Symmetries of second-order systems of ODEs and integrability

Canonical Forms and Their Integrability for Systems of Three 2nd-Order ODEs

“…For the construction of canonical forms the approach developed in [8,9,13] is employed. By utilizing realizations of underlying 3-dimensional Lie algebras admitted by system of three 2nd-order ordinary differential equations we find differential invariants associated with them.…”

“…There are various methods and approaches for finding differential invariants but we follow the peculiar approach developed by Ayub et al in [8,9,13]. In this approach invariants are interrelated via basic invariant.…”

“…Consider the second realization A 2 3,5 ≃ ⟨ , , + + ⟩ from [8]. We find the second prolongation of generators and then adopting the procedure of construction of differential invariants described in [8,9,13], we obtain the following differential invariants:…”

“…In classification problem related to system of differential equations, invariant interrelation plays a key role in different types of analysis, namely, integrability, Linearization, and singularity as well as some other aspects especially related to system of three 2nd-order ODEs (see [8,9,13]). By careful observations here we have obtained some relations between invariants.…”

“…After Lie, several researchers have been involved in analysis of different aspects of this work; see [1,6,[8][9][10][11][12]. Although the algebraic approach is more involved, it generates new cases and more insight to mathematical properties whose base is algebra (see [8,9,13,14]). Thus the classification of differential equations via algebraic approach is more efficient than other classical approaches.…”