# American Institute of Mathematical Sciences

ISSN:
2158-2491

eISSN:
2158-2505

All Issues

## Journal of Computational Dynamics

December 2020 , Volume 7 , Issue 2

Special issue on novel computational approaches and their applications

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2020, 7(2): ⅰ-ⅰ doi: 10.3934/jcd.2020007 +[Abstract](608) +[HTML](197) +[PDF](77.34KB)
Abstract:
2020, 7(2): 183-208 doi: 10.3934/jcd.2020008 +[Abstract](705) +[HTML](167) +[PDF](722.55KB)
Abstract:

We consider the following inverse problem for an ordinary differential equation (ODE): given a set of data points \begin{document}$P = \{(t_i,x_i),\; i = 1,\dots,N\}$\end{document}, find an ODE \begin{document}$x^\prime(t) = v (x)$\end{document} that admits a solution \begin{document}$x(t)$\end{document} such that \begin{document}$x_i \approx x(t_i)$\end{document} as closely as possible. The key to the proposed method is to find approximations of the recursive or discrete propagation function \begin{document}$D(x)$\end{document} from the given data set. Afterwards, we determine the field \begin{document}$v(x)$\end{document}, using the conjugate map defined by Schröder's equation and the solution of a related Julia's equation. Moreover, our approach also works for the inverse problems where one has to determine an ODE from multiple sets of data points.

We also study existence, uniqueness, stability and other properties of the recovered field \begin{document}$v(x)$\end{document}. Finally, we present several numerical methods for the approximation of the field \begin{document}$v(x)$\end{document} and provide some illustrative examples of the application of these methods.

2020, 7(2): 209-241 doi: 10.3934/jcd.2020009 +[Abstract](669) +[HTML](174) +[PDF](4606.91KB)
Abstract:

Dynamic Mode Decomposition (DMD) is a data-driven technique to identify a low dimensional linear time invariant dynamics underlying high-dimensional data. For systems in which such underlying low-dimensional dynamics is time-varying, a time-invariant approximation of such dynamics computed through standard DMD techniques may not be appropriate. We focus on DMD techniques for such time-varying systems and develop incremental algorithms for systems without and with exogenous control inputs. We build upon the work in [35] to scenarios in which high dimensional data are governed by low dimensional time-varying dynamics. We consider two classes of algorithms that rely on (ⅰ) a discount factor on previous observations, and (ⅱ) a sliding window of observations. Our algorithms leverage existing techniques for incremental singular value decomposition and allow us to determine an appropriately reduced model at each time and are applicable even if data matrix is singular. We apply the developed algorithms for autonomous systems to Electroencephalographic (EEG) data and demonstrate their effectiveness in terms of reconstruction and prediction. Our algorithms for non-autonomous systems are illustrated using randomly generated linear time-varying systems.

2020, 7(2): 243-269 doi: 10.3934/jcd.2020010 +[Abstract](571) +[HTML](163) +[PDF](2172.4KB)
Abstract:

A single queueing system with time-dependent exponentially distributed arrival processes and exponential machine processes (Kendall notation \begin{document}$M_t/M_t/1$\end{document}) is analyzed. Modeling the time evolution for the discrete queue-length distribution by a continuous drift-diffusion process a Smoluchowski equation on the half space is derived approximating the forward Kolmogorov equations. The approximate model is analyzed and validated, showing excellent agreement for the probabilities of all queue lengths and for all queuing utilizations, including ones that are very small and some that are significantly larger than one. Having an excellent approximation for the probability of an empty queue generates an approximation of the expected outflow of the queueing system. Comparisons to several well-established approximations from the literature show significant improvements in several numerical examples.

