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

Abstract

Continuing our study on the complete integrability of nonlinear ordinary differential equations, in this paper we consider the integrability of a system of coupled first order nonlinear ordinary differential equations (ODEs) of both autonomous and non-autonomous types. For this purpose, we modify the original Prelle-Singer 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 time dependent integrals) for two and three coupled first order ODEs by extending the Prelle-Singer(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 R$\ddot{o}$ssler system as well as other examples including non-autonomous systems in a straightforward way using this procedure. Finally, we develop a linearization procedure for coupled first order ODEs.

added a new section (section 8) and minor revision. Proc. R. Soc. London A (accepted)

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Cited by in corpus (11)

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• On the complete integrability and linearization of nonlinear ordinary differential equations - Part V: Linearization of coupled second order equations
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• Solutions by quadratures of complex Bernoulli differential equations and their quantum deformation
• Computer Assisted 'Proof' of the Global Existence of Periodic Orbits in the Rössler System
• On the symmetry solutions of two-dimensional systems not solvable by standard symmetry analysis
• A systematic method of finding linearizing transformations for nonlinear ordinary differential equations: I. Scalar case
• A Semi-Algorithmic Search for Lie Symmetries
• A systematic method of finding linearizing transformations for nonlinear ordinary differential equations: II. Extension to coupled ODEs