Stability of Lorenz System at the Second Equilibria Point based on Gardano’s Method

In this paper, stability conditions of the Lorenz system at the second equilibrium point are investigated by applying Gardano’s method where the system has three equilibria points. Most of the previous work focused their studies at the original point. A few studies demonstrated stability of dynamical systems at another equilibria points by use of conventional techniques. However, it is often unclear and based on numerical methods. This reason, motivate us to establish the stability conditions of the Lorenz system at a point which different from the origin point and compare between them. Finally, An illustrative example shows the effectiveness and feasibility of this method.


Introduction
In 1963, Lorenz found the first chaotic system, which is a third order autonomous system with only two multiplication-type quadratic terms, but displays very complex dynamical behaviors [1,2]. By definition Vanecek and Celikovsky the Lorenz system satisfies the condition > 0, where and are corresponding elements in the constant matrix = ( ) × for the linear part of system [3].
Lorenz system is not integrable and it is difficult to find an analytical solution for this system in three dimension parameters space, but special cases for Lorenz system are studied before studying periodic solutions, and Lorenz studied the system when = 10, = 8/3. From the definition of equilibrium points, it is easy to verify that, when ≤ 1 the Lorenz system has only one equilibrium point, which is the origin, but when > 1 it has three equilibria points: (0,0,0) , (± ( − 1), ± ( − 1) , − 1) [4,5], The Lorenz system has some simple properties such that this system has natural symmetry (x,y,z)  (-x,-y,z) and the z-axis is invariant [6,7]. [8,9] discussed the stability of Lorenz system about the equilibria points, and found the roots of characteristic equation for this system at origin, but at the second equilibrium point E ( β(r − 1), β(r − 1) , r − 1), the Ref [3] used Routh-Hurwitz test to investigated the stability without founding the roots, the Routh-Hurwitz paly important role in stability of dynamical systems  [10], while the Ref [5] depended on the value of r to investigated the stability without founding the roots.
In [9] studied the stability for system derived from the Lorenz system and depended on the roots to determine the stability at origin, The determination of the roots of a cubic equation in general is fairly difficult, but in the given case, one root is easily found [11]. We can find the roots of equations for third degree by numerical method, and these roots are proximal (not exact), but by using Gardano's method on the same equations we can find exact roots, and by these roots we can investigated the stability for any system.
In this paper, the stability conditions of Lorenz system at the second equilibrium point E is established by using the general formula Gardano's method to find the roots of the characteristic equation for this system. The Lorenz system is described by: x! = σ(y − x) y! = rx − y − xz z! = xy − βz (1) Where σ, r , β are positive parameters. Figure 1 and Figure 2 shows the attractors of the system (1). The approximating linear system (1) at E is: or the characteristic equation of the form: The solutions of Eq. 3 are λ , = .−σ − 1 ± (σ − 1) + 4σr 0, λ = −β (5) Now consider the system (1) at second equilibrium point E ( β(r − 1) , β(r − 1) , r − 1), Under the linear transformations (x, y, z) → (X, Y, Z), The system (1) becomes The approximating linear system (1) at equilibrium point E is: And the characteristic equation of the form:

Helping results Remark 1[5]:
When = 10, = 8/3 , the solutions of equation (9) depend on the parameter as follows: 1-For 1 < < ≅ 1.3456 , there are three negative real roots, 2-For < < ≅ 24.737, there are one negative real root and two complex roots with negative real parts, 3-For > there are one negative real root and two complex roots with positive real parts.

Remark 2[4]:
Let be a B × B matrix of constants. A equilibrium point for the system C ! = 9 is  aymptotically stable if all roots of has negative real parts  unstable if has at least one root with a positive real parts.

Remark 3[10]: Critical case
In critical cases when the real parts of all roots of the characteristic equation are non positive, with the real part of at least one root being zero.

Remark 4[3]: Critical value
Lorenz system has critical value which is r D = 1 at origin and r D = E(EFGF ) EHGH at the second equilibrium point, and this system is asymptotically stable if r lies between 1 and a critical value r D at E ( β(r − 1) , β(r − 1) , r − 1).
We will use the following theorem, which enables us to find the exact roots for cubic equation (three degree).

Theorem 1[12, 13]: (Gardano's method)
 If ∆= 0, then the second term of equation (9) has three roots, but one is multiple:  If ∆ < 0, then the equation (10) has three different real roots as:  If ∆ > 0, then the equation (10) has one real root and two complexes conjugate roots with non-vanishing imaginary parts as:  (1) when ∆ < 0 we obtain: λ , λ , λ are different real roots and these roots are negative when 1 < r ≤ r , (by Remark 1.1 and corollary1.1), hence satisfied Remark 2.1, therefore the system (1) is asymptotically stable, When ∆ > 0 we obtain: λ is a real root and λ , λ are complex conjugate roots and these roots are negative (negative real parts) when and r ∈ (r , r ) and satisfied remark 2.1, therefore the system (1) is asymptotically stable. Case 2: when ∆ > 0 , we have λ is a real root and λ , λ are complex conjugate roots and by remark 1.3.When r ∈ (r , ∞), we obtain that λ is negative and λ , λ are positive real part, hence satisfied remark 2,2 , therefore the system (1) is unstable. Case 3: When r = r we have λ is negative real root and Re λ = Re λ = 0 (Corollary1.2) and satisfied remark 3, hence the system (1) is a critical case, the proof is complete.
We can generalization Theorem (2) for any value of σ and β in the following theorem Theorem 3:The solutions of system (1) at second critical point E ( β(r − 1) , β(r − 1) , r − 1) are:  Asymptotically stable if the following cases hold: Proof: Case 1. By theorem (1) when ∆ < 0 we obtain: a three different real roots and λ , λ are negative (given), we must prove that λ is negative, Since the cubic equation (10) has a positive coefficients therefore, then at least one of these roots is negative real part, hence satisfied remark 2,1 and the system (1) is asymptotically stable, When ∆ > 0 , then we have Re λ < 0 also since Re λ = Re λ (two complex conjugate roots) and λ is negative real root (cubic equation with positive coefficients must have a negative real root), then we have one negative real root and complex roots with negative real part, hence the system (1) is asymptotically stable. Case 2. By the same theorem when ∆ > 0, we have Re λ > 0 since Re λ = Re λ (analogously as in proof of case 1 ) and λ is negative real root, then we have one negative real root and two complex roots with positive real part, hence satisfied remark 2,2 , the system (1) is unstable. Case 3. when ∆ > 0 and Re λ = 0 , then Re λ = 0 and λ is negative real part, hence satisfied remark 3, the system (1) is a critical case, the proof is complete.

Proposition:
The necessary and sufficient condition for a second critical point whose parameter is greater than the parameter plus one .

2-If <
then denominator of critical value is always negative therefore W is a negative (contradictions) the same reason of the first case. 3-If > , then critical value is positive and larger then 1(possible), but if = + 1 then critical value become W = k(kFlF ) m , (contradictions) not allowed dividing on a zero, therefore we must > + 1 only.

Nu.
At

Illustrative Examples:
In this section, we take two different systems, for example to show how to use the results obtained in this paper to analyze the stability of class chaotic systems.

Conclusions
In this paper, we have investigated the stability of Lorenz system at the second critical point by using a new method. By this method we justified the same results which found by previous methods. An illustrative examples show the effectiveness and feasibility of this method.