Darboux Integrability of a Generalized 3D Chaotic Sprott ET9 System

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Adnan Ali Jalal
Azad I. Amen
Nejmaddin A. Sulaiman


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 . All the Darboux polynomials for this system were derived together with its exponential factors. Using the weight homogenous polynomials helped us prove the process.


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Jalal AA, Amen AI, Sulaiman NA. Darboux Integrability of a Generalized 3D Chaotic Sprott ET9 System. Baghdad Sci.J [Internet]. [cited 2021Dec.4];19(3):0542. Available from: https://bsj.uobaghdad.edu.iq/index.php/BSJ/article/view/4474


Lu J, Chen G. A new chaotic attractor coined. Int J Bifurc Chaos. 2002;12(3):659–61.

Jalal AA, Amen AI, Sulaiman NA. Darboux integrability of the simple chaotic flow with a line equilibria differential system. Chaos, Solitons and Fractals [Internet]. 2020;135:109712. Available from: https://doi.org/10.1016/j.chaos.2020.109712

Jafari S, Sprott JC, Molaie M. A Simple Chaotic Flow with a Plane of Equilibria. Int J Bifurc Chaos. 2016;26(6):1–6.

Barati K, Jafari S, Sprott JC, Pham VT. Simple Chaotic Flows with a Curve of Equilibria. Int J Bifurc Chaos. 2016;26(12):1–6.

Barreira L, Valls C, Llibre J. Integrability and limit cycles of the Moon-Rand system. Int J Non Linear Mech [Internet]. 2015;69:129–36. Available from: http://dx.doi.org/10.1016/j.ijnonlinmec.2014.11.029

Llibre J, Oliveira R, Valls C. On the Darboux integrability of a three–dimensional forced–damped differential system. J Nonlinear Math Phys. 2017;24(4):473–94.

Zhang L, Yu J. Invariant algebraic surfaces of the FitzHugh-Nagumo system. J Math Anal Appl [Internet]. 2019;(November 2018). Available from: https://doi.org/10.1016/j.jmaa.2019.04.009

Oliveira R, Valls C. Global dynamical aspects of a generalized Chen–Wang differential system. Nonlinear Dyn. 2016;84(3):1497–516.

Sprott JC. Some simple chaotic flows. Phys Rev E [Internet]. 1994 Aug 1;50(2):R647–50. Available from: https://link.aps.org/doi/10.1103/PhysRevE.50.R647

Posch HA, Hoover WG, Vesely FJ. Canonical dynamics of the Nosé oscillator: Stability, order, and chaos. Phys Rev A [Internet]. 1986 Jun 1;33(6):4253–65. Available from: https://link.aps.org/doi/10.1103/PhysRevA.33.4253

Hoover WG. Canonical dynamics: Equilibrium phase-space distributions. Phys Rev A [Internet]. 1985 Mar 1;31(3):1695–7. Available from: https://link.aps.org/doi/10.1103/PhysRevA.31.1695

Messias M, Reinol AC. On the existence of periodic orbits and KAM tori in the Sprott A system: a special case of the Nosé–Hoover oscillator. Nonlinear Dyn [Internet]. 2018;92(3):1287–97. Available from: https://doi.org/10.1007/s11071-018-4125-1

Oliveira R, Valls C. Chaotic behavior of a generalized Sprott E differential dystem. Int J Bifurc Chaos. 2016;26(5):1–16.

Mahdi A, Valls C. Integrability of the Nosé-Hoover equation. J Geom Phys [Internet]. 2011;61(8):1348–52. Available from: http://dx.doi.org/10.1016/j.geomphys.2011.02.018

Sprott JC. Strange attractors with various equilibrium types. Eur Phys J Spec Top. 2015;224(8):1409–19.

Wang MJ, Liao XH, Deng Y, Li ZJ, Zeng YC, Ma ML. Bursting, Dynamics, and Circuit Implementation of a New Fractional-Order Chaotic System with Coexisting Hidden Attractors. J Comput Nonlinear Dyn. 2019;14(7):1–12.

