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Approximation of invariant foliations for stochastic dynamical systems  [PDF]
Xu Sun,Xingye Kan,Jinqiao Duan
Mathematics , 2011, DOI: 10.1142/S0219493712003614
Abstract: Invariant foliations are geometric structures for describing and understanding the qualitative behaviors of nonlinear dynamical systems. For stochastic dynamical systems, however, these geometric structures themselves are complicated random sets. Thus it is desirable to have some techniques to approximate random invariant foliations. In this paper, invariant foliations are approximated for dynamical systems with small noisy perturbations, via asymptotic analysis. Namely, random invariant foliations are represented as a perturbation of the deterministic invariant foliations, with deviation errors estimated.
Topological bifurcations of minimal invariant sets for set-valued dynamical systems  [PDF]
Jeroen S. W. Lamb,Martin Rasmussen,Christian S. Rodrigues
Mathematics , 2011,
Abstract: We discuss the dependence of set-valued dynamical systems on parameters. Under mild assumptions which are often satisfied for random dynamical systems with bounded noise and control systems, we establish the fact that topological bifurcations of minimal invariant sets are discontinuous with respect to the Hausdorff metric, taking the form of lower semi-continuous explosions and instantaneous appearances. We also characterise these transitions by properties of Morse-like decompositions.
Quivers, Geometric Invariant Theory, and Moduli of Linear Dynamical Systems  [PDF]
Markus Bader
Mathematics , 2007,
Abstract: We use geometric invariant theory and the language of quivers to study compactifications of moduli spaces of linear dynamical systems. A general approach to this problem is presented and applied to two well known cases: We show how both Lomadze's and Helmke's compactification arises naturally as a geometric invariant theory quotient. Both moduli spaces are proven to be smooth projective manifolds. Furthermore, a description of Lomadze's compactification as a Quot scheme is given, whereas Helmke's compactification is shown to be an algebraic Grassmann bundle over a Quot scheme. This gives an algebro-geometric description of both compactifications. As an application, we determine the cohomology ring of Helmke's compactification and prove that the two compactifications are not isomorphic when the number of outputs is positive.
A Geometric, Dynamical Approach to Thermodynamics  [PDF]
Hans Henrik Rugh
Physics , 1997, DOI: 10.1088/0305-4470/31/38/011
Abstract: We present a geometric and dynamical approach to the micro-canonical ensemble of classical Hamiltonian systems. We generalize the arguments in \cite{Rugh} and show that the energy-derivative of a micro-canonical average is itself micro-canonically observable. In particular, temperature, specific heat and higher order derivatives of the entropy can be observed dynamically. We give perturbative, asymptotic formulas by which the canonical ensemble itself can be reconstructed from micro-canonical measurements only. In a purely micro-canonical approach we rederive formulas by Lebowitz et al \cite{LPV}, relating e.g. specific heat to fluctuations in the kinetic energy. We show that under natural assumptions on the fluctuations in the kinetic energy the micro-canonical temperature is asymptotically equivalent to the standard canonical definition using the kinetic energy.
Set-based corral control in stochastic dynamical systems: Making almost invariant sets more invariant  [PDF]
Eric Forgoston,Lora Billings,Philip Yecko,Ira B. Schwartz
Physics , 2011, DOI: 10.1063/1.3539836
Abstract: We consider the problem of stochastic prediction and control in a time-dependent stochastic environment, such as the ocean, where escape from an almost invariant region occurs due to random fluctuations. We determine high-probability control-actuation sets by computing regions of uncertainty, almost invariant sets, and Lagrangian Coherent Structures. The combination of geometric and probabilistic methods allows us to design regions of control that provide an increase in loitering time while minimizing the amount of control actuation. We show how the loitering time in almost invariant sets scales exponentially with respect to the control actuation, causing an exponential increase in loitering times with only small changes in actuation force. The result is that the control actuation makes almost invariant sets more invariant.
Invariant foliations for stochastic partial differential equations with dynamic boundary conditions  [PDF]
Zhongkai Guo
Mathematics , 2013,
Abstract: Invariant foliations are complicated random sets useful for describing and understanding the qualitative behaviors of nonlinear dynamical systems. We will consider invariant foliations for stochastic partial differential equation with dynamical boundary condition.
Model sets: a survey  [PDF]
Robert V. Moody
Mathematics , 2000,
Abstract: This article surveys the mathematics of the cut and project method as applied to point sets, called here {\em model sets}. It covers the geometric, arithmetic, and analytical sides of this theory as well as diffraction and the connection with dynamical systems.
Coherent sets for nonautonomous dynamical systems  [PDF]
Gary Froyland,Simon Lloyd,Naratip Santitissadeekorn
Mathematics , 2009, DOI: 10.1016/j.physd.2010.03.009
Abstract: We describe a mathematical formalism and numerical algorithms for identifying and tracking slowly mixing objects in nonautonomous dynamical systems. In the autonomous setting, such objects are variously known as almost-invariant sets, metastable sets, persistent patterns, or strange eigenmodes, and have proved to be important in a variety of applications. In this current work, we explain how to extend existing autonomous approaches to the nonautonomous setting. We call the new time-dependent slowly mixing objects coherent sets as they represent regions of phase space that disperse very slowly and remain coherent. The new methods are illustrated via detailed examples in both discrete and continuous time.
Invariant probability measures and non-wandering sets for impulsive semiflows  [PDF]
Jose F. Alves,Maria Carvalho
Mathematics , 2014, DOI: 10.1007/s10955-014-1101-0
Abstract: We consider impulsive dynamical systems defined on compact metric spaces and their respective impulsive semiflows. We establish sufficient conditions for the existence of probability measures which are invariant by such impulsive semiflows. We also deduce the forward invariance of their non-wandering sets except the discontinuity points.
A simple approach to geometric realization of simplicial and cyclic sets  [PDF]
Amnon Besser
Mathematics , 2003,
Abstract: This is the same version that was previously only on my home page. We give a description of geometric realization which makes it evident that it commutes with products. A similar approach is used to treat cyclic sets. Our approach is similar to those of Drinfeld and Grayson.
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