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Search Results: 1 - 10 of 708 matches for " Mikko Stenlund "
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Asymptotic Expansion of the Homoclinic Splitting Matrix for the Rapidly, Quasiperiodically, Forced Pendulum
Mikko Stenlund
Physics , 2007, DOI: 10.1063/1.3398483
Abstract: We study a Hamiltonian describing a pendulum coupled with several anisochronous oscillators, devising an asymptotic expansion for the splitting (matrix) associated with a homoclinic point. This expansion consists of contributions that are manifestly exponentially small in the limit of vanishing hyperbolicity, by a shift-of-contour argument. Hence, we infer a similar upper bound on the splitting itself.
A strong pair correlation bound implies the CLT for Sinai Billiards
Mikko Stenlund
Mathematics , 2009, DOI: 10.1007/s10955-010-9987-7
Abstract: For Dynamical Systems, a strong bound on multiple correlations implies the Central Limit Theorem (CLT) [ChMa]. In Chernov's paper [Ch2], such a bound is derived for dynamically Holder continuous observables of dispersing Billiards. Here we weaken the regularity assumption and subsequently show that the bound on multiple correlations follows directly from the bound on pair correlations. Thus, a strong bound on pair correlations alone implies the CLT, for a wider class of observables. The result is extended to Anosov diffeomorphisms in any dimension.
A vector-valued almost sure invariance principle for Sinai billiards with random scatterers
Mikko Stenlund
Mathematics , 2012,
Abstract: Understanding the statistical properties of the aperiodic planar Lorentz gas stands as a grand challenge in the theory of dynamical systems. Here we study a greatly simplified but related model, proposed by Arvind Ayyer and popularized by Joel Lebowitz, in which a scatterer configuration on the torus is randomly updated between collisions. Taking advantage of recent progress in the theory of time-dependent billiards on the one hand and in probability theory on the other, we prove a vector-valued almost sure invariance principle for the model. Notably, the configuration sequence can be weakly dependent and non-stationary. We provide an expression for the covariance matrix, which in the non-stationary case differs from the traditional one. We also obtain a new invariance principle for Sinai billiards (the case of fixed scatterers) with time-dependent observables, and improve the accuracy and generality of existing results.
Construction of Whiskers for the Quasiperiodically Forced Pendulum
Mikko Stenlund
Mathematics , 2007, DOI: 10.1142/S0129055X07003127
Abstract: We study a Hamiltonian describing a pendulum coupled with several anisochronous oscillators, giving a simple construction of unstable KAM tori and their stable and unstable manifolds for analytic perturbations. We extend analytically the solutions of the equations of motion, order by order in the perturbation parameter, to a uniform neighbourhood of the time axis.
An almost sure ergodic theorem for quasistatic dynamical systems
Mikko Stenlund
Mathematics , 2015,
Abstract: We prove almost sure ergodic theorems for a class of systems called quasistatic dynamical systems. These results are needed, because the usual theorem due to Birkhoff does not apply in the absence of invariant measures. We also introduce the concept of a physical family of measures for a quasistatic dynamical system. These objects manifest themselves, for instance, in numerical experiments. We then verify the conditions of the theorems and identify physical families of measures for two concrete models, quasistatic expanding systems and quasistatic dispersing billiards.
Non-stationary compositions of Anosov diffeomorphisms
Mikko Stenlund
Mathematics , 2011, DOI: 10.1088/0951-7715/24/10/016
Abstract: Motivated by non-equilibrium phenomena in nature, we study dynamical systems whose time-evolution is determined by non-stationary compositions of chaotic maps. The constituent maps are topologically transitive Anosov diffeomorphisms on a 2-dimensional compact Riemannian manifold, which are allowed to change with time - slowly, but in a rather arbitrary fashion. In particular, such systems admit no invariant measure. By constructing a coupling, we prove that any two sufficiently regular distributions of the initial state converge exponentially with time. Thus, a system of the kind loses memory of its statistical history rapidly.
A local limit theorem for random walks in balanced environments
Mikko Stenlund
Mathematics , 2012,
Abstract: Central limit theorems for random walks in quenched random environments have attracted plenty of attention in the past years. More recently still, finer local limit theorems -- yielding a Gaussian density multiplied by a highly oscillatory modulating factor -- for such models have been obtained. In the one-dimensional nearest-neighbor case with i.i.d. transition probabilities, local limits of uniformly elliptic ballistic walks are now well understood. We complete the picture by proving a similar result for the only recurrent case, namely the balanced one, in which such a walk is diffusive. The method of proof is, out of necessity, entirely different from the ballistic case.
From limit cycles to strange attractors
William Ott,Mikko Stenlund
Physics , 2010, DOI: 10.1007/s00220-010-0994-y
Abstract: We define a quantitative notion of shear for limit cycles of flows. We prove that strange attractors and SRB measures emerge when systems exhibiting limit cycles with sufficient shear are subjected to periodic pulsatile drives. The strange attractors possess a number of precisely-defined dynamical properties that together imply chaos that is both sustained in time and physically observable.
Positive Lyapunov exponent by a random perturbation
Zeng Lian,Mikko Stenlund
Physics , 2010, DOI: 10.1080/14689367.2012.667787
Abstract: We study the effect of a random perturbation on a one-parameter family of dynamical systems whose behavior in the absence of perturbation is ill understood. We provide conditions under which the perturbed system is ergodic and admits a positive Lyapunov exponent, with an explicit lower bound, for a large and controlled set of parameter values.
Multi-Gaussian modes of diffusion in a quenched random medium
Tapio Simula,Mikko Stenlund
Physics , 2010, DOI: 10.1103/PhysRevE.82.041125
Abstract: We have studied a model of a random walk in a quenched random environment. In addition to featuring anomalous diffusion and localization, for special regimes of disorder parameters the particle density decomposes into multi-Gaussian structure while its cumulative distribution is normal. We explain the observed fine structure of the density and point out its significance to experiments.
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