Dynamical Systems

Generalized hyperbolicity and the shadowing property

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In this talk we discuss hyperbolicity, expansivity, the shadowing property and structural stability for operators on Banach spaces. In particular, we discuss relations between these notions and some open problems. All concepts will be gently introduced and the talk will be accessible to graduate students and non-specialists.

Strange attractors in a seasonally forced epidemic SIR model

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We analyse the forced SIR model to investigate the influence of seasonality on the disease dynamics. Seasonality is often introduced in epidemic models via a transmission rate of sinusoidal type. In this talk, we present two results related to this model:

Holder continuity for Lyapunov exponents of random linear cocycles

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In 2017, Baraviera and Duarte extended a classical theorem from Le Page. They obtained an elegant proof for the local Holder continuity of the Lyapunov exponents of random linear cocycles defined over the Bernoulli Shift under generic hypothesis. The authors proved local Holder continuity with respect to the cocycle, with a fixed measure. The main tools are Furstenberg’s Formula and regularity properties from the stationary measure.

Uniform convergence rate for Birkhoff means of certain uniquely ergodic toral maps

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In this talk, we will present an estimate on the uniform convergence rate of the Birkhoff average of a continuous observable over torus translations. This convergence rate depends explicitly on the modulus of continuity of the observable and on the arithmetic properties of the frequency defining the transformation. Furthermore, we obtained similar results for affine skew product toral transformations and, in the case of one dimensional torus translation, these estimates are nearly optimal. This is a joint work with Xiao- Chuan Liu and Silvius Klein.

On Ergodic Theory of Impulsive Semiflows

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Impulsive Dynamical Systems (IDS) can be seen as suitable mathematical models of real world phenomena that display abrupt changes in their behavior.

More precisely, an IDS is described by three objects: a continuous semiflow on a space X; a set D contained in X where the flow undergoes sudden perturbations; and an impulsive function from D to X, which determines the change in the trajectory each time it collides with the impulsive set D.