In this paper a model reduction technique is introduced for piecewise-smooth (PWS) vector fields, whose trajectories fall into a Banach space, but the domain of definition of the vector fields is a non-dense subset of the Banach space. The vector fields depend on a parameter that can assume different discrete values in two parts of the phase space and a continuous family of values on the boundary that separates the two parts of the phase space. In essence the parameter parametrizes the possible vector fields on the boundary. The problem is to find one or more values of the parameter so that the solution of the PWS system on the boundary satisfies certain requirements. In this paper we require continuous solutions. Motivated by the properties of applications, we assume that when the parameter is forced to switch between the two discrete values, trajectories become discontinuous. Discontinuous trajectories exist in systems whose domain of definition is non-dense. It is shown that under our assumptions the trajectories of such PWS systems have unique forward-time continuation when the parameter of the system switches. A finite-dimensional reduced order model is constructed, which accounts for the discontinuous trajectories. It is shown that this model retains uniqueness of solutions and other properties of the original PWS system. The model reduction technique is illustrated on a nonlinear bowed string model.
We propose a class of locally Lipschitz functions with piecewise structure for use as Lyapunov functions for hybrid dynamical systems. Subject to some regularity of the dynamics, we show that Lyapunov inequalities can be checked only on a dense set and thus we avoid checking them at points of nondifferentiability of the Lyapunov function. Connections to other classes of locally Lipschitz or piecewise regular functions are also discussed and applications to hybrid dynamical systems are included.
We introduce a weak transversality condition for piecewise C^{1+α} and piecewise hyperbolic maps which admit a C^{1+α} stable distribution. We show good bounds on the essential spectral radius of the associated transfer operators acting on classical anisotropic Sobolev spaces of Triebel-Lizorkin type. In many cases, we obtain a spectral gap from which we deduce the existence of finitely many physical measures with basin of total measure. The analysis relies on standard techniques (in particular complex interpolation) and applies also to piecewise expanding maps and to Anosov diffeomorphisms, g…
We consider piecewise cone hyperbolic systems satisfying a bunching condition and we obtain a bound on the essential spectral radius of the associated weighted transfer operators acting on anisotropic Sobolev spaces. The bunching condition is always satisfied in dimension two, and our results give a unifying treatment of the work of Demers-Liverani and our previous work. When the complexity is subexponential, our bound implies a spectral gap for the transfer operator corresponding to the physical measures in many cases (for example if $T$ preserves volume, or if the stable dimension is equal t…
One of the fundamental results of ergodic optimisation asserts that for any dynamical system on a compact metric space $X$ and for any Banach space of continuous real-valued functions on $X$ which embeds densely in $C(X)$ there exists a residual set of functions in that Banach space for which the maximising measure is unique. We extend this result by showing that this residual set is additionally prevalent, answering a question of J. Bochi and Y. Zhang.
We study transfer operators associated to piecewise monotone interval transformations and show that the essential spectrum is large whenever the Banach space bounds $L^\infty$ and the transformation fails to be Markov. Constructing a family of Banach spaces we show that the lower bound on the essential spectral radius is optimal. Indeed, these Banach spaces realise an essential spectral radius as close as desired to the theoretical best possible case.
I introduce Banach spaces on which it is possible to precisely characterize the spectrum of the transfer operator associated to a piecewise expanding map with Hölder weight.
R. Jin showed that whenever A and B are sets of integers having positive upper Banach density, the sumset A+B is piecewise syndetic. This result was strengthened by Bergelson, Furstenberg, and Weiss to conclude that A+B must be piecewise Bohr. We generalize the latter result to cases where A has Banach density 0, giving a new proof of the previous results in the process.
The collection of all non-degenerate, continuous, two-piece, piecewise-linear maps on $\mathbb{R}^2$ can be reduced to a four-parameter family known as the two-dimensional border-collision normal form. We prove that throughout an open region of parameter space this family has an attractor satisfying Devaney's definition of chaos. This strengthens existing results on the robustness of chaos in piecewise-linear maps. We further show that the stable manifold of a saddle fixed point, despite being a one-dimensional object, densely fills an open region containing the attractor. Finally we identify …
In an abstract Banach space we study conditions for the existence of piecewise continuous, almost periodic solutions for semi-linear impulsive differential equation with fixed and non-fixed moments of impulsive action
We introduce a family of Banach spaces of measures, each containing the set of measures with density of bounded variation. These spaces are suitable for the study of weighted transfer operators of piecewise-smooth maps of the interval where the weighting used in the transfer operator is not better than piecewise Hölder continuous and the partition on which the map is continuous may possess a countable number of elements. For such weighted transfer operators we give upper bounds for both the spectral radius and for the essential spectral radius.
We consider semiflows in general Banach spaces motivated by monotone cyclic feedback systems or differential equations with integer-valued Lyapunov functionals. These semiflows enjoy strong monotonicity properties with respect to cones of high ranks, which imply order-related structures on the $ω$-limit sets of precompact semi-orbits. We show that for a pseudo-ordered precompact semi-orbit the $ω$-limit set $Ω$ is either ordered, or is contained in the set of equilibria, or possesses a certain ordered homoclinic property. In particular, we show that if $Ω$ contains no equilibrium, then $Ω$ its…
For a class of piecewise hyperbolic maps in two dimensions, we propose a combinatorial definition of topological entropy by counting the maximal, open, connected components of the phase space on which iterates of the map are smooth. We prove that this quantity dominates the measure theoretic entropies of all invariant probability measures of the system, and then construct an invariant measure whose entropy equals the proposed topological entropy. We prove that our measure is the unique measure of maximal entropy, that it is ergodic, gives positive measure to every open set, and has exponential…
As the parameters of a map are varied an attractor may vary continuously in the Hausdorff metric. The purpose of this paper is to explore the continuation of chaotic attractors. We argue that this is not a helpful concept for smooth unimodal maps for which periodic windows fill parameter space densely, but that for piecewise-smooth maps it provides a way to delineate structure within parameter regions of robust chaos and form a stronger notion of robustness. We obtain conditions for the continuity of an attractor and demonstrate the results with coupled skew tent maps, the Lozi map, and the bo…
Given any smooth Anosov map we construct a Banach space on which the associated transfer operator is quasi-compact. The peculiarity of such a space is that in the case of expanding maps it reduces exactly to the usual space of functions of bounded variation which has proven particularly successful in studying the statistical properties of piecewise expanding maps. Our approach is based on a new method of studying the absolute continuity of foliations which provides new information that could prove useful in treating hyperbolic systems with singularities.
We prove that if a quasi-isometry of warped cones is induced by a map between the base spaces of the cones, the actions must be conjugate by this map. The converse is false in general, conjugacy of actions is not sufficient for quasi-isometry of the respective warped cones. For a general quasi-isometry of warped cones, using the asymptotically faithful covering constructed in a previous work with Jianchao Wu, we deduce that the two groups are quasi-isometric after taking Cartesian products with suitable powers of the integers. Secondly, we characterise geometric properties of a group (coarse e…
We use the method of atomic decomposition and a new family of Banach spaces to study the action of transfer operators associated to piecewise-defined maps. It turns out that these transfer operators are quasi-compact even when the associated potential, the dynamics and the underlying phase space have very low regularity. In particular it is often possible to obtain exponential decay of correlations, the Central Limit Theorem and almost sure invariance principle for fairly general observables, including unbounded ones.
