Darboux integrability of the simple chaotic flow with a line equilibria differential system

Jalal, Adnan A.; Amen, Azad I.; Sulaiman, Nejmaddin A.

doi:10.1016/j.chaos.2020.109712

Cited by 14 publications

(11 citation statements)

References 23 publications

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“…We recall some definitions given in [5,7]. Some well-known results on the Darboux theory of integrability and analytic first integrals may be found in [1,2,5,6]. We characterize here integrability and non-integrability of system (1).…”

Section: Elementary Resultsmentioning

confidence: 99%

Integrability Analysis of the Smallest 3D Biochemical Reaction Model

Amen¹

2023

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In this paper the complex dynamics of a smallest biochemical system model in three-dimensional systems with the reaction scheme. This model is described by a system of three nonlinear ordinary differential equations with five positive real parameters, are analyzed and studied. We present a thorough analysis of their invariant algebraic surfaces and exponential factors and investigate the integrability and nonintegrabilty of this model. Particularly, we show the non-existence of polynomial, rational, Darboux and local analytic first integrals in a neighborhood of the equilibrium. Moreover, we prove that, the model is not integrable in the sense of Bogoyavlensky in the class of rational functions.

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Section: Elementary Resultsmentioning

confidence: 99%

Integrability Analysis of the Smallest 3D Biochemical Reaction Model

Amen¹

2023

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“…To prove Theorem 2 we use the weight-homogeneous polynomials. Where the weight-homogeneous polynomial has been used extensively in several standard systems [20], [21], and [22].…”

Section: Integrability For Smooth Systemmentioning

confidence: 99%

Limit cycles and Integrability of a continuous system with a line of equilibrium points

Abdulkareem

Amen

Hussein

2023

Preprint

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We focus on a chaotic differential system in 3-dimension, including an absolute term and a line of equilibrium points. Which describes in the following ẋ = y , ẏ = −ax + yz , ż = by − cxy − x2. This system has an implementation in electronic components. The first purpose of this paper is to provide sufficient conditions for the existence of a limit cycle bifurcating from the zero-Hopf equilibrium point located at the origin of the coordinates. The second aim is to study the integrability of each differential system, one defined in half–space y ≥ 0 and the other in half–space y < 0. We prove that these two systems have no polynomial, rational, or Darboux first integrals for any value of a, b, and c. Furthermore, we provide a formal series and an analytic first integral of these systems. We also classify Darboux polynomials and exponential factors.

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“…In recent decades, there have been reports in the literature of several chaotic differential systems such as [1][2][3][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] .…”

Section: Introductionmentioning

confidence: 99%