# American Institute of Mathematical Sciences

ISSN:
2158-2491

eISSN:
2158-2505

All Issues

## Journal of Computational Dynamics

December 2019 , Volume 6 , Issue 2

Special issue in honor of Reinout Quispel

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2019, 6(2): i-v doi: 10.3934/jcd.2019007 +[Abstract](2206) +[HTML](689) +[PDF](194.73KB)
Abstract:
2019, 6(2): 147-169 doi: 10.3934/jcd.2019008 +[Abstract](2625) +[HTML](356) +[PDF](3361.15KB)
Abstract:

The objective of this work is the introduction and investigation of favourable time integration methods for the Gross-Pitaevskii equation with rotation term. Employing a reformulation in rotating Lagrangian coordinates, the equation takes the form of a nonlinear Schrödinger equation involving a space-time-dependent potential. A natural approach that combines commutator-free quasi-Magnus exponential integrators with operator splitting methods and Fourier spectral space discretisations is proposed. Furthermore, the special structure of the Hamilton operator permits the design of specifically tailored schemes. Numerical experiments confirm the good performance of the resulting exponential integrators.

2019, 6(2): 171-198 doi: 10.3934/jcd.2019009 +[Abstract](6393) +[HTML](653) +[PDF](13381.36KB)
Abstract:

We consider recent work of [18] and [9], where deep learning neural networks have been interpreted as discretisations of an optimal control problem subject to an ordinary differential equation constraint. We review the first order conditions for optimality, and the conditions ensuring optimality after discretisation. This leads to a class of algorithms for solving the discrete optimal control problem which guarantee that the corresponding discrete necessary conditions for optimality are fulfilled. The differential equation setting lends itself to learning additional parameters such as the time discretisation. We explore this extension alongside natural constraints (e.g. time steps lie in a simplex). We compare these deep learning algorithms numerically in terms of induced flow and generalisation ability.

2019, 6(2): 199-222 doi: 10.3934/jcd.2019010 +[Abstract](1974) +[HTML](335) +[PDF](239.67KB)
Abstract:

Aromatic B-series are a generalization of B-series. Some of the algebraic structures on B-series can be defined analogically for aromatic B-series. This paper derives combinatorial formulas for the composition and substitution laws for aromatic B-series.

2019, 6(2): 223-237 doi: 10.3934/jcd.2019011 +[Abstract](2619) +[HTML](330) +[PDF](12453.66KB)
Abstract:

A parameter-dependent class of Hamiltonian (generalized) Lotka–Volterra systems is considered. We prove that this class contains Liouville integrable as well as superintegrable cases according to particular choices of the parameters. We determine sufficient conditions which result in integrable behavior, while we numerically explore the complementary cases, where these analytically derived conditions are not satisfied.

2019, 6(2): 239-249 doi: 10.3934/jcd.2019012 +[Abstract](1926) +[HTML](318) +[PDF](298.49KB)
Abstract:

We explore a new asymptotic-numerical solver for the time-dependent wave equation with an interaction term that is oscillating in time with a very high frequency. The method involves representing the solution as an asymptotic series in inverse powers of the oscillation frequency. Using the new scheme, high accuracy is achieved at a low computational cost. Salient features of the new approach are highlighted by a numerical example.

2019, 6(2): 251-276 doi: 10.3934/jcd.2019013 +[Abstract](2243) +[HTML](356) +[PDF](5084.81KB)
Abstract:

Principal Geodesic Analysis is a statistical technique that constructs low-dimensional approximations to data on Riemannian manifolds. It provides a generalization of principal components analysis to non-Euclidean spaces. The approximating submanifolds are geodesic at a reference point such as the intrinsic mean of the data. However, they are local methods as the approximation depends on the reference point and does not take into account the curvature of the manifold. Therefore, in this paper we develop a specialization of principal geodesic analysis, Principal Symmetric Space Analysis, based on nested sequences of totally geodesic submanifolds of symmetric spaces. The examples of spheres, Grassmannians, tori, and products of two-dimensional spheres are worked out in detail. The approximating submanifolds are geometrically the simplest possible, with zero exterior curvature at all points. They can deal with significant curvature and diverse topology. We show that in many cases the distance between a point and the submanifold can be computed analytically and there is a related metric that reduces the computation of principal symmetric space approximations to linear algebra.

