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In this paper 2D discrete time dynamical system is presented. The fixed points were found. The stability of fixed points is measured by characteristic roots, jury criteria, Lyapunov function. All show that the system is unstable, and analyzing the dynamic behavior of the system finds bifurcation diagrams at the bifurcation parameter. Newton’s Raphson numerical method was used the roots of the system with the minimum error. Then, chaoticity is measured by the phase space; maximum Lyapunov exponent is obtain as (L
_{max}=2.394569); Lyapunov dimension is obtain as (D
_{L}=3.366413); binary test (0 - 1) is obtain as (k = 0.982). All show that the system is chaotic. Finally, the adaptive control was performed. Moreover, theoretical and graphical results of the system after control show the system is stable and Lyapunov exponent is obtained as: L
_{1}=-0.390000, L
_{2}=-0.500000, so the system is regular.

In the last two decades, the interest in dynamical systems has increased, because they are an important concept in describing the behavior of many models and in various fields. Some studies and research have focused on discrete dynamical systems in which the systems are described in the form of difference equations [

Accordingly, the research was arranged as follows: system description, system analysis and finding fixed points [

In this work, a two-dimensional discrete time dynamical system was taken [

{ x t + 1 = x t + a 1 x t ( 1 − x t ) − a 2 x t y t y t + 1 = y t + a 2 y t ( x t − y t ) (1)

a 1 = 2.4 , a 2 = 2 (2)

where x t represents the prey society and y t represents the predator community in discrete time (t), and that a 1 , a 2 represent the parameters of the system and are positive, the part: x t [ 1 + a 1 ( 1 − x t ) ] is represents the rate of increase of the prey community in the absence of the predation community, and that Part a 2 x t y t represents the rate of decline of the prey community due to the presence of the predation community, and a 2 represents the predation parameter, while the part y t [ 1 + a 2 ( x t − y t ) ] represents the variance in the size of the predation population which depends on the size of the prey community.

For 2D discrete dynamical system (1) with continuous differentiable transition function g 1 and g 2 , given by

x t + 1 = g 1 ( x t , y t ) y t + 1 = g 2 ( x t , y t )

The Jacobian Matrix of system (1) is:

J ( x t , y t ) = [ ∂ g 1 ∂ x t ∂ g 1 ∂ y t ∂ g 2 ∂ x t ∂ g 2 ∂ y t ] = [ 1 + a 1 − 2 a 1 x t − a 2 y t − a 2 x t a 2 y t 1 + a 2 x t − 2 a 2 y t ] (3)

In this section we find the fixed points of system (1), assume that g 1 ( x t , y t ) = x t , g 2 ( x t , y t ) = y t

x t = x t + a 1 x t ( 1 − x t ) − a 2 x t y t (4)

y t = y t + a 2 y t ( x t − y t ) (5)

from Equations (4) and (5) we get the following fixed points:

q 0 = ( 0 , 0 ) , q 1 = ( 1 , 0 ) , q 2 = ( a 1 a 1 + a 2 , a 1 a 1 + a 2 )

Theorem (1): Let

G ( λ ) = a 2 λ 2 + a 1 λ + a 0 = 0 (6)

A characteristic equation of (3), the following cases are true:

1) If the absolute value of the roots of Equation (6) is less than one, then the fixed point of the system (1) is locally asymptotically stable and is called the sink.

2) If the absolute value of the roots of Equation (6) is greater than one, then the fixed point of the system (1) is unstable and is usually called the source, but if at least one of the values of the roots of Equation (6) is greater than one, then the fixed point is called the Saddle.

3) If the absolute value of the roots of Equation (6) is equal to one, then the fixed point of the system (1) is called the non-hyperbolic point. But if there are no roots of values equal to one, then the fixed point is called the hyperbolic point.

In this section, the stability of the fixed points of the system (1) using the following criteria:

Substituting the point q 0 into (3) we get:

J ( 0 , 0 ) = [ 1 + a 1 0 0 1 ]

det ( λ I − J ) = 0

| λ − 1 − a 1 0 0 λ − 1 | = 0

And from Equation (2) we get:

λ 2 − 4.4 λ + 3.4 = 0 (7)

so the roots of quadratic Equation (7) are: λ 1 = 1 , λ 2 = 3.4

since | λ 1 | = 1 , | λ 2 | = 3.4 , so by theorem (1) we get system (1) is unstable at q 0 , similarly we test the points q 1 and q 2 , the results shown in

Lemma (1):

Let the characteristic equation of system (1)

