Liouvillian and Analytic First Integrals for the Brusselator System

Abstract

We characterize the Liouvillian and analytic first integrals for the polynomial differential systems of the form x = a − (b + 1)x + x 2 y, y = bx − x 2 y, with a, b ∈ R, called the Brusselator differential systems.

On the Liouville integrability of Edelstein’s reaction system in R3

“…From equation (8) with i = 3 we obtain three different situations in order to obtain a polynomial expression for g 3 :…”

“…Edelstein's system has one such first integral, namely H = y + z. With some exceptions the non-linear first integrals have rarely been considered, see [1,10,9] for some general considerations and [8,6,7,12] for specific examples. Non-linear first integral are often useful for studying the dynamics of the system, see for example [12,9].…”

Conservation Laws in Biochemical Reaction Networks

“…In general, it is difficult to prove the existence or non-existence of Darboux 35 first integrals, which often depends on the reaction rate constants in complicated ways [11,12,17,10]. We hope that this paper would inspire further work to characterize the existence and form of non-linear first integrals for reaction networks.…”

“…Although the use of linear conservation laws in reaction network theory is common, the non-linear first integrals have rarely been considered, with the exception of some special cases [15,1,17,11,12,20]. Quadratic first integrals have been characterized in [20].…”

“…which implies the constraint (17), if λ 1 = 0. Similarly, the equation λ 2 k 1 − λ n k n = 0 also implies the constraint (17), if λ 2 = 0.…”