Akgul A, Moroz I, Pehlivan I, Vaidyanathan S. A new four-scroll chaotic attractor and its engineering applications. Optik (Stuttg) [Internet]. 2016;127(13):5491–9. Available from: http://dx.doi.org/10.1016/j.ijleo.2016.02.066

Moysis L, Volos C, Pham V, Goudos S, Stouboulos I, Gupta MK, et al. Analysis of a Chaotic System with Line Equilibrium and Its Application to Secure Communications Using a Descriptor Observer †. Technologies. 2019;7(4):76.

Sambas A, Vaidyanathan S, Mamat M, Sanjaya Ws M, Yuningsih SH, Zakaria K. Analysis, Control and Circuit Design of a Novel Chaotic System with Line Equilibrium. J Phys Conf Ser. 2018;1090(1).

Dumortier F, Llibre J, Artés JC. Qualitative theory of planar differential systems. Springer, Berlin, Heidelberg; 2006.

Canada A, Drabek P, Fonda A. Handbook of Differential Equations Ordinary Differential Equations. 1st ed. Netherlands: Elsevier B.V; 2004. 708 p.

Llibre J, Valls C. Darboux integrability of generalized Yang-Mills Hamiltonian system. J Nonlinear Math Phys [Internet]. 2016;23(2):234–42. Available from: https://doi.org/10.1080/14029251.2016.1175820

Llibre J, Valls C. On the Darboux integrability of the Painlevé II equations. J Nonlinear Math Phys [Internet]. 2015;22(1):60–75. Available from: http://dx.doi.org/10.1080/14029251.2015.996441

Llibre J, Zhang X. Darboux theory of integrability in Cn taking into account the multiplicity. J Differ Equ [Internet]. 2009;246(2):541–51. Available from: http://dx.doi.org/10.1016/j.jde.2008.07.020

Llibre J, Yu J, Zhang X. On polynomial integrability of the Euler equations on so(4). J Geom Phys [Internet]. 2015;96:36–41. Available from: http://dx.doi.org/10.1016/j.geomphys.2015.06.001

Zhang X. Integrability of Dynamical Systems: Algebra and Analysis [Internet]. Vol. 47. Singapore Springer; 2017. Available from: http://link.springer.com/10.1007/978-981-10-4226-3

Zhang X. Liouvillian integrability of polynomial differential systems. Am Math Soc. 2016;368(1):607–20.

Darboux G. Mémoire sur les équations différentielles algébriques du premier ordre et du premier degré. 1878;2(1):60–96.

Fraleigh JB. A first course in abstract algebra [Internet]. seventh ed. Vol. 3. Pearson Education Limited; 2014. 461 p. Available from: https://www.amazon.com/First-Course-Abstract-Algebra-7th/dp/B009NGC1UO

Christopher C, Llibre J, Pereira JV. Multiplicity of Invariant Algebraic Curves in Polynomial Vector Fields. Pacific J Math [Internet]. 2007;229(1):63–117. Available from: 34C05

Christopher C, Llibre J, Pantazi C, Walcher S. On planar polynomial vector fields with elementary first integrals. J Differ Equ [Internet]. 2019;267(8):4572–88. Available from: https://doi.org/10.1016/j.jde.2019.05.007

Li W, Llibre J, Zhang X. Local first integrals of differential systems equations and diffeomorphism. Zeitschrift für Angew Math und Phys ZAMP. 2003;54(2):235–55.

Llibre J, Zhang X. Darboux theory of integrability for polynomial vector fields in Rn taking into account the multiplicity at infinity. Bull des Sci Math [Internet]. 2009;133(7):765–78. Available from: http://dx.doi.org/10.1016/j.bulsci.2009.06.002

Llibre J, Audia CL. On the integrability of the 5-dimensional lorenz system for the Gravity-Wave activity. Proceedings of the American Mathematical Society.2017;145(2):665–79.

Lü T, Zhang X. Darboux polynomials and algebraic integrability of the Chen system. Int J Bifurc Chaos. 2007;17(8):2739–48.