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Introduction to the Modern Theory of Dynamical Systems

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An Introduction To The Theory Of Point Processes

The Bootstrap and Edgeworth Expansion

An Introduction to Probability Theory and its Applications, Volume I

We review work on extreme events, their causes and consequences, by a group of European and American researchers involved in a three-year project on these topics The review covers theoretical aspects of time series analysis and of extreme value theory, as well as of the deterministic modeling of extreme events, via continuous and discrete dynamic models The applications include climatic, seismic and socio-economic events, along with their prediction Two important results refer to (i) the complementarity of spectral analysis of a time series in terms of the continuous and the discrete part of its power spectrum; and (ii) the need for coupled modeling of natural and socio-economic systems Both these results have implications for the study and prediction of natural hazards and their human impacts

https://npg.copernicus.org/articles/18/295/2011/npg-18-295-2011.pdf

Extreme events: dynamics, statistics and prediction

The object of this paper is twofold. From one side we study the dichotomy, in terms of the Extremal Index of the possible Extreme Value Laws, when the rare events are centred around periodic or non periodic points. Then we build a general theory of Extreme Value Laws for randomly perturbed dynamical systems. We also address, in both situations, the convergence of Rare Events Point Processes. Decay of correlations against $L^1$ observables will play a central role in our investigations.

/pdf/laws-of-rare-events-for-deterministic-and-random-dynamical-1pfxum7hjk.pdf

Laws of rare events for deterministic and random dynamical systems

The object of this paper is twofold. From one side we study the dichotomy, in terms of the Extremal Index of the possible Extreme Value Laws, when the rare events are centred around periodic or non periodic points. Then we build a general theory of Extreme Value Laws for randomly perturbed dynamical systems. We also address, in both situations, the convergence of Rare Events Point Processes. Decay of correlations against L 1 observables will play a central role in our investigations.

/pdf/laws-of-rare-events-for-deterministic-and-random-dynamical-45bvdbp099.pdf

We present a review of recent results regarding the existence of extreme value laws for stochastic processes arising from dynamical systems. We gather all the conditions on the dependence structure of stationary stochastic processes in order to obtain both the distributional limit for partial maxima and the convergence of point processes of rare events. We also discuss the existence of clustering which can be detected by an extremal index less than 1 and relate it with the occurrence of rare events around periodic points. We also present the connection between the existence of extreme value laws for certain dynamically defined stationary stochastic processes and the existence of hitting times statistics (or return times statistics). Finally, we make a complete description of the extremal behaviour of expanding and piecewise expanding systems by giving a dichotomy regarding the types of extreme value laws that apply. Namely, we show that around periodic points we have an extremal index less than 1 and at v...

/pdf/extremal-behaviour-of-chaotic-dynamics-2i7xk3zwqn.pdf

Extremal behaviour of chaotic dynamics

We study the distribution of maxima (extreme value statistics) for sequences of observables computed along orbits generated by random transformations. The underlying, deterministic, dynamical system can be regular or chaotic. In the former case, we show that, by perturbing rational or irrational rotations with additive noise, an extreme value law appears, regardless of the intensity of the noise, while unperturbed rotations do not admit such limiting distributions. In the case of deterministic chaotic dynamics, we will consider observables specially designed to study the recurrence properties in the neighbourhood of periodic points. Hence, the exponential limiting law for the distribution of maxima is modified by the presence of the extremal index, a positive parameter not larger than one, whose inverse gives the average size of the clusters of extreme events. The theory predicts that such a parameter is unitary when the system is perturbed randomly. We perform sophisticated numerical tests to assess how strong the impact of noise level is when finite time series are considered. We find agreement with the asymptotic theoretical results but also non-trivial behaviour in the finite range. In particular, our results suggest that, in many applications where finite datasets can be produced or analysed, one must be careful in assuming that the smoothing nature of noise prevails over the underlying deterministic dynamics.

/pdf/extreme-value-statistics-for-dynamical-systems-with-noise-5aotdgq7gq.pdf

Extreme value statistics for dynamical systems with noise

We consider families of transitive multimodal interval maps with polynomial growth of the derivative along the critical orbits. For these maps Bruin and Todd have shown the existence and uniqueness of equilibrium states for the potential t : x −t log |Df(x)|, for t close to 1. We show that these equilibrium states vary continuously in the weak* topology within such families. Moreover, in the case t = 1, when the equilibrium states are absolutely continuous with respect to Lebesgue, we show that the densities vary continuously within these families.

/pdf/the-statistical-stability-of-equilibrium-states-for-interval-5d6f99yb0n.pdf

Jorge Milhazes Freitas

Papers

Laws of rare events for deterministic and random dynamical systems

Laws of rare events for deterministic and random dynamical systems

Extremal behaviour of chaotic dynamics

Extreme value statistics for dynamical systems with noise

The statistical stability of equilibrium states for interval maps