The Completely Integrable Differential Systems are Essentially Linear Differential Systems

Abstract

Abstract. Letẋ = f (x) be a C k autonomous differential system with k ∈ N ∪ {∞, ω} defined in an open subset Ω of R n . Assume that the systemẋ = f (x) is C r completely integrable, i.e. there exist n − 1 functionally independent first integrals of class C r with 2 ≤ r ≤ k. If the divergence of systemẋ = f (x) is non-identically zero, then any Jacobian multiplier is functionally independent of the n − 1 first integrals. Moreover the systemẋ = f (x) is C r−1 orbitally equivalent to the linear differential systemẏ = y in a full Lebesgue measure subset of Ω. For Darboux and polynomial integrable polynomial differential systems we characterize their type of Jacobian multipliers.

Meromorphic Non-Integrability of Several 3D Dynamical Systems

“…Moreover, the topological structure of a completely integrable system is also simple. Indeed, if system (1) admits n − 1 functionally independent first integrals of class C r with r > 2, then it is C r−1 orbitally equivalent to a linear differential system, for more details see [32]. If system (1) admits 0 < m < n − 1 first integrals, i.e., "partially integrable", then we can replace the considered system with an n − m-dimension reduced one.…”

Poisson and integrable systems through the Nambu bracket and its Jacobi multiplier

“…This process generates automatically, and without extra computation, the Jacobian multiplier (a generalisation of an integrating factor) associated to the original system of differential equation (1). Therefore, we partially obtain in Theorem 1.1, part (ii), an alternative proof of a recent theorem in [13], that provides a linearisation of completely integrable systems. Note that in the case of considering several sets of functionally independent integrals, the theorem leaves open which one might be more convenient in each application, since the corresponding Jacobian multiplier defines the reparametrisation given in (4).…”

“…A Jacobian multiplier is an integrating factor for the two dimensional case of system (1). In addition, it plays a central role in the linearisation of completely integrable systems proposed in [13]. Due to the connection with this paper we maintain the notation given there for the benefit of the reader.…”

“…Conditions to guarantee that an ordinary differential equation may be written exactly (without reparametrisation) in the Nambu form are given in [17]. We also provide the Jacobi multiplier in case the original system has non zero divergence, which in addition is an alternative for the claim (a) in Theorem 2 of [13]. For an application example see Section 3.2.…”

The First Integrals of a Second Order Ordinary Differential Equation and Application

“…4) where a1(x), a2(x), a3(x) and a4(x) are functions of x to be determined. Inserting (3.3) and (3.4) into the third equation of (3.2), we have a polynomial of y with degree 2 which is zero if and only if each variable coefficient is set to zero…”