Quantitative recurrence statistics and convergence to an extreme value distribution for non-uniformly hyperbolic dynamical systems

For non-uniformly hyperbolic dynamical systems we consider the time series of maxima along typical orbits. Using ideas based upon quantitative recurrence time statistics we prove convergence of the maxima (under suitable normalization) to an extreme value distribution, and obtain estimates on the rate of convergence. We show that our results are applicable to a range of examples, and include new results for Lorenz maps, certain partially hyperbolic systems, and non-uniformly expanding systems with sub-exponential decay of correlations. For applications where analytic results are not readily available we show how to estimate the rate of convergence to an extreme value distribution based upon numerical information of the quantitative recurrence statistics. We envisage that such information will lead to more efficient statistical parameter estimation schemes based upon the block-maxima method.Comment: This article is a revision of the previous article titled: "On the convergence to an extreme value distribution for non-uniformly hyperbolic dynamical systems." Relative to this older version, the revised article includes new and up to date results and developments (based upon recent advances in the field

Symmetries in the Lorenz-96 model

The Lorenz-96 model is widely used as a test model for various applications, such as data assimilation methods. This symmetric model has the forcing $F\in\mathbb{R}$ and the dimension $n\in\mathbb{N}$ as parameters and is $\mathbb{Z}_n$ equivariant. In this paper, we unravel its dynamics for $F<0$ using equivariant bifurcation theory. Symmetry gives rise to invariant subspaces, that play an important role in this model. We exploit them in order to generalise results from a low dimension to all multiples of that dimension. We discuss symmetry for periodic orbits as well. Our analysis leads to proofs of the existence of pitchfork bifurcations for $F<0$ in specific dimensions $n$: In all even dimensions, the equilibrium $(F,\ldots,F)$ exhibits a supercritical pitchfork bifurcation. In dimensions $n=4k$, $k\in\mathbb{N}$, a second supercritical pitchfork bifurcation occurs simultaneously for both equilibria originating from the previous one. Furthermore, numerical observations reveal that in dimension $n=2^qp$, where $q\in\mathbb{N}\cup\{0\}$ and $p$ is odd, there is a finite cascade of exactly $q$ subsequent pitchfork bifurcations, whose bifurcation values are independent of $n$. This structure is discussed and interpreted in light of the symmetries of the model.Comment: 31 pages, 9 figures and 3 table

On Max-Semistable Laws and Extremes for Dynamical Systems

Suppose [Formula: see text] is a measure preserving dynamical system and [Formula: see text] a measurable observable. Let [Formula: see text] denote the time series of observations on the system, and consider the maxima process [Formula: see text]. Under linear scaling of [Formula: see text] , its asymptotic statistics are usually captured by a three-parameter generalised extreme value distribution. This assumes certain regularity conditions on the measure density and the observable. We explore an alternative parametric distribution that can be used to model the extreme behaviour when the observables (or measure density) lack certain regular variation assumptions. The relevant distribution we study arises naturally as the limit for max-semistable processes. For piecewise uniformly expanding dynamical systems, we show that a max-semistable limit holds for the (linear) scaled maxima process

Wave propagation in the Lorenz-96 model

In this paper we study the spatiotemporal properties of waves in the Lorenz-96 model and their dependence on the dimension parameter <i>n</i> and the forcing parameter <i>F</i>. For <i>F</i> &gt; 0 the first bifurcation is either a supercritical Hopf or a double-Hopf bifurcation and the periodic attractor born at these bifurcations represents a traveling wave. Its spatial wave number increases linearly with <i>n</i>, but its period tends to a finite limit as <i>n</i> → ∞. For <i>F</i> &lt; 0 and odd <i>n</i>, the first bifurcation is again a supercritical Hopf bifurcation, but in this case the period of the traveling wave also grows linearly with <i>n</i>. For <i>F</i> &lt; 0 and even <i>n</i>, however, a Hopf bifurcation is preceded by either one or two pitchfork bifurcations, where the number of the latter bifurcations depends on whether <i>n</i> has remainder 2 or 0 upon division by 4. This bifurcation sequence leads to stationary waves and their spatiotemporal properties also depend on the remainder after dividing <i>n</i> by 4. Finally, we explain how the double-Hopf bifurcation can generate two or more stable waves with different spatiotemporal properties that coexist for the same parameter values <i>n</i> and <i>F</i>