On Calculation of the First Integrals for Three-dimensional ODE Systems

Abstract

The search for solutions of nonlinear stationary systems of ordinary differential equations (ODE) is sometimes very complicated. It is not always possible to obtain a general solution in an analytical form. As a consequence, a qualitative theory of nonlinear dynamical systems has been developed. Its methods allow us to investigate the properties of solutions without finding a general solution. Numerical methods of investigation are also widely used.In the case when it is impossible to find an analytically general solution of the ODE system, sometimes, nevertheless, it is possible to find its first integral. There is a number of known results that make it possible to obtain the first integral for certain special cases.The article deals with the method for obtaining the first integrals of ODE systems of the third order, based on the fact of integrability of the involutive distribution.The method proposed in the paper allows us to obtain the first integral of a nonlinear ODE system of the third order in the case when a vector field, which generates an involutive distribution of dimension 2 together with the vector field of the right-hand side of a given ODE system, is known. In this case, the solution of a certain sequence of Cauchy problems allows us to construct a level surface of the function of the first integral containing the given point of the state space of the system. Using the method of least squares, in a number of cases it is possible to obtain an analytic expression for the first integral.The article gives examples of the method application to two ODE systems, namely to a simple nonlinear third-order system and to the Lorentz system with special parameter values. The article shows how the first integrals can be obtained analytically using the method developed for the two systems mentioned above.