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A three-dimensional system is presented with unknown parameters that employs two nonlinearities terms. The basic characteristics of the system are studied. The stability is measured by Characteristic equation roots, Routh stability criteria, Hurwitz stability criteria and Lapiynov function, all show that the system unstable. Then, Chaoticity is measured by maximum Lapiynov exponent of (*L*_{max}=2.509426) and “Kaplan-Yorke” dimension (*D _{L}*=2.22349544). The system is controlled effectively and synchronized by designed adaptive controllers. Furthermore, the theoretical and graphic results of the system before and after control are compared.

Researches in recent years on chaotic phenomena have increased a lot, because of the increasing frontiers of applications of chaos in engineering and non-engineering systems. “Chaos is a phenomenon which results from the exhibits sensitivity to perturbation in the structural parameters and initial conditions of some classes of dynamical systems” [

“In the context of stability and stabilization, the principle of Lapiynov stability continued to enjoy large applications; it can effectively stabilize the dissipative systems” [

This paper is organized as: Section 2, present a description of 3-d system. Section 3, basic analysis such as stability, dissipativity, Lapiynov dimension ‘‘Kaplan-Yοrke dimension’’. Section 4, we designed adaptive control law of the chaotic system. Section 5, a comparison of the analysis results before and after control. Section 6, we derive results for the adaptive synchronization of identical highly chaotic system. Finally Section 7, summarization of the main results.

A three-dimensional dynamical system [

x ˙ 1 = ρ ( x 2 − x 1 ) x ˙ 2 = a x 1 − δ x 1 x 3 x ˙ 3 = φ x 1 x 2 − x 3 (1)

The parameters values are taken as

ρ = 10 , δ = 40 , a = 296.5 , φ = 10 (2)

In this section essential, the system (1) is invested and has the following characteristics.

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 )

“A necessary and sufficient condition for the system to be stable is that the real parts of the characteristic equation have negative real parts”.

When the parameters values are taken as in (2), the Jacobian matrix of system (1) at E 0 = ( 0 , 0 , 0 ) is:

J = [ − 10 10 0 296.5 0 0 0 0 − 1 ]

det ( J − λ I ) = 0

⇒ λ 3 + 11 λ 2 − 2955 λ − 2965 = 0 (3)

By using Horner’s Ruffini method [

λ 1 = − 1 , λ 2 = 49.68089 , λ 3 = − 59.68089

Similarly, we find Jacobian matrix at E 1 and E 2 , then we obtain the eigenvalues, as shown in

So the system (1) is unstable.

“The system is considered stable by the Routh stability states (all poles in OLHP (Open Loop Half plane)) if and only if all elements of the first column in the Routh array are positive. In addition, number of poles not in the OLHP is equal to the number of sign changes in the first column” [

a 0 = − 2965

a 1 = − 2955

a 2 = 11

a 3 = 1

b 1 = a 1 − a 3 a 0 a 2 = − 2685.4545

Since, there are two negative elements in the first column. Therefore, the system (1) is unstable.

“This criterion is applied using determinants formed from coefficients of the characteristic equation. If the small minors of the square matrix J of the system (1) are all positive then the system (1) is stable, otherwise it’s unstable” [

If n = 3 (n denote the degree of the square matrix)

From Equation (3) we find:

Δ 1 = a 2 = 11 > 0

Δ 2 = | a 2 a 0 a 3 a 1 | = a 2 a 1 − a 3 a 0 = − 29.540 < 0

Δ 3 = | a 2 a 0 0 a 3 a 1 0 0 a 2 a 0 | = a 2 a 1 a 0 − a 0 2 a 3 = 87586100 > 0

Since the values of one minors is less than zero, so the system (1) is unstable.

We can use quadratic function for system (1).

We assume that

V ( x 1 , x 2 , x 3 ) = 1 2 ( x 1 2 + x 2 2 + x 3 2 )

V ˙ ( x 1 , x 2 , x 3 ) = x 1 x ˙ 1 + x 2 x ˙ 2 + x 3 x ˙ 3 (4)

If V ˙ ( x 1 , x 2 , x 3 ) < 0 then the system is stable.

By substituting (1) in Equation (4) we get:

V ˙ ( x 1 , x 2 , x 3 ) = − 10 x 1 2 + 306.5 x 1 x 2 − 30 x 1 x 2 x 3 − x 3 2

Since V ˙ ( x 1 , x 2 , x 3 ) > 0 therefore the system (1) is unstable.

