Nonlinear Dynamics

, Volume 48, Issue 4, pp 417–422

Linearization criteria for a system of second-order quadratically semi-linear ordinary differential equations

Original Article

DOI: 10.1007/s11071-006-9095-z

Cite this article as:
Mahomed, F.M. & Qadir, A. Nonlinear Dyn (2007) 48: 417. doi:10.1007/s11071-006-9095-z


Conditions are derived for the linearizability via invertible maps of a system of n second-order quadratically semi-linear differential equations that have no lower degree lower order terms in them, i.e., for the symmetry Lie algebra of the system to be sl(n + 2, ℝ). These conditions are stated in terms of the coefficients of the equations and hence provide simple invariant criteria for such systems to admit the maximal symmetry algebra. We provide the explicit procedure for the construction of the linearizing transformation. In the simplest case of a system of two second-order quadratically semi-linear equations without the linear terms in the derivatives, we also provide the construction of the linearizing point transformation using complex variables. Examples are given to illustrate our approach for two- and three-dimensional systems.


Lie symmetry algebraLinearizationSystem of second-order ordinary differential equations

Copyright information

© Springer Science + Business Media B.V. 2007

Authors and Affiliations

  1. 1.Centre for Differential Equations, Continuum Mechanics and Applications, School of Computational and Applied MathematicsUniversity of the WitwatersrandJohannesbergSouth Africa
  2. 2.Centre for Advanced Mathematics and PhysicsNational University of Sciences and TechnologyRawalpindiPakistan
  3. 3.Department of Mathematical SciencesKing Fahd University of Petroleum and MineralsDhahranSaudi Arabia