On the complete integrability and linearization of nonlinear ordinary differential equations. III. Coupled first-order equations

Abstract

Continuing our study on the complete integrability of nonlinear ordinary differential equations (ODEs), in this paper we consider the integrability of a system of coupled first-order nonlinear ODEs of both autonomous and non-autonomous types. For this purpose, we modify the original Prelle-Singer (PS) procedure so as to apply it to both autonomous and non-autonomous systems of coupled first-order ODEs. We briefly explain the method of finding integrals of motion (time-independent as well as timedependent integrals) for two and three coupled first-order ODEs by extending the PS method. From this we try to answer some of the open questions in the original PS method. We also identify integrable cases for the two-dimensional Lotka-Volterra system and three-dimensional Rössler system as well as other examples including nonautonomous systems in a straightforward way using this procedure. Finally, we develop a linearization procedure for coupled first-order ODEs.

On the complete integrability and linearization of nonlinear ordinary differential equations. IV. Coupled second-order equations

“…By introducing four integrating factors, one can deduce the relevant determining equations by following the procedure given by us in the earlier paper, part III (Chandrasekar et al 2009), to the above system of first-order ODEs. However, after examining several examples, we find that it is more advantageous to solve the system (2.1) in the second-order form itself rather than introducing more variables.…”

On integrals, Hamiltonian and metriplectic formulations of polynomial systems in 3D

“…Any other Casimir of J has to be linearly dependent to the potential function H 1 since the kernel is one dimensional. Substitution of the general solution (11) of J into the Hamilton's equations (8) results witḣ…”

“…While writing a non-autonomous system in form of the Hamilton's equations (8), inevitably, one of the two, Poisson vector or Hamiltonian function, must depend explicitly on the time variable t. The calculation…”

Interplay of symmetries, null forms, Darboux polynomials, integrating factors and Jacobi multipliers in integrable second-order differential equations

“…As we have pointed out in the introduction, once we know the quantities S and R, one can find all the other quantities using the relations given in Eqs. (10) and (12) respectively. For example to obtain the expression for X from S one has to integrate the first order partial differential equation (10).…”

“…(10) and (12) respectively. For example to obtain the expression for X from S one has to integrate the first order partial differential equation (10). As far as the present example is concerned we make an ansatz for X of the form…”