Integrability and linearizability of a family of three-dimensional quadratic systems

Abstract

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 systems. Some techniques like Darboux method are used to prove the sufficiency of the obtained conditions. For a particular three parametric subfamily we provide conditions to guarantee the non existence of a polynomial first integral.

Integrability Analysis of the Smallest 3D Biochemical Reaction Model

“…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). Thus to prove the main results, we use Darboux theorem of integrability in order to find invariant algebraic surfaces, exponential factors and characterize its local analytic first integrals of system (1).…”

“…According to our knowledge, the integrability and nonintegrabilty problems for the system have not studied. This study focus for studying some types of first integrals of system (1) in the analysis we use Darboux Theorem and some preliminary results. All mathematical analysis, particularly, solving partial differential equations are verified with the help of Maple.…”

“…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).…”

“…We characterize here integrability and non-integrability of system (1). Thus to prove the main results, we use Darboux theorem of integrability in order to find invariant algebraic surfaces, exponential factors and characterize its local analytic first integrals of system (1). By χ we denote the corresponding vector field of system (1)…”

Some Applications of Affine in Velocities Lagrangians in Two-Dimensional Systems

“…Section 4 is devoted to analyse the twodimensional inverse problem for systems of first-order differential equations. As the main result is to be expressed in terms of a Jacobi multiplier of the given system, we first review in Subsection 4.1 the theory of Jacobi multipliers in geometrical terms [2,3,10,18,19,21,28,33,40,41,42,43,44,45,46,50,61,62,63]. The main result asserts that in order to have a Lagrangian description for a given system of two firstorder differential equations it is necessary and sufficient to establish the existence of a Jacobi multiplier for the system.…”