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 ]

The wave-form x 1 ( t ) , x 2 ( t ) , x 3 ( t ) for the system (1) is characteristic with non-periodic shape, shown in Figures 1(a)-(c) which is one of the basic characteristic behaviors of chaotic dynamical system.

Figures 2(a)-(d) and Figures 3(a)-(c) are shows chaotic attractor for system (1) in ( x 1 , x 2 , x 3 ) , ( x 1 , x 3 , x 2 ) , ( x 2 , x 1 , x 3 ) , ( x 3 , x 1 , x 2 ) space, and 2-D attractor of system (1) in ( x 1 , x 3 ) , ( x 1 , x 2 ) , ( x 2 , x 3 ) plane.

The orbit is dense in each graph which means the system exhibit two-scroll hyper chaotic attractor.

“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

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 ] .

See Tables 1-6.

Equilibrium point | Before control | After control |
---|---|---|

( 0 , 0 , 0 ) | λ 1 = − 1 λ 2 = 49.68089 λ 3 = − 59.68089 | λ 1 = − 30 λ 2 = − 50 λ 3 = − 30 |

( 593 2 20 , 593 2 20 , 593 80 ) | λ 1 = 1.39097 λ 2 = 42.622 λ 3 = − 55.013 | λ 1 = − 30 λ 2 = − 50 λ 3 = − 30 |

( − 593 2 20 , − 593 2 20 , 593 80 ) | λ 1 = 1.39097 λ 2 = 42.622 λ 3 = − 55.013 | λ 1 = − 30 λ 2 = − 50 λ 3 = − 30 |

Equilibrium point | Before control | After control |
---|---|---|

( 0 , 0 , 0 ) | Δ 1 = 11 Δ 2 = − 29.540 Δ 3 = 87586100 | Δ 1 = 110 Δ 2 = 384000 Δ 3 = 1728 × 10 10 |

( 593 2 20 , 593 2 20 , 593 80 ) | Δ 1 = 11 Δ 2 = − 29243.5 Δ 3 = − 95377675.25 | Δ 1 = 110 Δ 2 = 384000 Δ 3 = 1728 × 10 10 |

( − 593 2 20 , − 593 2 20 , 593 80 ) | Δ 1 = 11 Δ 2 = − 29243.5 Δ 3 = − 95377675.25 | Δ 1 = 110 Δ 2 = 384000 Δ 3 = 1728 × 10 10 |

Equilibrium point | λ | Before Control | After Control | ||
---|---|---|---|---|---|

( 0 , 0 , 0 ) | λ 3 | 1 | −2955 | 1 | 3900 |

λ 2 | 11 | −2965 | 110 | 45,000 | |

λ 1 | −2685.5 | 0 | 3490.9 | 0 | |

λ 0 | −2965 | 0 | 45,000 | 0 | |

( 593 2 20 , 593 2 20 , 593 80 ) | λ 3 | 1 | −2362 | 1 | 3900 |

λ 2 | 11 | 3261.5 | 110 | 45,000 | |

λ 1 | −5634.5 | 0 | 3490.9 | 0 | |

λ 0 | 3261.5 | 0 | 45,000 | 0 | |

( − 593 2 20 , − 593 2 20 , 593 80 ) | λ 3 | 1 | −2362 | 1 | 3900 |

λ 2 | 11 | 3261.5 | 110 | 45,000 | |

λ 1 | −5634.5 | 0 | 3490.9 | 0 | |

λ 0 | 3261.5 | 0 | 45,000 | 0 |

Before control | After control | |

44.79278685 | −1486.699813 | |

565 | −6925 |

Before control | After control |
---|---|

After control | Before control |
---|---|

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

The drive system is

where

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

where

The synchronization error is defined by

then the error dynamics is obtained as

The adaptive control functіοns

where the constants

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

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

We note that (27) is negative definite on

Hence, by Lapiynov stability [

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

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

A three-dimensional dynamical system is dealt in this paper, it has quadratic cross-product nonlinear terms. The basic characteristics are analyzed by equilibrium points, stability analysis (such as characteristic equation roots, Routh criterion, Hurwitz criterion and Lapiynov function) all methods of stability shows that the system is unstable. Then, dissipativity analysis indicates system (1) is dissipative for positive values of the parameter

parameters taken as

The authors declare no conflicts of interest regarding the publication of this paper.

Aziz, M.M. and Merie, D.M. (2020) Stability and Adaptive Control with Sychronization of 3-D Dynamical System. Open Access Library Journal, 7: e6075. https://doi.org/10.4236/oalib.1106075