2020, 7(2): 271-290 doi: 10.3934/jcd.2020011 +[Abstract](566) +[HTML](167) +[PDF](2671.46KB)
Abstract:

We consider a simple neural field model in which the state variable is dendritic voltage, and in which somas form a continuous one-dimensional layer. This neural field model with dendritic processing is formulated as an integro-differential equation. We introduce a computational method for approximating solutions to this nonlocal model, and use it to perform numerical simulations for neuro-biologically realistic choices of anatomical connectivity and nonlinear firing rate function. For the time discretisation we adopt an Implicit-Explicit (IMEX) scheme; the space discretisation is based on a finite-difference scheme to approximate the diffusion term and uses the trapezoidal rule to approximate integrals describing the nonlocal interactions in the model. We prove that the scheme is of first-order in time and second order in space, and can be efficiently implemented if the factorisation of a small, banded matrix is precomputed. By way of validation we compare the outputs of a numerical realisation to theoretical predictions for the onset of a Turing pattern, and to the speed and shape of a travelling front for a specific choice of Heaviside firing rate. We find that theory and numerical simulations are in excellent agreement.

2020, 7(2): 291-312 doi: 10.3934/jcd.2020012 +[Abstract](587) +[HTML](163) +[PDF](5646.28KB)
Abstract:

We consider the initial-boundary value problem for the 3D regularized compressible isothermal Navier–Stokes–Cahn–Hilliard equations describing flows of a two-component two-phase mixture taking into account capillary effects. We construct a new numerical semi-discrete finite-difference method using staggered meshes for the main unknown functions. The method allows one to improve qualitatively the computational flow dynamics by eliminating the so-called parasitic currents and keeping the component concentration inside the physically reasonable range \begin{document}$(0,1)$\end{document}. This is achieved, first, by discretizing the non-divergent potential form of terms responsible for the capillary effects and establishing the dissipativity of the discrete full energy. Second, a logarithmic (or the Flory–Huggins potential) form for the non-convex bulk free energy is used. The regularization of equations is accomplished to increase essentially the time step of the explicit discretization in time. We include 3D numerical results for two typical problems that confirm the theoretical predictions.

2020, 7(2): 313-337 doi: 10.3934/jcd.2020013 +[Abstract](705) +[HTML](185) +[PDF](2929.21KB)
Abstract:

The Finite-Time Lyapunov Exponent (FTLE) is a well-established numerical tool for assessing stretching rates of initial parcels of fluid, which are advected according to a given time-varying velocity field (which is often available only as data). When viewed as a field over initial conditions, the FTLE's spatial structure is often used to infer the nonhomogeneous transport. Given the measurement and resolution errors inevitably present in the unsteady velocity data, the computed FTLE field should in reality be treated only as an approximation. A method which, for the first time, is able for attribute spatially-varying errors to the FTLE field is developed. The formulation is, however, confined to two-dimensional flows. Knowledge of the errors prevent reaching erroneous conclusions based only on the FTLE field. Moreover, it is established that increasing the spatial resolution does not improve the accuracy of the FTLE field in the presence of velocity uncertainties, and indeed has the opposite effect. Stochastic simulations are used to validate and exemplify these results, and demonstrate the computability of the error field.

2020, 7(2): 339-368 doi: 10.3934/jcd.2020014 +[Abstract](627) +[HTML](161) +[PDF](619.12KB)
Abstract:

In this paper we introduce a procedure, based on the method of equivariant moving frames, for formulating continuous Galerkin finite element schemes that preserve the Lie point symmetries of initial value problems for ordinary differential equations. Our methodology applies to projectable and non-projectable symmetry group actions, to ordinary differential equations of arbitrary order, and finite element approximations of arbitrary polynomial degree. Several examples are included to illustrate various features of the symmetry-preserving process. We summarise extensive numerical experiments showing that symmetry-preserving finite element schemes may provide better long term accuracy than their non-invariant counterparts and can be implemented on larger elements.

