In this paper we study local stable/unstable sets of sensitive homeomorphisms with the shadowing property defined on compact metric spaces. We prove that local stable/unstable sets always contain a compact and perfect subset of the space. As a corollary we generalize results in [6] and [11] proving that positively countably expansive homeomorphisms defined on compact metric spaces satisfying either transitivity and the shadowing property, or the L-shadowing property, can only be defined in countably spaces.
Nonintrusive shadowing algorithms efficiently compute $v$, the difference between shadowing trajectories, then use $v$ to compute derivatives of averaged objectives of chaos with respect to parameters of the dynamical system. However, previous proofs of shadowing methods wrongly assume that shadowing trajectories are representative. In contrast, the linear response formula is proved rigorously, but is more difficult to compute. We prove that $v$ gives only a part, called the shadowing contribution, of the linear response; hence, the other part, the unstable contribution, is the systematic erro…
This paper develops a variant of the Least Squares Shadowing (LSS) method, which has successfully computed the derivative for several chaotic ODEs and PDEs. The development in this paper aims to simplify Least Squares Shadowing method by improving how time dilation is treated. Instead of adding an explicit time dilation term as in the original method, the new variant uses windowing, which can be simpler to implement, especially for PDEs.
Errors in numerical simulations of gravitating systems can be magnified exponentially over short periods of time. Numerical shadowing provides a way of demonstrating that the dynamics represented by numerical simulations are representative of true dynamics. Using the Sitnikov Problem as an example, it is demonstrated that unstable orbits of the 3-body problem can be shadowed for long periods of time. In addition, it is shown that the stretching of phase space near escape and capture regions is a cause for the failure of the shadowing refinement procedure.
In this computational paper, we perform sensitivity analysis of long-time (or ensemble) averages in the chaotic regime using the shadowing algorithm. We introduce automatic differentiation to eliminate the tangent/adjoint equation solvers used in the shadowing algorithm. In a gradient-based optimization, we use the computed shadowing sensitivity to minimize different long-time averaged functionals of a chaotic time-delayed system by optimal parameter selection. In combined state and parameter estimation for data assimilation, we use the computed sensitivity to predict the optimal trajectory gi…
We study various types of shadowing properties and their implication for C1 generic vector fields. We show that, generically, any of the following three hypotheses implies that an isolated set is topologically transitive and hyperbolic: (i) the set is chain transitive and satisfies the (classical) shadowing property, (ii) the set satisfies the limit shadowing property, or (iii) the set satisfies the (asymptotic) shadowing property with the additional hypothesis that stable and unstable manifolds of any pair of critical orbits intersect each other. In our proof we essentially rely on the proper…
For a parameterized hyperbolic system $\frac{du}{dt}=f(u,s)$ the derivative of the ergodic average $\langle J \rangle = \lim_{T \to \infty}\frac{1}{T}\int_0^T J(u(t),s)$ to the parameter $s$ can be computed via the Least Squares Shadowing algorithm (LSS). We assume that the sytem is ergodic which means that $\langle J \rangle$ depends only on $s$ (not on the initial condition of the hyperbolic system). After discretizing this continuous system using a fixed timestep, the algorithm solves a constrained least squares problem and, from the solution to this problem, computes the desired derivative…
Computational methods for sensitivity analysis are invaluable tools for scientists and engineers investigating a wide range of physical phenomena. However, many of these methods fail when applied to chaotic systems, such as the Kuramoto-Sivashinsky (K-S) equation, which models a number of different chaotic systems found in nature. The following paper discusses the application of a new sensitivity analysis method developed by the authors to a modified K-S equation. We find that least squares shadowing sensitivity analysis computes accurate gradients for solutions corresponding to a wide range o…
In this paper we present a systematic study of shadowing properties with average error in tracing such as (asymptotic) average shadowing, $\underline{d}$-shadowing, $\overline{d}$-shadowing and almost specification. As the main tools we provide a few equivalent characterizations of the average shadowing property, which also partly apply to other notions of shadowing. We prove that almost specification on the whole space induces this property on the measure center. Next, we show that always (e.g. without assumption that the map is onto) almost specification implies asymptotic average shadowing,…
Computational methods for sensitivity analysis are invaluable tools for aerodynamics research and engineering design. However, traditional sensitivity analysis methods break down when applied to long-time averaged quantities in turbulent fluid flow fields, specifically those obtained using high-fidelity turbulence simulations. This is because of a number of dynamical properties of turbulent and chaotic fluid flows, most importantly high sensitivity of the initial value problem, popularly known as the "butterfly effect". The recently developed least squares shadowing (LSS) method avoids the iss…
For a given a normally hyperbolic invariant manifold, whose stable and unstable manifolds intersect transversally, we consider several tools and techniques to detect trajectories with prescribed itineraries: the scattering map, the transition map, the method of correctly aligned windows, and the shadowing lemma. We provide an user's guide on how to apply these tools and techniques to detect unstable orbits in Hamiltonian systems. This consists in the following steps: (i) computation of the scattering map and of the transition map for a flow, (ii) reduction to the scattering map and to the tran…
We obtain sufficient conditions under which the limit of a sequence of functions exhibits a particular dynamical behaviour at a point like expansivity, shadowing, mixing, sensitivity and transitivity. We provide examples to show that the set of all expansive, positively expansive and sensitive points are neither open nor closed in general. We also observe that the set of all transitive and mixing points are closed but not open in general. We give examples to show that properties like expansivity, sensitivity, shadowing, transitivity and mixing at a point need not be preserved under uniform con…
We present a frequency-domain method for computing the sensitivities of time-averaged quantities of chaotic systems with respect to input parameters. Such sensitivities cannot be computed by conventional adjoint analysis tools, because the presence of positive Lyapunov exponents leads to exponential growth of the adjoint variables. The proposed method is based on the least-square shadowing (LSS) approach [1], that formulates the evaluation of sensitivities as an optimisation problem, thereby avoiding the exponential growth of the solution. However, all existing formulations of LSS (and its var…
Accurate approximations of the change of system's output and its statistics with respect to the input are highly desired in computational dynamics. Ruelle's linear response theory provides breakthrough mathematical machinery for computing the sensitivity of chaotic dynamical systems, which enables a better understanding of chaotic phenomena. In this paper, we propose an algorithm for sensitivity analysis of discrete chaos with an arbitrary number of positive Lyapunov exponents. We combine the concept of perturbation space-splitting regularizing Ruelle's original expression together with measur…
Generalized hyperbolicity for diffeomorphisms of Banach spaces
Pontificia Universidade Catolica do Rio de Janeiro (PUC-Rio)（天主教里约热内卢 Pontifical University）；Institute of Mathematics, Statistics and Computer Science, University of São Paulo (IME-USP)（圣保罗大学数学、统计和计算机科学学院）
We present a computable reformulation of Ruelle's linear response formula for chaotic systems. The new formula, called Space-Split Sensitivity or S3, achieves an error convergence of the order ${\cal O}(1/\sqrt{N})$ using $N$ phase points. The reformulation is based on splitting the overall sensitivity into that to stable and unstable components of the perturbation. The unstable contribution to the sensitivity is regularized using ergodic properties and the hyperbolic structure of the dynamics. Numerical examples of uniformly hyperbolic attractors are used to validate the S3 formula against a …
