We examine the presence of general (nonlinear) time-independent Lie point symmetries in dynamical systems, and especially in bifurcation problems. A crucial result is that center manifolds are invariant under these symmetries: this fact, which may be also useful for explicitly nding the center manifold, implies that Lie point symmetries are inherited by the "reduced" bifurcation equation (a res...

We investigate Lie symmetries of general Yang-Mills equations. For this purpose, we first write down the second prolongation of the symmetry generating vector fields, and compute its action on the Yang-Mills equations. Determining equations are then obtained, and solved completely. Provided that Yang-Mills equations are locally solvable, this allows for a complete classification of their Lie sy...

The reduction of nonlinear ordinary differential equations by a combination of first integrals and Lie group symmetries is investigated. The retention, loss or even gain in symmetries in the integration of a nonlinear ordinary differential equation to a first integral are studied for several examples. The differential equations and first integrals are expressed in terms of the invariants of Lie...

A fast diffusion equation is investigated with symmetry point of view. The set of exact non-Lie solutions constructed by M.L. Gandarias [Phys. Lett. A, 2001,V.286, 153-160] are supplemented with the similar ones. The non-classical symmetries of the corresponding potential equation are completely classified with respect to the Lie symmetry group of this equation. As a result, we obtain new wide ...

We prove that any potential symmetry of a system of evolution equations reduces to a Lie symmetry through a nonlocal transformation of variables. Based on this fact is our method of group classification of potential symmetries of systems of evolution equations having non-trivial Lie symmetry. Next, we modify the above method to generate more general nonlocal symmetries, which yields a purely al...

Given a planar vector field U which generates the Lie symmetry of some other vector field X , we prove a new criterion to control the stability of the periodic orbits of U . The problem is linked to a classical problem proposed by A.T. Winfree in the seventies about the existence of isochrons of limit cycles (the question suggested by the study of biological clocks), already answered by Guckenh...

Lie symmetry method has been and still is successfully applied in different problems of physics for about a hundred years, but its application in epidemiology has been rare perhaps because the ordinary differential equations studied in this field are generally of first-order in contrast with those in physics which are usually of second-order. Here we exemplify the use of Lie symmetry method in ...

We show that biholomorphic automorphisms of a real analytic hypersurface in I C can be considered as (pointwise) Lie symmetries of a holomorphic completely overdetermined involutive second order PDE system defining its Segre family. Using the classical S.Lie method we obtain a complete description of infinitesimal symmetries of such a system and give a new proof of some well known results of CR...

The discrete heat equation is worked out in order to illustrate the search of symmetries of difference equations. It is paid an special attention to the Lie structure of these symmetries, as well as to their dependence on the derivative discretization. The case of q–symmetries for discrete equations in a q–lattice is also considered.

The application of computer algebra for determining Lie and Lie–Bäcklund (LB) symmetries of differential equations is considered. Algorithms for calculating the symmetries are developed and implemented on the basis of computer algebra systems REDUCE, AMP and FORMAC. The most effective and advanced program is written in FORMAC. It finds LB symmetries completely automatically. In many cases the p...