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In this work, we focus on studying the integrability of the following three-dimensional Van der Pol–Duffing system x ˙ = − m ( x 3 − μ x − y ) , y ˙ = x − y − z , z ˙ = β y . More precisely, if m β ≠ 0, then the above system has no analytic and nor Darboux first integrals at the neighborhood of the origin. Also, the stability and instability of the...
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 homog...
In this paper, we investigate the first integrals of the following system ̇ ̇ ̇ where and ,. This kind of system is a special case of three-dimensional polynomial cubic differential systems. Generally, several methods can be used to investigate the first integrals, but unfortunately, most of them are not enabled for finding first integrals. In this...
We provide necessary and sufficient conditions for both integrability and linearizability of a three dimensional vector field with quadratic nonlinearities. For our investigation we consider the case of (1:−2:1)–resonance at the origin and in general non of the axes planes is invariant. Hence, we deal with a nine parametric family of quadratic syst...
In this paper, we study the first integrals of Darboux type of the differential system x˙=y,y˙=−x+yz,z˙=−x−axy−bxz, which exhibits chaotic phenomena for suitable chosen values of the real parameters a and b. We show that the system has no polynomial, rational, or Darboux first integrals for any value of a and b. All the Darboux polynomials of the s...
We investigate the global stability, existence and nonexistence of limit cycles of general Sel’kov model using different approach such as qualitative analysis, Dulac’s criterion, Lyapunov function and Poincaré-Bendixson theorem. We also study the nonexistence of analytic first integrals of the general Sel’kov model.
We investigate the global stability, existence and nonexistence of limit cycles of general Sel’kov model using different approach such as qualitative analysis, Dulac’s criterion, Lyapunov function and Poincaré-Bendixson theorem. We also study the nonexistence of analytic first integrals of the general Sel'kov model.
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 th...
The main aim of this paper is to construct Bendixson-Dulac and Dulac-Cherkas functions to study the maximum number of limit cycles for several families of planar dynamical system. We also apply the results to Lienard and biochemistry reaction systems.
We prove that near the bifurcation point unstable limit cycle arises from the Lorenz system. In the analysis, we use the method of local bifurcation theory, especially the center manifold and the normal form theorem. A computer algebra system using Maple to derive all the formulas and verify the results presented in this paper.
W. H. Aziz and. Multiplicity of Periodic Solutions by Word Problems, J. Zanco. Pure & Applied Scince, Salahaddin University, V 16, No.1 (2004),73-82.
We study the polynomial, rational, analytic and Darboux first integrals. We also try to find Darboux polynomials for the system along with its exponential factors.