Darboux Integrability of a Generalized 3D Chaotic Sprott ET9 System

: In this paper, the first integrals of Darboux type of the generalized Sprott ET9 chaotic system will be studied. This study showed that the system has no polynomial, rational, analytic and Darboux first integrals for any value of 𝑎 and 𝑏 . All the Darboux polynomials for this system were derived together with its exponential factors. Using the weight homogenous polynomials helped us prove the process.


Introduction:
In recent decades, there have been reports in the literature of several chaotic differential systems such as 1-4 and many others. Recently, even differential systems with only one equilibrium points to have chaotic behavior also have been demonstrated [5][6][7][8] .
In 1994, Sprott 9 displayed 19 distinct simple three-dimensional autonomous ordinary differential equations of chaotic flows and quadratic non-linearities, based on two real parameters, and then described their properties. These examples are simpler than Lorenz and Rössler system. Among these 19 systems, Case A is the simplest one that 10,11 used to describe a one-dimensional Nose-Hoover mechanical system exhibiting chaos. Some systems attracted much attention see [12][13][14] . Twenty one years later, in 2015, Sprott 15 made generalization of Nose-Hoover oscillator, revealing 11 cases with strange attractors (hidden or selfexcited) among these he introduced a chaotic system ET9 with only one non-hyperbolic equilibrium point. This equilibrium is nonlinearity unstable and strange attractor is self-excited. Selfexcited attractor are examples such as van der Pol, Belousov-Zhabotinskii, Lorenz, Rössler, Chen, Chua, Lu, Jerk or Sprott's system (case B-S). Various studies of systems with self-excited attractors have been conducted in different science areas especially in engineering applications, like in the design electronic circuits, communications, control systems, and artificial intelligence [16][17][18][19] , Nevertheless, in these systems with the self-excited attractors, there are still various issues that invite further research.
This research modifies the Sprott ET9 system 15 by considering two parameters in the nonlinear portion, which are prospective to a more chaotic system of behavior. More specifically, the following generalized system will be studied ̇= , ̇= − + , (1) ̇= − − − , where and are nonzero real parameters. In 15 , Sprott presented system (1) with = 4 and = −1 as ET9 ( Fig. 1) among eleven different autonomous systems having chaotic conduct.
To the best of our knowledge, this rich dynamical system (1) has never been investigated from the integrability perspective. The key objective of this work involves the characterization of the rational and Darboux first integrals. For this, the invariant algebraic surfaces need to be fully characterized based on their parameters. To achieve invariant algebraic surfaces as such, the theory of Darboux integrability needs to be utilized, for further details on this theory see [20][21][22][23][24][25][26][27] .
In 1878 Darboux 28 demonstrated how the first integrals of 2D differential system could have been formed with enough invariant algebraic curves. In particular, he had shown that if a polynomial autonomous system with degree This research provides the invariant of system (1) which consists of the rational and Darboux first integral. The analytic and polynomial first integrals are also provided. For our system, one first integral reduces the complexity of its dynamics, and the presence of two irreducible first integrals solves entirely the issue of determining its phase portraits.