2019, 6(2): 277-306 doi: 10.3934/jcd.2019014 +[Abstract](1762) +[HTML](313) +[PDF](478.78KB)
Abstract:

In this paper we construct a family of integrable reductions of the dressing chain, described in its Lotka-Volterra form. For each \begin{document}$k, n\in \mathbb N$\end{document} with \begin{document}$n \geqslant 2k+1$\end{document} we obtain a Lotka-Volterra system \begin{document}$\hbox{LV}_b(n, k)$\end{document} on \begin{document}$\mathbb {R}^n$\end{document} which is a deformation of the Lotka-Volterra system \begin{document}$\hbox{LV}(n, k)$\end{document}, which is itself an integrable reduction of the \begin{document}$2m+1$\end{document}-dimensional Bogoyavlenskij-Itoh system \begin{document}$\hbox{LV}({2m+1}, m)$\end{document}, where \begin{document}$m = n-k-1$\end{document}. We prove that \begin{document}$\hbox{LV}_b(n, k)$\end{document} is both Liouville and non-commutative integrable, with rational first integrals which are deformations of the rational first integrals of \begin{document}$\hbox{LV}({n}, {k})$\end{document}. We also construct a family of discretizations of \begin{document}$\hbox{LV}_b(n, 0)$\end{document}, including its Kahan discretization, and we show that these discretizations are also Liouville and superintegrable.

2019, 6(2): 307-323 doi: 10.3934/jcd.2019015 +[Abstract](2458) +[HTML](327) +[PDF](1446.74KB)
Abstract:

Finite difference schemes that preserve two conservation laws of a given partial differential equation can be found directly by a recently-developed symbolic approach. Until now, this has been used only for equations with quadratic nonlinearity.

In principle, a simplified version of the direct approach also works for equations with polynomial nonlinearity of higher degree. For the modified Korteweg-de Vries equation, whose nonlinear term is cubic, this approach yields several new families of second-order accurate schemes that preserve mass and either energy or momentum. Two of these families contain Average Vector Field schemes of the type developed by Quispel and co-workers. Numerical tests show that each family includes schemes that are highly accurate compared to other mass-preserving methods that can be found in the literature.

2019, 6(2): 325-343 doi: 10.3934/jcd.2019016 +[Abstract](1935) +[HTML](334) +[PDF](408.22KB)
Abstract:

A QRT map is the composition of two involutions on a biquadratic curve: one switching the \begin{document}$x$\end{document}-coordinates of two intersection points with a given horizontal line, and the other switching the \begin{document}$y$\end{document}-coordinates of two intersections with a vertical line. Given a QRT map, a natural question is to ask whether it allows a decomposition into further involutions. Here we provide new answers to this question and show how they lead to a new class of maps, as well as known HKY maps and quadrirational Yang-Baxter maps.

2019, 6(2): 345-360 doi: 10.3934/jcd.2019017 +[Abstract](2503) +[HTML](326) +[PDF](379.81KB)
Abstract:

Classical mechanical systems are defined by their kinetic and potential energies. They generate a Lie algebra under the canonical Poisson bracket. This Lie algebra, which is usually infinite dimensional, is useful in analyzing the system, as well as in geometric numerical integration. But because the kinetic energy is quadratic in the momenta, the Lie algebra obeys identities beyond those implied by skew symmetry and the Jacobi identity. Some Poisson brackets, or combinations of brackets, are zero for all choices of kinetic and potential energy, regardless of the dimension of the system. Therefore, we study the universal object in this setting, the 'Lie algebra of classical mechanics' modelled on the Lie algebra generated by kinetic and potential energy of a simple mechanical system with respect to the canonical Poisson bracket. We show that it is the direct sum of an abelian algebra \begin{document}$\mathfrak{X}$\end{document}, spanned by 'modified' potential energies and isomorphic to the free commutative nonassociative algebra with one generator, and an algebra freely generated by the kinetic energy and its Poisson bracket with \begin{document}$\mathfrak{X}$\end{document}. We calculate the dimensions \begin{document}$c_n$\end{document} of its homogeneous subspaces and determine the value of its entropy \begin{document}$\lim_{n\to\infty} c_n^{1/n}$\end{document}. It is \begin{document}$1.8249\dots$\end{document}, a fundamental constant associated to classical mechanics. We conjecture that the class of systems with Euclidean kinetic energy metrics is already free, i.e., that the only linear identities satisfied by the Lie brackets of all such systems are those satisfied by the Lie algebra of classical mechanics.

