## Lie systems: theory and applications

#### Dr Cristina Sardón Muñoz (Instituto de Ciencias Matemáticas, Madrid)

Frank Adams Room 1,

Lie systems are nonautonomous systems of first-order ordinary differential equations that admit a (generally nonlinear) *superposition rule*, i.e., the general solution is described in terms of a generic family of particular solutions and a set of constants related to initial conditions. Geometrically, these systems are described by the so-called *Vessiot-Guldberg Lie algebra*, which is a finite dimensional Lie algebra of vector fields.

But there exists a restricted number of Lie systems in the literature, since most differential equations cannot be described as a curve in a finite-dimensional Lie algebra of vector fields. Nonetheless, the few existing ones are very important in the study of relevant physical models, Mathematics, Biology or Control theory.

Lie systems may also admit Vessiot-Guldberg Lie algebras of Hamiltonian vector fields with respect to a Poisson structure, these are known as Lie-Hamilton systems. We will pay particular attention to this class of Lie systems. We give their classification and research their properties on the plane.

References

- [1] J.F. Cariñena, J. de Lucas, Lie systems: theory, generalisations, and applications, Dissertationes Math. (Rozprawy Mat.)
**479**, 1-162 (2011). - [2] C. Sardon, Lie systems, Lie symmetries and reciprocal transformations, http://arxiv.org/abs/1508.00726 (2015).