Some Definitions and Preliminary Results
This section begins with a brief overview of the integrability problem, the Darboux method, and the auxiliary results. To prove the main results of this paper, few basic definitions and theorems are given as a background to this study. Let = ( , , ) be a real polynomial defined as a Darboux polynomial for the system (1) if (2) for a real polynomial ( , , ), that is a cofactor of with a degree of almost one. As a result, the cofactor form can be assumed as follow ( , , ) = 0 + 1 + 2 + 3 , (3) where ∈ ℂ for = 0, . . ,3 if ( , , ) is a Darboux polynomial of the differential system (1), then the invariant algebraic surface in ℝ 3 is = 0. It is called as such because of the fact that when a solution of system (1) includes a point on the invariant algebraic surface, then the entire solution is contained in it 20 .
It is known if a Darboux polynomial is with zero cofactor then it is defined as a polynomial first integral of system (1). i.e.
(4) Once the function is satisfying Eq. (4) and is also rational (analytic) then it is a rational (analytic) first integral. Definition 29 A nonconstant polynomial ( ) ∈ [ ] is irreducible over the field or is an irreducible polynomial in [ ] if ( ) cannot be expressed as a product ( ) * ℎ( ) of two polynomials ( ) and ℎ( ) in [ ] both of lower degree than the degree of ( ). If ( ) ∈ [ ] is a nonconstant polynomial that is not irreducible over , then ( ) is reducible over . First, depending on the following two Propositions that have been proved in 20,30,31 :  (1) indicates the occurrence of either a polynomial first integral or two Darboux polynomials with the identical non-zero cofactor.
To validate system (1) that does not have analytic first integral in the neighborhood of the origin, the following result is necessary which is due to Li, Llibre and Zhang in 32 . It is obvious to us that an exponential factor of system (1) is defined as an exponential function of the form = ( /ℎ) ∉ ℂ with , ℎ ∈ ℂ[ , , ] and let and are coprime in the ring ℂ[ , , ], and satisfying for certain polynomials = ( , , ) having degree at most 1, is said to be the cofactor of . The result below provides a geometrical meaning of the exponential factor concept, which can be found in 30 for the plane and 24,33 for higher dimension systems.
also an exponential factor of system (1) with cofactor = ∑ =1 . For the proof of the above result, see 5 .
The first integral of system (1) can be considered a Darboux type when expressed as: , where 1 , … , are Darboux polynomials, 1 , … , are exponential factors, and and are complex numbers, for = 1, … , and = 1, … , . Theorem 1.6. A polynomial system (1) if and only if the function of Darboux type is a first integral of system (1). For proof of Theorem 1.6 and more information, refer to 20,33 . The weight homogeneous polynomials can now be defined, which is used in the proof of Theorem 2.3. The invariant algebraic surfaces of many popular systems, such as Lorenz system 34 , Chen system 35 , Moon-Rand system 5 , and et al. have been commonly used in this procedure.
A polynomial ( ) with ∈ ℝ is considered as weight homogeneous if there was any = ( 1 , … , ) ∈ ℕ and ∈ ℕ such that for all > 0, ( 1 1 , 2 2 , … , ) = ( ), where ℕ signifies the set of positive integers. The variable s refers to the weight exponent of , and denotes the weight degree of with the weight exponent .

Main Results and their Proofs of the Chaotic ET9 System
The study of Darboux integrability is presented in this section. These results are expected to prove that system (1) has only one irreducible Darboux polynomial, where the parameter is zero. It is also anticipated to demonstrate that the system has neither a polynomial first integral nor a rational first integral. Subsequently, it can be proven that the system contains only one exponential factor when is not zero. Finally, the system will be proven that is not Darboux integrable. 3D and 2D projections of system (1) were plotted for a given set of initial conditions by selecting the parameters = 4 and = −1 with initial condition [0, 1, 0.4]. These projections were subjected to detailed numerical and theoretical analysis (Fig. 1). Firstly, the study begins with the following two lemmas: can be written, where each is a polynomial of and only. Then from substituting into Eq.
The terms of +1 in Eq.
Solving the above equation leads to ℎ 0 ( ) = , where is a constant. This is contradiction. This completes the proof. Case 2. If ℎ ( ) = 0 , then ℎ = 0, this means that = ℎ 0 ( , ). The arguments of the proof for this case is similar also to case 1.□ Theorem 2.5. For System (1) there is no first integrals of rational type. Proof. Based on the Propositions 1.1 and 1.2. system (1) has a rational first integral if it has a polynomial first integral or it has two Darboux polynomials with the same cofactor. But by Theorems 2.3 and 2.4, system (1) does not have any polynomial first integrals and it has only one Darboux polynomial. Thus system (1) has no rational first integral. This completes the proof.□ Theorem 2.6. The next two statements hold for system (1). a) For ≠ 0, the distinctive independent exponential factors of the autonomous system (1)  where = 0 + 1 + 2 + 3 (18). The solution of above equation is where −1 ( ) denotes a polynomial of . Since −1 is a polynomial then must be = 0. Therefore ( , , ) = 0 ( , ) + 1 ( , ) . Equation (17) becomes Compute the coefficients of , = 0,1,2 , the following equations will obtain: i=2: A characteristic equation of is ( ) = 3 + 2 + + 1 = 0. Then the eigenvalues of are 1 = −1, 2 = , and 3 = − . Since there is not exist 1 , 2 , and 3 positive integers such that