A geometrical method towards first integrals for dynamical systems

We develop a method, based on Darboux’ and Liouville’s works, to find first integrals and/or invariant manifolds for a physically relevant class of dynamical systems, without making any assumption on these elements’ form. We apply it to three dynamical systems: Lotka–Volterra, Lorenz and Rikitake. I. HISTORICAL OVERVIEW. In, Roger Liouville and A. Tresse developed a method for deciding whether two differential equations of the form dy dx + a1(x, y) ( dy dx )3 + 3 a2(x, y) ( dy dx )2 + 3 a3(x, y) dy dx + a4(x, y) = 0 (1) where the ai are arbitrary functions of the real or complex variables x and y, are geometrically equivalent, i.e. can be transformed into each other by the most general dependent and independent variable change x′ = φ(x, y), y′ = ψ(x, y) (2) PACS numbers: 02.30, 02.40, 02.90 1 FIRST INTEGRALS FROM GEOMETRICAL EQUIVALENCE 2 This method was based on the construction of a “relative invariant” function called ν5 of the ai and of their derivatives, such that in any transformation (2) it becomes ν ′ 5 = J(x, y) ν5 where J(x, y) is the Jacobian of the transformation. In the general case, two equations such that their ν5 are non-zero and proportional to each other are indeed equivalent. If ν5 = 0 for both, however, one cannot conclude at first, and other invariants, involving higher derivatives of the ai, must be calculated in order to decide. As an application, Liouville proposed the effective reduction of Equation (1) into its simplest canonical form, which in most cases leads to an explicit integration. Here we will adopt another point of view. We have derived from these theories a method for finding out first integrals for a wide and physically important class of dynamical systems without having to make any ansatz on their functional form. In the rest of this section, we shall recall some mathematical results of Darboux, Liouville and Tresse. Then we explain our method in Section II, and apply it in Section III to three well-known dynamical systems. Finally, Section IV discusses our results summarised in Table I. A. Essentials of Liouville theory. Consider a differential equation like (1). Liouville defined the following functions — which are seen as functions of (x, y), forgetting the supposed relation between those variables. L2 = ∂ ∂x (