Schedule

Abstract: In this talk, I analyse a FitzHugh-Nagumo (1,1)-fast-slow system and characterise its dynamics with respect to the parameters of the reduced equation. I perform a bifurcation analysis and establish the presence of canard orbits. Afterwards, I construct a (2,2)-fast–slow system by coupling two FitzHugh-Nagumo systems within the slow equations. Through numerical simulations, I observe the emergence of trajectories with both low and high amplitude periodic orbits, referred to as mixed-mode oscillations. Additionally, I identify trajectories that consist of both low and high amplitude orbits, but do not alternate periodically, leading to what is known as chaotic mixed-mode oscillations.

Abstract: In this talk I will discuss some recent results regarding the existence of invariant curves for piecewise isometries as well as renormalization of such transformations.

Abstract: We obtain a weak version of Lusin's theorem in the Sobolev-(1,p) (0<p<1) closure of volume preserving Lipschitz homeomorphisms on closed and connected dimensional manifolds, d≥2. With this result at hand we will be able to prove that such conservative homeomorphisms are ergodic from a generic viewpoint. This establishes a version of Oxtoby and Ulam theorem for this Sobolev class. We also prove that within the Sobolev-(1,p) class (now considering 0<p< d-1) the topological transitive maps are generic. Join work with Assis Azevedo, Mário Bessa (CMUP) and Maria Joana Torres.

Abstract: In this talk, we are concerned to understand the velocity tracking control problem for a class of stochastic non-Newtonian fluids. More precisely, we consider stochastic third-grade fluid equations perturbed by infinite-dimensional additive white noise defined on the two-dimensional torus. The control acts as a distributed random external force. Making use of the infinite-dimensional Ornstein-Uhlenbeck process, we convert the stochastic system into an equivalent pathwise deterministic system which assists us to prove the well-posedness of the original stochastic system. Then, we establish the existence and uniqueness of solutions to the corresponding linearized state equation and adjoint equation. Furthermore, we derive an appropriate stability result for the state equation and verify that the Gˆateaux derivative of the control-to-state mapping coincides with the solution of the linearized state equation. Finally, we formulate the first-order optimality conditions and prove the existence of an optimal solution.