2019, 6(2): 361-383 doi: 10.3934/jcd.2019018 +[Abstract](2475) +[HTML](329) +[PDF](1316.05KB)
Abstract:

We propose a Fourier pseudo-spectral scheme for the space-fractional nonlinear Schrödinger equation. The proposed scheme has the following features: it is linearly implicit, it preserves two invariants of the equation, its unique solvability is guaranteed without any restrictions on space and time step sizes. The scheme requires solving a complex symmetric linear system per time step. To solve the system efficiently, we also present a certain variable transformation and preconditioner.

2019, 6(2): 385-400 doi: 10.3934/jcd.2019019 +[Abstract](2070) +[HTML](356) +[PDF](793.78KB)
Abstract:

In this paper, we exploit the gradient flow structure of continuous-time formulations of Bayesian inference in terms of their numerical time-stepping. We focus on two particular examples, namely, the continuous-time ensemble Kalman–Bucy filter and a particle discretisation of the Fokker–Planck equation associated to Brownian dynamics. Both formulations can lead to stiff differential equations which require special numerical methods for their efficient numerical implementation. We compare discrete gradient methods to alternative semi-implicit and other iterative implementations of the underlying Bayesian inference problems.

2019, 6(2): 401-408 doi: 10.3934/jcd.2019020 +[Abstract](1959) +[HTML](309) +[PDF](589.36KB)
Abstract:

Kahan discretization is applicable to any quadratic vector field and produces a birational map which approximates the shift along the phase flow. For a planar quadratic Hamiltonian vector field with a linear Poisson tensor and with a quadratic Hamilton function, this map is known to be integrable and to preserve a pencil of conics. In the paper "Three classes of quadratic vector fields for which the Kahan discretization is the root of a generalised Manin transformation" by P. van der Kamp et al. [5], it was shown that the Kahan discretization can be represented as a composition of two involutions on the pencil of conics. In the present note, which can be considered as a comment to that paper, we show that this result can be reversed. For a linear form \begin{document}$\ell(x,y)$\end{document}, let \begin{document}$B_1,B_2$\end{document} be any two distinct points on the line \begin{document}$\ell(x,y) = -c$\end{document}, and let \begin{document}$B_3,B_4$\end{document} be any two distinct points on the line \begin{document}$\ell(x,y) = c$\end{document}. Set \begin{document}$B_0 = \tfrac{1}{2}(B_1+B_3)$\end{document} and \begin{document}$B_5 = \tfrac{1}{2}(B_2+B_4)$\end{document}; these points lie on the line \begin{document}$\ell(x,y) = 0$\end{document}. Finally, let \begin{document}$B_\infty$\end{document} be the point at infinity on this line. Let \begin{document}$\mathfrak E$\end{document} be the pencil of conics with the base points \begin{document}$B_1,B_2,B_3,B_4$\end{document}. Then the composition of the \begin{document}$B_\infty$\end{document}-switch and of the \begin{document}$B_0$\end{document}-switch on the pencil \begin{document}$\mathfrak E$\end{document} is the Kahan discretization of a Hamiltonian vector field \begin{document}$f = \ell(x,y)\begin{pmatrix}\partial H/\partial y \\ -\partial H/\partial x \end{pmatrix}$\end{document} with a quadratic Hamilton function \begin{document}$H(x,y)$\end{document}. This birational map \begin{document}$\Phi_f:\mathbb C P^2\dashrightarrow\mathbb C P^2$\end{document} has three singular points \begin{document}$B_0,B_2,B_4$\end{document}, while the inverse map \begin{document}$\Phi_f^{-1}$\end{document} has three singular points \begin{document}$B_1,B_3,B_5$\end{document}.

