Seminar, Dmitry Sinelschikov

12:15h. Salón de Grados ETSII

Dynamical systems: integrability, bifurcations and applications.

In the first part of the talk we consider a family of cubic, with respect to the first derivative, second-order differential equations. Certain members of this family appear frequently in various applications in mechanics, physics and biology. In addition, this family of equations is a projection of equations for the geodesics of a two-dimensional Riemannian manifold and, therefore, these equations are often called projective. We propose an approach for constructing integrability criteria for the projective equations, which is based on the solution of equivalence problems for this family of equations and its integrable subcases, including linear and Painleve-type equations. We demonstrate how these criteria can be used to construct integrable dynamical systems and (super)integrable Riemannian metrics. In addition, we propose an approach for finding transcendental invariant curves for two-dimensional dynamical systems and demonstrate its applications for establishing non-Liouvillian integrability. This part is partially done in collaboration with Jaume Giné.

The second part of the talk is devoted to the nonlinear dynamics in the models of encapsulated micrometer-size gas bubbles in a liquid. Such bubbles are currently used as ultrasound contrast agents and there are other potential applications such as targeted drug delivery and sonoporation. Different types of bubbles oscillations can be either beneficial or undesirable, depending on application. We consider models of several interacting gas bubbles and classify the dependence of their dynamics on the control parameters. We propose several scenarios for the onset of hyperchaotic oscillations. These scenarios appear to be typical routes to hyperchaos in other multidimensional dynamical systems. We also discuss other types of dynamics in the model, multistability and synchronisation. This part is done a collaboration with Ivan Garashuck and Alexey Kazakov.