Darboux Integrability of a Generalized 3D Chaotic Sprott ET9 System

في هذا البحث تم دراسة التكامل الاول من نوع داربوكس لتعميم النظام الفوضوي الثلاثي الابعاد Sprott ET9 . حيث وضحنا ان النظام لايمتلك متعددة حدود . دالة كسرية, تحليلية والداربوكس للتكامل الاول لاي قيمتين a و b. كما استطعنا ابجاد متعددة  حدود داربوكس لهذا النظام بقرب المفكوك الاسي. باستخدام وزن متعددة الحدود المتجانسة التي ساعدتنا في برهان الطريقة.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

Zero-Hopf Bifurcation in the Generalized Stretch-Twist-Fold Flow

A zero-Hopf equilibrium in a three-dimensional system is an isolated equilibrium point which has a zero eigenvalue and a simple pair of purely imaginary eigenvalues. In general, for such an equilibrium, there is no theory for finding when some periodic solutions are bifurcated by perturbing the parameters of the system. In this work, we describe the values of the parameters for which a zero-Hopf equilibrium occurs at the equilibrium points in the generalized stretch-twist-fold flow. Thus, only one condition for parameters of the generalized stretch-twist-fold flow introduced in a system (Eq. 1) is found for which the equilibrium point is a zero-Hopf equilibrium. For this condition, we use the averaging method to provide the existence of a periodic solution, which bifurcates from the zero-Hopf equilibrium point. The main result in this paper is Theorem 1, which gives a periodic solution of the generalized stretch-twist-fold flow