2019, 6(2): 409-427 doi: 10.3934/jcd.2019021 +[Abstract](2819) +[HTML](342) +[PDF](1664.33KB)
Abstract:

We consider the dynamics and numerical simulation of systems of linked rigid bodies (chains). We describe the system using the moving frame method approach of [18]. In this framework, the dynamics of the \begin{document}$j$\end{document}th body is described in a frame relative to the \begin{document}$(j-1)$\end{document}th one. Starting from the Lagrangian formulation of the system on \begin{document}${{\rm{SO}}}(3)^{N}$\end{document}, the final dynamic formulation is obtained by variational calculus on Lie groups. The obtained system is solved by using unit quaternions to represent rotations and numerical methods preserving quadratic integrals.

2019, 6(2): 429-448 doi: 10.3934/jcd.2019022 +[Abstract](2101) +[HTML](369) +[PDF](3261.54KB)
Abstract:

In plasma simulations, numerical methods with high computational efficiency and long-term stability are needed. In this paper, symplectic methods with adaptive time steps are constructed for simulating the dynamics of charged particles under the electromagnetic field. With specifically designed step size functions, the motion of charged particles confined in a Penning trap under three different magnetic fields is studied, and also the dynamics of runaway electrons in tokamaks is investigated. The numerical experiments are performed to show the efficiency of the new derived adaptive symplectic methods.

2019, 6(2): 449-467 doi: 10.3934/jcd.2019023 +[Abstract](2390) +[HTML](318) +[PDF](380.73KB)
Abstract:

We conjecture recurrence relations satisfied by the degrees of some linearizable lattice equations. This helps to prove linear degree growth of these equations. We then use these recurrences to search for lattice equations that have linear growth and hence are linearizable.

2019, 6(2): 469-483 doi: 10.3934/jcd.2019024 +[Abstract](2070) +[HTML](318) +[PDF](2220.88KB)
Abstract:

A system of ordinary differential equations of a predator–prey type, depending on nine parameters, is studied. We have included in this model a nonmonotonic response function and time periodic perturbation. Using numerical continuation software, we have detected three codimension two bifurcations for the unperturbed system, namely cusp, Bogdanov-Takens and Bautin bifurcations. Furthermore, we concentrate on two regions in the parameter space, the region where the Bogdanov-Takens and the region where Bautin bifurcations occur. As we turn on the time perturbation, we find strange attractors in the neighborhood of invariant tori of the unperturbed system.

2019, 6(2): 485-511 doi: 10.3934/jcd.2019025 +[Abstract](2051) +[HTML](317) +[PDF](829.7KB)
Abstract:

The theory of moving frames has been used successfully to solve one dimensional (1D) variational problems invariant under a Lie group symmetry. In the one dimensional case, Noether's laws give first integrals of the Euler–Lagrange equations. In higher dimensional problems, the conservation laws do not enable the exact integration of the Euler–Lagrange system. In this paper we use the theory of moving frames to help solve, numerically, some higher dimensional variational problems, which are invariant under a Lie group action. In order to find a solution to the variational problem, we need first to solve the Euler Lagrange equations for the relevant differential invariants, and then solve a system of linear, first order, compatible, coupled partial differential equations for a moving frame, evolving on the Lie group. We demonstrate that Lie group integrators may be used in this context. We show first that the Magnus expansions on which one dimensional Lie group integrators are based, may be taken sequentially in a well defined way, at least to order 5; that is, the exact result is independent of the order of integration. We then show that efficient implementations of these integrators give a numerical solution of the equations for the frame, which is independent of the order of integration, to high order, in a range of examples. Our running example is a variational problem invariant under a linear action of \begin{document}$SU(2)$\end{document}. We then consider variational problems for evolving curves which are invariant under the projective action of \begin{document}$SL(2)$\end{document} and finally the standard affine action of \begin{document}$SE(2)$\end{document}.

2020 CiteScore: 1