2020, 7(2): 369-399 doi: 10.3934/jcd.2020015 +[Abstract](676) +[HTML](170) +[PDF](6099.9KB)
Abstract:

Coherent structures are spatially varying regions which disperse minimally over time and organise motion in non-autonomous systems. This work develops and implements algorithms providing multilayered descriptions of time-dependent systems which are not only useful for locating coherent structures, but also for detecting time windows within which these structures undergo fundamental structural changes, such as merging and splitting events. These algorithms rely on singular value decompositions associated to Ulam type discretisations of transfer operators induced by dynamical systems, and build on recent developments in multiplicative ergodic theory. Furthermore, they allow us to investigate various connections between the evolution of relevant singular value decompositions and dynamical features of the system. The approach is tested on models of periodically and quasi-periodically driven systems, as well as on a geophysical dataset corresponding to the splitting of the Southern Polar Vortex.

2020, 7(2): 401-423 doi: 10.3934/jcd.2020016 +[Abstract](581) +[HTML](161) +[PDF](1078.9KB)
Abstract:

Degree assortativity refers to the increased or decreased probability of connecting two neurons based on their in- or out-degrees, relative to what would be expected by chance. We investigate the effects of such assortativity in a network of theta neurons. The Ott/Antonsen ansatz is used to derive equations for the expected state of each neuron, and these equations are then coarse-grained in degree space. We generate families of effective connectivity matrices parametrised by assortativity coefficient and use SVD decompositions of these to efficiently perform numerical bifurcation analysis of the coarse-grained equations. We find that of the four possible types of degree assortativity, two have no effect on the networks' dynamics, while the other two can have a significant effect.

2020, 7(2): 425-460 doi: 10.3934/jcd.2020017 +[Abstract](652) +[HTML](164) +[PDF](643.35KB)
Abstract:

Birkhoff normal forms are commonly used in order to ensure the so called "effective stability" in the neighborhood of elliptic equilibrium points for Hamiltonian systems. From a theoretical point of view, this means that the eventual diffusion can be bounded for time intervals that are exponentially large with respect to the inverse of the distance of the initial conditions from such equilibrium points. Here, we focus on an approach that is suitable for practical applications: we extend a rather classical scheme of estimates for both the Birkhoff normal forms to any finite order and their remainders. This is made for providing explicit lower bounds of the stability time (that are valid for initial conditions in a fixed open ball), by using a fully rigorous computer-assisted procedure. We apply our approach in two simple contexts that are widely studied in Celestial Mechanics: the Hénon-Heiles model and the Circular Planar Restricted Three-Body Problem. In the latter case, we adapt our scheme of estimates for covering also the case of resonant Birkhoff normal forms and, in some concrete models about the motion of the Trojan asteroids, we show that it can be more advantageous with respect to the usual non-resonant ones.

2020, 7(2): 461-468 doi: 10.3934/jcd.2020018 +[Abstract](557) +[HTML](173) +[PDF](966.48KB)
Abstract:

We describe a renormalization method in maps of the plane \begin{document}$(x, y)$\end{document}, with constant Jacobian \begin{document}$b$\end{document} and a second parameter \begin{document}$a$\end{document} acting as a bifurcation parameter. The method enables one to organize high period periodic attractors and thus find hordes of them in quasi-conservative maps (i.e. \begin{document}$|b| = 1-\varepsilon$\end{document}), when sharing the same rotation number. Numerical challenges are the high period, and the necessary extreme vicinity of many such different points, which accumulate on a hyperbolic periodic saddle. The periodic points are organized, in the \begin{document}$(x, y, a)$\end{document} space, in sequences of diverging period, that we call "branches". We define a renormalization approach, by "hopping" among branches to maximize numerical convergence. Our numerical renormalization has met two kinds of numerical instabilities, well localized in certain ranges of the period for the parameter \begin{document}$a$\end{document} (see [3]) and in other ranges of the period for the dynamical plane \begin{document}$(x, y)$\end{document}. For the first time we explain here how specific numerical instabilities depend on geometry displacements in dynamical plane \begin{document}$(x, y)$\end{document}. We describe how to take advantage of such displacement in the sequence, and of the high period, by moving forward from one branch to its image under dynamics. This, for high period, allows entering the hyperbolicity neighborhood of a saddle, where the dynamics is conjugate to a hyperbolic linear map.

