American Institute of Mathematical Sciences

ISSN:
2158-2491

eISSN:
2158-2505

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Journal of Computational Dynamics

June 2020 , Volume 7 , Issue 1

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2020, 7(1): 1-33 doi: 10.3934/jcd.2020001 +[Abstract](906) +[HTML](408) +[PDF](1159.16KB)
Abstract:

We study and compare two different model reduction techniques for bilinear systems, specifically generalized balancing and \begin{document}$\mathcal{H}_2$\end{document}-based model reduction, and apply it to semi-discretized controlled Fokker-Planck and Liouville–von Neumann equations. For this class of transport equations, the control enters the dynamics as an advection term that leads to the bilinear form. A specific feature of the systems is that they are stable, but not asymptotically stable, and we discuss aspects regarding structure and stability preservation in some depth as these aspects are particularly relevant for the equations of interest. Another focus of this article is on the numerical implementation and a thorough comparison of the aforementioned model reduction methods.

2020, 7(1): 35-56 doi: 10.3934/jcd.2020002 +[Abstract](1312) +[HTML](443) +[PDF](6147.66KB)
Abstract:

Dynamic mode decomposition (DMD) gives a practical means of extracting dynamic information from data, in the form of spatial modes and their associated frequencies and growth/decay rates. DMD can be considered as a numerical approximation to the Koopman operator, an infinite-dimensional linear operator defined for (nonlinear) dynamical systems. This work proposes a new criterion to estimate the accuracy of DMD on a mode-by-mode basis, by estimating how closely each individual DMD eigenfunction approximates the corresponding Koopman eigenfunction. This approach does not require any prior knowledge of the system dynamics or the true Koopman spectral decomposition. The method may be applied to extensions of DMD (i.e., extended/kernel DMD), which are applicable to a wider range of problems. The accuracy criterion is first validated against the true error with a synthetic system for which the true Koopman spectral decomposition is known. We next demonstrate how this proposed accuracy criterion can be used to assess the performance of various choices of kernel when using the kernel method for extended DMD. Finally, we show that our proposed method successfully identifies modes of high accuracy when applying DMD to data from experiments in fluids, in particular particle image velocimetry of a cylinder wake and a canonical separated boundary layer.

2020, 7(1): 57-81 doi: 10.3934/jcd.2020003 +[Abstract](775) +[HTML](336) +[PDF](672.31KB)
Abstract:

Methods have previously been developed for the approximation of Lyapunov functions using radial basis functions. However these methods assume that the evolution equations are known. We consider the problem of approximating a given Lyapunov function using radial basis functions where the evolution equations are not known, but we instead have sampled data which is contaminated with noise. We propose an algorithm in which we first approximate the underlying vector field, and use this approximation to then approximate the Lyapunov function. Our approach combines elements of machine learning/statistical learning theory with the existing theory of Lyapunov function approximation. Error estimates are provided for our algorithm.

2020, 7(1): 83-121 doi: 10.3934/jcd.2020004 +[Abstract](684) +[HTML](300) +[PDF](643.91KB)
Abstract:

We study the mean-median map as a dynamical system on the space of finite sets of piecewise-affine continuous functions with rational coefficients. We determine the structure of the limit function in the neighbourhood of a distinctive family of rational points, the local minima. By constructing a simpler map which represents the dynamics in such neighbourhoods, we extend the results of Cellarosi and Munday [2] by two orders of magnitude. Based on these computations, we conjecture that the Hausdorff dimension of the graph of the limit function of the set \begin{document}$[0,x,1]$\end{document} is greater than \begin{document}$1$\end{document}.

2020, 7(1): 123-158 doi: 10.3934/jcd.2020005 +[Abstract](470) +[HTML](250) +[PDF](1911.47KB)
Abstract:

This work develops validated numerical methods for linear stability analysis at an equilibrium solution of a system of delay differential equations (DDEs). In addition to providing mathematically rigorous bounds on the locations of eigenvalues, our method leads to validated counts. For example we obtain the computer assisted theorems about Morse indices (number of unstable eigenvalues). The case of a single constant delay is considered. The method downplays the role of the scalar transcendental characteristic equation in favor of a functional analytic approach exploiting the strengths of numerical linear algebra/techniques of scientific computing. The idea is to consider an equivalent implicitly defined discrete time dynamical system which is projected onto a countable basis of Chebyshev series coefficients. The projected problem reduces to questions about certain sparse infinite matrices, which are well approximated by \begin{document}$N \times N$\end{document} matrices for large enough \begin{document}$N$\end{document}. We develop the appropriate truncation error bounds for the infinite matrices, provide a general numerical implementation which works for any system with one delay, and discuss computer-assisted theorems in a number of example problems.

2020, 7(1): 159-181 doi: 10.3934/jcd.2020006 +[Abstract](351) +[HTML](195) +[PDF](1879.86KB)
Abstract:

This paper is devoted to the theoretical and numerical analysis of the non-stationary Rayleigh-Bénard-Marangoni (RBM) system. We analyze the existence of global weak solutions for the non-stationary RBM system in polyhedral domains of \begin{document}$\mathbb{R}^3$\end{document} and the convergence, in the norm of \begin{document}$L^{2}(\Omega),$\end{document} to the corresponding stationary solution. Additionally, we develop a numerical scheme for approximating the weak solutions of the non-stationary RBM system, based on a mixed approximation: finite element approximation in space and finite differences in time. After proving the unconditional well-posedness of the numerical scheme, we analyze some error estimates and establish a convergence analysis. Finally, we present some numerical simulations to validate the behavior of our scheme.