G ( λ ) = a 2 λ 2 + a 1 λ + a 0 = 0

Then it’s jury

the description | Characteristic Equation Roots | Fixed points |
---|---|---|

unstable | λ 1 = 1 , λ 2 = 3.4 | q 0 = ( 0 , 0 ) |

unstable | λ 1 = 2.2283 , λ 2 = 0.6283 | q 1 = ( 1 , 0 ) |

unstable | λ 1 , 2 = 1.2907 | q 2 = ( 0.5454 , 0.5454 ) |

λ 2 | λ 1 | λ 0 |
---|---|---|

a 2 | a 1 | a 0 |

a 0 | a 1 | a 2 |

b 1 | b 0 | |

b 0 | b 1 | |

c 0 |

Such that

b k = | a 0 a n − k a n a k | , k = 0 , 1 , n = 2

c k = | b 0 b n − 1 − k b n − 1 b k | , k = 0 , n = 2

We say that the fixed point of the system (1) is stable if the satisfies following conditions:

G ( 1 ) > 0

( − 1 ) n G ( − 1 ) > 0

| a 0 | < a n , | b 0 | > | b n − 1 | , | c 0 | > | c n − 2 |

Otherwise, fixed points are unstable.

We test the stability of point q 0 , by using lemma (1) and values from eq. (7) we get the jury

The fixed points are said to be stable using the criterion of the Lyapunov function if Δ V ≤ 0 except for the origin, which is stable, and to study the stability of the fixed points of the system (1) we impose the quadratic equation of the following Lyapunov function:

V ( x , y ) = x 2 + y 2 > 0

Where x, y are not equal to zero, and using ∆V we get:

Δ V ( x t , y t ) = V ( x t + 1 , y t + 1 ) 2 − V ( x t , y t ) 2 = ( x t + a 1 x t ( 1 − x t ) − a 2 x t y t ) 2 + ( y t + a 2 y t ( x t − y t ) ) 2 − ( x t ) 2 − ( y t ) 2 (8)

By substituting (2) in Equation (8) and test the point q 2 = ( 0.5454 , 0.5454 ) we get:

λ 0 | λ 1 | λ 2 |
---|---|---|

3.4 | −4.4 | 1 |

1 | −4.4 | 3.4 |

10.56 | −10.56 | |

−10.56 | 10.56 | |

0 |

Δ V ( 0.5454 , 0.5454 ) = ( 0.5454 + 2.4 × 0.5454 × ( 1 − 0.5454 ) − 2 × 0.5454 × 0.5454 ) 2 + ( 0.5454 + 2 × 0.5454 × ( 0.5454 − 0.5454 ) ) 2 − ( 0.5454 ) 2 − ( 0.5454 ) 2 = 0.1438 > 0

It is clear that V is positive definite and ∆V is positive, hence the fixed point q 2 is unstable and so system (1) is unstable.

Newton’s Raphson method is one of the important numerical methods for finding the roots of the difference equations. By using a written programme on MATLAB we get the best result obtained for the system (1) is ( x , y ) = ( 0.5 , 0.5 ) with minimum error (0.0001).

In this section the time behavior of the system (1) was studied and the parameters were fixed at the values a 1 = 2.4 , a 2 = 2 with the values ( x , y ) = ( 0.5 , 0.5 ) and for (1000) iterations, shown in

In this section, the phase space of the system (1) was found, the parameters were fixed at the values a 1 = 2.4 , a 2 = 2 with the values ( x , y ) = ( 0.5 , 0.5 ) and shown in

shows that the system (1) generates trajectories chaotic for x t and y t at some values parameter a 1 .

In this section the bifurcation diagrams of system (1) are found at the bifurcation parameter a 1 , the parameters a 2 = 2 with the values ( x , y ) = ( 0.5 , 0.5 ) and the a 1 parameter ranging from 1.8 to 2.7 showing the behavior chaotic of the system (1), also the internal balance of the parameter a 1 on the period [2.6, 2.65], as shown in

The Lyapunov exponent is one of the important tests in detecting the chaotic behavior of dynamic systems, as the system is said to be a chaotic system if one of the values of the Lyapunov exponent is greater than zero, and by using a mathematical program in MATLAB, the Lyapunov exponent of the system (1) was tested and the following values were obtained:

L 1 = 2.394569 , L 2 = − 1.011898

Since one of the values of the Lyapunov exponent of system (1) is a positive value, then system is chaotic, show in

To calculate the Lyapunov dimension of the system (1), we use the following law:

D L = p + L 1 | L 2 | = 1 + 2.394569 | 1.011898 | = 3.366413

In this section, the binary test was used to analysis the chaos of the system (1), a time series x t was generated from the regularity (1) at the parameters a 1 = 2.4 , a 2 = 2 and the values ( x , y ) = ( 0.5 , 0.5 ) For (1000) iterations, and by using a mathematical program in MATLAB, the binary selection of the system (1), we calculate p c ( t ) with q c ( t ) for t = 100 and chose(c) is random value within the period (0, π), show in

To achieve the stability of the chaotic system (1) we will design an adaptive control law with the unknown parameter a 1 .