The subtle interplay of branches and of the hyperbolic neighborhood can hopefully help visualize the renormalization approach theoretically discussed in [7] for highly dissipative systems.

2020, 7(2): 469-487 doi: 10.3934/jcd.2020019 +[Abstract](617) +[HTML](171) +[PDF](669.04KB)
Abstract:

In this research, we investigate the application of Dynamic Mode Decomposition combined with Kalman Filtering, Smoothing, and Wavelet Denoising (DMD-KF-W) for denoising time-resolved data. We also compare the performance of this technique with state-of-the-art denoising methods such as Total Variation Diminishing (TV) and Divergence-Free Wavelets (DFW), when applicable. Dynamic Mode Decomposition (DMD) is a data-driven method for finding the spatio-temporal structures in time series data. In this research, we use an autoregressive linear model resulting from applying DMD to the time-resolved data as a predictor in a Kalman Filtering-Smoothing framework for the purpose of denoising. The DMD-KF-W method is parameter-free and runs autonomously. Tests on numerical phantoms show lower error metrics when compared to TV and DFW, when applicable. In addition, DMD-KF-W runs an order of magnitude faster than DFW and TV. In the case of synthetic datasets, where the noise-free datasets were available, our method was shown to perform better than TV and DFW methods (when applicable) in terms of the defined error metric.

2020, 7(2): 489-510 doi: 10.3934/jcd.2020020 +[Abstract](625) +[HTML](169) +[PDF](2136.24KB)
Abstract:

We consider the bifurcation diagram in a suitable parameter plane of a quadratic vector field in \begin{document}$\mathbb{R}^3$\end{document} that features a homoclinic flip bifurcation of the most complicated type. This codimension-two bifurcation is characterized by a change of orientability of associated two-dimensional manifolds and generates infinite families of secondary bifurcations. We show that curves of secondary \begin{document}$n$\end{document}-homoclinic bifurcations accumulate on a curve of a heteroclinic bifurcation involving infinity.

We present an adaptation of the technique known as Lin's method that enables us to compute such connecting orbits to infinity. We first perform a weighted directional compactification of \begin{document}$\mathbb{R}^3$\end{document} with a subsequent blow-up of a non-hyperbolic saddle at infinity. We then set up boundary-value problems for two orbit segments from and to a common two-dimensional section: the first is to a finite saddle in the regular coordinates, and the second is from the vicinity of the saddle at infinity in the blown-up chart. The so-called Lin gap along a fixed one-dimensional direction in the section is then brought to zero by continuation. Once a connecting orbit has been found in this way, its locus can be traced out as a curve in a parameter plane.

2020, 7(2): 511-536 doi: 10.3934/jcd.2020021 +[Abstract](543) +[HTML](150) +[PDF](2953.16KB)
Abstract:

Algorithms proposed for solving high-dimensional optimization problems with no derivative information frequently encounter the "curse of dimensionality, " becoming ineffective as the dimension of the parameter space grows. One feature of a subclass of such problems that are effectively low-dimensional is that only a few parameters (or combinations thereof) are important for the optimization and must be explored in detail. Knowing these parameters/combinations in advance would greatly simplify the problem and its solution. We propose the data-driven construction of an effective (coarse-grained, "trend") optimizer, based on data obtained from ensembles of brief simulation bursts with an "inner" optimization algorithm, that has the potential to accelerate the exploration of the parameter space. The trajectories of this "effective optimizer" quickly become attracted onto a slow manifold parameterized by the few relevant parameter combinations. We obtain the parameterization of this low-dimensional, effective optimization manifold on the fly using data mining/manifold learning techniques on the results of simulation (inner optimizer iteration) burst ensembles and exploit it locally to "jump" forward along this manifold. As a result, we can bias the exploration of the parameter space towards the few, important directions and, through this "wrapper algorithm, " speed up the convergence of traditional optimization algorithms.