Interplay of symmetries and other integrability quantifiers in finite-dimensional integrable nonlinear dynamical systems

Abstract

In this work, we establish a connection between the extended Prelle-Singer procedure and other widely used analytical methods to identify integrable systems in the case of nth-order nonlinear ordinary differential equations (ODEs). By synthesizing these methods, we bring out the interlink between Lie point symmetries, contact symmetries, λ-symmetries, adjoint symmetries, null forms, Darboux polynomials, integrating factors, the Jacobi last multiplier and generalized λ-symmetries corresponding to the nth-order ODEs. We also prove these interlinks with suitable examples. By exploiting these interconnections, the characteristic quantities associated with different methods can be deduced without solving the associated determining equations.

The search for Invariants for 3D Systems of 1ODEs - a new Method and Integrability Analysis

“…The main theoretical improvement is embodied by the new method introduced in section (3), where we use the S-function on the core of the new approach. The S-function was first introduced on [3] and later on used on many different ways by us and other people [8,9,10,11,12]. In [7], we have produced a new approach to deal with reducing 2odes also using the S-function as the corner stone.…”

Some Applications of Affine in Velocities Lagrangians in Two-Dimensional Systems

“…[10,61,62]) and alternative Lagrangians [13]. For more information on the rôle of Jacobi multipliers in integrability and with other approaches to integrability see e.g [55,56,57,58]. In Section 5 we exhibit explicit Lagrangians for some important examples of differential equations, as mechanical systems, and interesting results on biological examples [39], as a generalisation of the Lotka-Volterra model [29,38,74] and a host-parasite model [73], are derived from this new perspective and their Hamiltonian functions and a set of canonical variables are also given.…”

Solving 1ODEs with functions

“…In [19], we have picked it up again and we have developed an new approach to deal with 3D-systems of 1ODEs, this time basing our method on our (so-called) S-function (introduced in [13] and used by us and other people [20,21,22,23,24,25,26,27]), that improves greatly the efficiency of the method introduced in [2], for many cases.…”