables are, (i = 1, 2, 3).

The synchronization error is defined by

then the error dynamics is obtained as

The adaptive control functіοns define as

where the constants greater than zero, and is the estimated value of the parameter α.

Substitute (19) into (18), to obtain the error dynamics as

Now, the parameter estimation error is

By substituting (21) into (20), the error dynamics simplifies to

From Lapiynov approach we derive the updated law to adjust the estimation of the parameter.

The quadratic lapiynov function is

which be a positive definite on.

Note that

Differentiating V and substituting (22) & (24) in it, we get:

update the estimated parameter in Equation (25) by the following

where the constant is greater than zero.

From (25) and (26), we obtain:

We note that (27) is negative definite on.

Hence, by Lapiynov stability [14], “it is immediate that the parameter error and synchronization error decay exponentially to zero with time for all initial values”.

Thus, we proved the results below.

Proposition 2. The drive and response identical chaotic systems (15) and (16) with unknown parameter α are synchronized for all initial values by adaptive control law (19), where the estimated parameter given by (26) and are constants greater than zero.

To get the results numerically, we used the 4th-order Runge-Kutta method to solve systems (15) & (16), and solve system (18) with adaptive control law (19).

We take and as initial states of the drive system (15) and the response system (16) respectively. Also take α = 296.5 and for i = 1, 2, 3, 4.

Figure 6 shows adaptive synchronization of the highly chaotic system.

Figure 7 shows the convergent for system (18) with controller (19).