A necessary and sufficient condition for the stability of the system is that the characteristic equation’s eigenvalues have negative real components. Following is the Jacobian matrix for the new system (1) up to E 0 = ( 0 , 0 ) :

J = [ 35.5 0 0 − 4.2 ] , (3)

The characteristic equation is:

λ 2 − 31.3 λ − 149.1 = 0 , (4)

Roots of the characteristic equation:

λ 1 = 35.5 , λ 2 = − 4.2 ,

Thus, the system is unsteady.

A system meets the Routh requirement for stability (all poles in the half-loop level), if and only if the components in the first column of the Routh row have only positive values for all of their values. The number of sign changes in the first column multiplied by the sum of the non-OLHP columns [25]. Regarding the Roth stability test, see Table 1.

a 0 = − 149.1 , a 1 = − 31.3 , a 2 = 1 ,

The system is unstable because the first column has four negative elements.

Determinants generated from the coefficients of the characteristic equation are used to implement this criterion. System (1) is stable if the tiny minors of its square matrix J are all positive; if not, it is unstable [25].

From Equation (3):

Δ 1 = a n − 1 = a 1 = − 31.3 < 0 ,

Δ 2 = | a n − 1 a n − 3 a n a n − 2 | = | a 1 0 a 2 a 0 | = | − 31.3 0 1 149.1 | = − 4666.83 < 0 ,

System (1) is unstable because some of the values of the determinants are less than zero.

Where we assume the Lyapunov function is:

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

V ˙ ( x 1 , x 2 ) = ∂ v ∂ x 1 ∂ x 1 ∂ t + ∂ v ∂ x 2 ∂ x 2 ∂ t , (5)

The system is stable, 0 > V ˙ if

We get: (5) in Equation (1) substituting.

Since V ˙ > 0 , as a result, new system (1) is unstable.

By creating a continuous fraction from the odd and even parts of the equation, the characteristic equation of a continuous system is subjected to this condition. The distinguishing equation:

λ 2 − 31.3 λ − 149.1 = 0 ,

By taking the even terms and then the odd terms, respectively, we have:

Q 1 ( λ ) = λ 2 − 149.1 (6)

Q 2 ( λ ) = − 31.3 λ (7)

After dividing the even terms by the odd terms and using algebraic steps, we get the following results:

h 1 = − 0.031 ,   h 2 = − 0.209.

Since some values of h are negative, the equation of the system has some positive real roots, so system (1) is chaotic.

Suppose that

f 1 = d x d t , f 2 = d y d t .

The obtained vector field,

( x ˙ , y ˙ ) T = ( f 1 , f 2 ) T

∇ ⋅ ( x ˙ , y ˙ ) T = ∂ f 1 ∂ x + ∂ f 2 ∂ y = b − 1.207 s − d + 0.004 e s = f .

Note that, f = − b − 1.207 s − d + 0.004 e s = − 0.504 , for all values that are positive and greater than zero, the system (1) dissipates

Here is the exponential rate:

d V d t = f V ⇒ V ( t ) = V 0 e f t = V 0 e − 0.504 t

By flowing into ( V 0 e − 0.504 t ), the volume element ( V 0 ) from the previous equation is condensed at the time (t).

The fourth and fifth order Runge-Kotta method is used to solve the system (1). Initial values included

x | x ( 0 ) , y ( 0 ) = [ 0.5 , 1 ]

The waveform exhibits aperiodic structure, the primary defining feature of chaotic systems. x ( t ) and y ( t ) for system (1) (as showed in Figure 1).

In this paragraph, the strange attractor for the system (1) in (x, y) space is shown along with the chaotic strange attractor for the system (1), shown in Figure 2.

Since the orbit in each graph looks to be dense, the new system exhibits a chaotic attractor.

The typical exponential growth rates of almost divergent trajectories in phase space are frequently referred to as the Lyapunov exponent. The new system is regarded as chaotic if it has at least one positive Lyapunov exponent. Values of the Lyapunov exponent are:

( L 1 = 1.523 , L 2 = − 2.388 ) .

As a result, the system’s “Kaplan-Yorke dimension” or Lyapunov dimension is as follows:

D L = 1 + L 1 | L 2 | = 1.637

Figure 3 shows that system (1) is very Chaotic.

To stabilize a chaotic system (1) use the sufficiency control law generalized with an unknown parameter c as follows:

x ˙ = 35.5 x − 29.4 x y + u 1 y ˙ = − 4.2 y + 29.4 e x y + u 2 (8)

where [ u 1 , u 2 ] T are feedback controllers.

We now consider the following adaptive control procedures to make sure the managed system (7) converges asymptotically to the origin.

u 1 = − 35.5 x + 29.4 x y − μ 1 x u 2 = 4.2 y − 29.4 e ^ x y − μ 2 y (9)

where μ 1 , μ 2 are constants, c ^ is an estimator of the parameter c.

Substituting (8) into (7), we get:

x ˙ = − μ 1 x y ˙ = 29.4 x y ( e − e ^ ) − μ 2 y (10)

Let the estimation error of the parameter be:

e c = c − c ^ (11)

Using (10), system (9) can be written as:

x ˙ = − μ 1 x , y ˙ = 29.4 e e x y − μ 2 y , (12)

The parameter estimates c ^ is changed using the Lyapunov method of obtaining the updated law. It is thought that the quadratic Lyapunov function:

V ( x 1 , x 2 ) = 1 2 ( x 1 2 + x 2 2 + e e 2 ) , (13)

Which definite, positive-in ℝ 3 .

Also

e ˙ c = − c ^ ˙ (14)

Differentiate V & substituting (11) and (13), we get:

V ˙ = − μ 1 x 2 − μ 2 y 2 + e e ( 29.4 x y 2 − e ^ ˙ )

Assume that:

e ^ ˙ = x y + μ 3 e e (15)

where μ 3 is higher than 0 in value.

Substitute (14) into V ˙ ˙ , we get:

V ˙ = − μ 1 x 2 − μ 2 y 2 − μ 3 e e 2 + 29.4 e e y 2 x − e e x y (16)

Which is negative-definite on ℝ 3 .

The outcome is as follows because of Lyapunov stability, Eigenvalues, and the Routh array criteria.

Proposition 1. Byadaptive control (10), where c ^ ˙ = x y + μ 3 e c and μ 1 , μ 2 , μ 3 are positive constants, The chaotic system (8) is stabilized for x ( 0 ) ∈ ℝ 2 .

The controlled extremely chaotic system (8) was simulated using

x | x 1 ( 0 ) , x 2 ( 0 ) = [ 3 , 9 ]

μ 2 , μ 1 = [ 30 , 15 ] and e c = 25.3 .

The new system (1)’s-controlled state trajectories are displayed in Figure 4.