Recent zbMATH articles in MSC 37M https://www.zbmath.org/atom/cc/37M 2021-11-25T18:46:10.358925Z Werkzeug Stability domains for quadratic-bilinear reduced-order models https://www.zbmath.org/1472.34105 2021-11-25T18:46:10.358925Z "Kramer, Boris" https://www.zbmath.org/authors/?q=ai:kramer.boris Fourier multipliers and transfer operators https://www.zbmath.org/1472.37026 2021-11-25T18:46:10.358925Z "Pollicott, Mark" https://www.zbmath.org/authors/?q=ai:pollicott.mark The author gives a rigorous proof of a conjectured numerical value proposed by \textit{X. Chen} and \textit{H. Volkmer} [J. Fractal Geom. 5, No. 4, 351--386 (2018; Zbl 1400.37026)] which estimates a quantity related to the spectral radius of a transfer operator. The problem is significantly connected to the theory of Fourier multipliers. More specifically, the author takes the bounded linear operator $$\mathcal{L}: C^{0}([0, 1])\rightarrow C^{0}([0, 1])$$ defined by $(\mathcal{L}u)(t) = \frac{1}{3} \sum_{i=0}^{3}\left|\sin\left(\frac{\pi (t+i)}{3}\right)\right|u\left(\frac{t+i}{3}\right).$ For estimating the conjectured numerical value $$c=\lim_{n\rightarrow +\infty}||\mathcal{L}^{n}||^{1/n}$$, the following complex function is used: $d(z)=\exp\bigg(-\sum_{n=1}^{\infty}\frac{z^{n}}{n}\frac{1}{3^{n}-1}\sum_{j=0}^{3^{n}-1}\prod_{k=0}^{n-1}\sin\bigg(\frac{3^{k}j\pi}{3^{n}-1}\bigg)\bigg), \quad z\in \mathbb{C}.$ Note that $$d(z)$$ extends analytically to $$\mathbb{C}$$. The smallest positive zero $$\alpha>0$$ is the reciprocal of the spectral radius $$c$$, i.e., $$c=1/\alpha$$. He describes a rigorous computation to determine a better estimate of $$c$$, namely $c= 0.648314752798325682324771447 \dots \pm 10^{-27}.$ The author also considers a more general form of the above bounded linear operator $$\mathcal{L}$$ and estimates its spectral radius. He gives two applications to justify the importance of his results. Erratum to: Rethinking the definition of rate-induced tipping'' https://www.zbmath.org/1472.37083 2021-11-25T18:46:10.358925Z "Hoyer-Leitzel, Alanna" https://www.zbmath.org/authors/?q=ai:hoyer-leitzel.alanna "N. Nadeau, Alice" https://www.zbmath.org/authors/?q=ai:nadeau.alice-n From the text: Through this Erratum, we are correcting two statements made in describing the hyperbolicity of the single globally defined solution at the critical rate for the system $\dot{x} = -(x-\lambda(rt))(x-\lambda(rt)-\mu)$ and the system $\dot{x} = -(x-\lambda(rt))(x-\lambda(rt)-\mu)(x-\lambda(rt) + \mu)$ as described in Secs. III A 2 and III A 3, respectively, of our previously published paper [ibid. 31, No. 5, 053133, 10 p. (2021; Zbl 1470.37104)]. Categorizing with catastrophic radii in numerical minimization https://www.zbmath.org/1472.65071 2021-11-25T18:46:10.358925Z "Levy, Adam B." https://www.zbmath.org/authors/?q=ai:levy.adam-b Summary: We introduce and develop a notion of catastrophic radii'' to identify where a minimization method may require an arbitrarily large number of steps to approximate a minimizer of an objective function, and we use this notion to categorize the performance of method/objective combinations. In order to investigate the different categories, we explore simple examples where explicit formulas can be used, and we discuss several ways that simulation can be used to investigate catastrophic radii for other method/objective combinations. An extension of the entropic chaos degree and its positive effect https://www.zbmath.org/1472.65160 2021-11-25T18:46:10.358925Z "Inoue, Kei" https://www.zbmath.org/authors/?q=ai:inoue.kei "Mao, Tomoyuki" https://www.zbmath.org/authors/?q=ai:mao.tomoyuki "Okutomi, Hidetoshi" https://www.zbmath.org/authors/?q=ai:okutomi.hidetoshi "Umeno, Ken" https://www.zbmath.org/authors/?q=ai:umeno.ken Summary: The Lyapunov exponent is used to quantify the chaos of a dynamical system, by characterizing the exponential sensitivity of an initial point on the dynamical system. However, we cannot directly compute the Lyapunov exponent for a dynamical system without its dynamical equation, although some estimation methods do exist. Information dynamics introduces the entropic chaos degree to measure the strength of chaos of the dynamical system. The entropic chaos degree can be used to compute the strength of chaos with a practical time series. It may seem like a kind of finite space Kolmogorov-Sinai entropy, which then indicates the relation between the entropic chaos degree and the Lyapunov exponent. In this paper, we attempt to extend the definition of the entropic chaos degree on a $$d$$-dimensional Euclidean space to improve the ability to measure the stength of chaos of the dynamical system and show several relations between the extended entropic chaos degree and the Lyapunov exponent. Fast construction of forward flow maps using Eulerian based interpolation schemes https://www.zbmath.org/1472.65161 2021-11-25T18:46:10.358925Z "You, Guoqiao" https://www.zbmath.org/authors/?q=ai:you.guoqiao "Leung, Shingyu" https://www.zbmath.org/authors/?q=ai:leung.shingyu Summary: We propose a modification to a recently developed Eulerian interpolation scheme for constructing the flow map for autonomous, periodic and aperiodic dynamical systems. We show that the proposed methods significantly improve the computational efficiency when a large number of flow maps are needed, yet retain second-order accuracy as in the original approach. The idea is to pre-compute and to store a carefully selected set of intermediate flow maps. When the initial and the final time of a required flow map are known, our proposed methods can simply load these pre-processed short-time flow maps for flow map construction. Numerical examples are included to validate our theoretical prediction, and demonstrate the effectiveness of these proposed Eulerian interpolation schemes as a simple numerical tool for other applications such as the finite-time Lyapunov exponent in determining the Lagrangian coherent structure of dynamical systems and the geometrical optics problems. Asymptotic behaviour of orbit determination for hyperbolic maps https://www.zbmath.org/1472.70050 2021-11-25T18:46:10.358925Z "Marò, Stefano" https://www.zbmath.org/authors/?q=ai:maro.stefano "Bonanno, Claudio" https://www.zbmath.org/authors/?q=ai:bonanno.claudio Summary: We deal with the orbit determination problem for hyperbolic maps. The problem consists in determining the initial conditions of an orbit and, eventually, other parameters of the model from some observations. We study the behaviour of the confidence region in the case of simultaneous increase in the number of observations and the time span over which they are performed. More precisely, we describe the geometry of the confidence region for the solution, distinguishing whether a parameter is added to the estimate of the initial conditions or not. We prove that the inclusion of a dynamical parameter causes a change in the rate of decay of the uncertainties, as suggested by some known numerical evidences.