x t + 1 = x t + a 1 x t ( 1 − x t ) − 2 x t y t + u 1 y t + 1 = y t + 2 y t ( x t − y t ) + u 2 (9)

where u 1 , u 2 are the controllers for the adaptive and are known as follows:

u 1 = − x t − a ^ 1 x t ( 1 − x t ) + 2 x t y t − M 1 x t u 2 = − y t − 2 y t ( x t − y t ) − M 2 y t (10)

where M 1 , M 2 are positive constants and the parameter a ^ 1 is an approximate parameter of the parameter a 1 , and by substituting (10) in (9) we get:

x t + 1 = ( a 1 − a ^ 1 ) x t ( 1 − x t ) − M 1 x t y t + 1 = − M 2 y t (11)

Let the error for the discretionary parameter be defined as follows:

e a = a 1 − a ^ 1 (12)

Substituting (12) into (11) we get:

x t + 1 = e a x t ( 1 − x t ) − M 1 x t y t + 1 = − M 2 y t (13)

In this section, we will test the stability of the fixed points of the system (1) in the controlled system (11) with the values ( x , y ) = ( 0.5 , 0.5 ) and M 1 = 0.4 , M 2 = 0.5 and the parameter a ^ 1 is an estimated parameter of the parameter a 1 , and let a ^ 1 = 2.39 .

From (11) the Jacobian matrix is:

J ( x , y ) = [ − 0.39 − 0.02 x t 0 0 − 0.5 ] (14)

We test the stability of the fixed point q 0 = ( 0 , 0 ) , from characteristic equation λ 2 + 0.89 λ + 0.195 = 0 at point q 0 we get a 0 = 0.195 , a 1 = 0.89 , a 2 = 1

Then we get Jury

Δ V ( x t , y t ) = ( x t + 1 ) 2 + ( y t + 1 ) 2 − ( x t ) 2 − ( y t ) 2

From the system (11) we get

Δ V ( x t , y t ) = ( ( a 1 − a ^ 1 ) x t ( 1 − x t ) − M 1 x t ) 2 + ( − M 2 y t ) 2 − ( x t ) 2 − ( y t ) 2

Substituting the parameters a ^ 1 , a 1 , M 1 , M 2 we obtain

Δ V ( x t , y t ) = ( 0.01 x t ( 1 − x t ) − 0.4 x t ) 2 + ( − 0.5 y t ) 2 − ( x t ) 2 − ( y t ) 2

We test the fixed points q 0 , q 1 , q 2 , it is clear that V is positive definite and ∆V is negative definite, consequently the adaptive strategy success to control system (1), and

the description | Characteristic Equation Roots | Fixed points |
---|---|---|

stable | λ 1 = 0.39 , λ 2 = 0.5 | q 0 = ( 0 , 0 ) |

stable | λ 1 = 0.41 , λ 2 = 0.5 | q 1 = ( 1 , 0 ) |

stable | λ 1 = 0.4008 , λ 2 = 0.5 | q 2 = ( 0.5454 , 0.5454 ) |

λ 0 | λ 1 | λ 2 |
---|---|---|

0.195 | 0.89 | 1 |

1 | 0.89 | 0.195 |

-0.962 | -0.7165 | |

-0.7165 | -0.962 | |

0.4121 |

the description | Results of the Lyapunov function test | Fixed points |
---|---|---|

stable | Δ V = 0 | q 0 = ( 0 , 0 ) |

stable | Δ V = − 0.84 | q 1 = ( 1 , 0 ) |

stable | Δ V = − 0.4647 | q 2 = ( 0.5454 , 0.5454 ) |

In this section, the Lyapunov exponent test of the system was performed and the values were obtained L 1 = − 0.390000 , L 2 = − 0.500000 , accordingly, the system is regular and

In this research, 2D discrete - time dynamical system was taken, the system was analyzed, fixed points were found, and the stability analyzed for fixed points using (roots characteristic equation, Jury test, Lyapunov function test). The roots of the system were found using Newton’s Raphson numerical method, and the dynamic behavior was analyzed and studied. The phase space of the system shows that the system is unstable at parameter a 1 = 2.4 . For the chaos analysis of the system, the bifurcation diagrams of the bifurcation parameter a 1 = 2.4 are found for the system, and the Lyapunov exponent test was used and the value ( L max = 2.394569 ) was obtained, which is an indication of the chaos of the system. And Lyapunov dimension was calculated as ( D L = 3.366413 ). When using the binary test (0 - 1), it was found that the value of (k = 0.982) with the parameter a 1 = 2.4 which is an indication of the chaos of the system. Finally, the adaptive control of the system and the stability test of the system after the control was performed which showed us that the system is stable, and when we tested the Lyapunov exponent ( L 1 = − 0.390000 , L 2 = − 0.500000 ) shown the regular behavior of the system.

We would like sincerely thank and also acknowledge to the university of Mosul and college of computer science and mathematics for the support and encouragement that help us to improve the quality of this work.

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

Aziz, M.M. and Jihad, O.M. (2021) Stability, Chaos Diagnose and Adaptive Control of Two Dimensional Discrete - Time Dynamical System. Open Access Library Journal, 8: e7270. https://doi.org/10.4236/oalib.1107270