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Discrete & Continuous Dynamical Systems - B

September 2018 , Volume 23 , Issue 7

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Preface: "Structural Dynamical Systems: Computational aspects"
Nicoletta Del Buono, Cinzia Elia, Roberto Garrappa and Alessandro Pugliese
2018, 23(7): i-i doi: 10.3934/dcdsb.201807i +[Abstract](139) +[HTML](39) +[PDF](92.02KB)
In memory of Timo Eirola, 1951-2016
Luca Dieci
2018, 23(7): ⅲ-ⅲ doi: 10.3934/dcdsb.20187iii +[Abstract](175) +[HTML](56) +[PDF](63.5KB)
A comparison of boundary correction methods for Strang splitting
Lukas Einkemmer and Alexander Ostermann
2018, 23(7): 2641-2660 doi: 10.3934/dcdsb.2018081 +[Abstract](536) +[HTML](369) +[PDF](571.3KB)

In this paper we investigate splitting methods in the presence of non-homogeneous boundary conditions. In particular, we consider the corrections that have been described and analyzed in Einkemmer, Ostermann 2015 and Alonso-Mallo, Cano, Reguera 2016. The latter method is extended to the non-linear case, and a rigorous convergence analysis is provided. We perform numerical simulations for diffusion-reaction, advection-reaction, and dispersion-reaction equations in order to evaluate the relative performance of these two corrections. Furthermore, we introduce an extension of both methods to obtain order three locally and evaluate under what circumstances this is beneficial.

Positive symplectic integrators for predator-prey dynamics
Fasma Diele and Carmela Marangi
2018, 23(7): 2661-2678 doi: 10.3934/dcdsb.2017185 +[Abstract](1448) +[HTML](593) +[PDF](940.26KB)

We propose novel positive numerical integrators for approximating predator-prey models. The schemes are based on suitable symplectic procedures applied to the dynamical system written in terms of the log transformation of the original variables. Even if this approach is not new when dealing with Hamiltonian systems, it is of particular interest in population dynamics since the positivity of the approximation is ensured without any restriction on the temporal step size. When applied to separable M-systems, the resulting schemes are proved to be explicit, positive, Poisson maps. The approach is generalized to predator-prey dynamics which do not exhibit an M-system structure and successively to reaction-diffusion equations describing spatially extended dynamics. A classical polynomial Krylov approximation for the diffusive term joint with the proposed schemes for the reaction, allows us to propose numerical schemes which are explicit when applied to well established ecological models for predator-prey dynamics. Numerical simulations show that the considered approach provides results which outperform the numerical approximations found in recent literature.

Effect of perturbation in the numerical solution of fractional differential equations
Roberto Garrappa, Eleonora Messina and Antonia Vecchio
2018, 23(7): 2679-2694 doi: 10.3934/dcdsb.2017188 +[Abstract](1297) +[HTML](626) +[PDF](446.48KB)

The equations describing engineering and real-life models are usually derived in an approximated way. Thus, in most cases it is necessary to deal with equations containing some kind of perturbation. In this paper we consider fractional differential equations and study the effects on the continuous and numerical solution, of perturbations on the given function, over long-time intervals. Some bounds on the global error are also determined.

On the stability of $\vartheta$-methods for stochastic Volterra integral equations
Dajana Conte, Raffaele D'Ambrosio and Beatrice Paternoster
2018, 23(7): 2695-2708 doi: 10.3934/dcdsb.2018087 +[Abstract](516) +[HTML](341) +[PDF](584.98KB)

The paper is focused on the analysis of stability properties of a family of numerical methods designed for the numerical solution of stochastic Volterra integral equations. Stability properties are provided with respect to the basic test equation, as well as to the convolution test equation. For each equation, stability properties are intended both in the mean-square and in the asymptotic sense. Stability regions are also provided for a selection of methods. Numerical experiments confirming the theoretical study are also given.

Two-step collocation methods for fractional differential equations
Angelamaria Cardone, Dajana Conte and Beatrice Paternoster
2018, 23(7): 2709-2725 doi: 10.3934/dcdsb.2018088 +[Abstract](613) +[HTML](363) +[PDF](465.33KB)

We propose two-step collocation methods for the numerical solution of fractional differential equations. These methods increase the order of convergence of one-step collocation methods, with the same number of collocation points. Moreover, they are continuous methods, i.e. they furnish an approximation of the solution at each point of the time interval. We describe the derivation of two-step collocation methods and analyse convergence. Some numerical experiments confirm theoretical expectations.

Pseudospectral reduction to compute Lyapunov exponents of delay differential equations
Dimitri Breda and Sara Della Schiava
2018, 23(7): 2727-2741 doi: 10.3934/dcdsb.2018092 +[Abstract](551) +[HTML](435) +[PDF](1987.12KB)

A recent pseudospectral collocation is used to reduce a nonlinear delay differential equation to a system of ordinary differential equations. Standard methods are then applied to compute Lyapunov exponents. The validity of this simple approach is shown experimentally. Matlab codes are also included.

Periodic orbits of planar discontinuous system under discretization
Luca Dieci, Timo Eirola and Cinzia Elia
2018, 23(7): 2743-2762 doi: 10.3934/dcdsb.2018103 +[Abstract](457) +[HTML](298) +[PDF](1253.06KB)

We consider a model planar system with discontinuous right-hand side possessing an attracting periodic orbit, and we investigate what happens to a Euler discretization with stepsize \begin{document}$ τ$\end{document} of this system. We show that, in general, the resulting discrete dynamical system does not possess an invariant curve, in sharp contrast to what happens for smooth problems. In our context, we show that the numerical trajectories are forced to remain inside a band, whose width is proportional to the discretization stepsize \begin{document}$ τ$\end{document}. We further show that if we consider an event-driven discretization of the model problem, whereby the solution is forced to step exactly on the discontinuity line, then there is a discrete periodic solution near the one of the original problem (for sufficiently small \begin{document}$ τ$\end{document}). Finally, we consider what happens to the Euler discretization of the regularized system rewritten in polar coordinates, and give numerical evidence that the discrete solution now undergoes a period doubling cascade with respect to the regularization parameter \begin{document}$ \epsilon$\end{document}, for fixed \begin{document}$ τ$\end{document}.

Numerical preservation of long-term dynamics by stochastic two-step methods
Raffaele D'Ambrosio, Martina Moccaldi and Beatrice Paternoster
2018, 23(7): 2763-2773 doi: 10.3934/dcdsb.2018105 +[Abstract](491) +[HTML](298) +[PDF](328.86KB)

The paper aims to explore the long-term behaviour of stochastic two-step methods applied to a class of second order stochastic differential equations. In particular, the treatment focuses on preserving long-term statistics related to the dynamics of a linear stochastic damped oscillator whose velocity, in the stationary regime, is distributed as a Gaussian variable and uncorrelated with the position. By computing the solution of a very simple matrix equality, we a-priori determine the long-term statistics characterizing the numerical dynamics and analyze the behaviour of a selection of methods.

Domain-growth-induced patterning for reaction-diffusion systems with linear cross-diffusion
Anotida Madzvamuse and Raquel Barreira
2018, 23(7): 2775-2801 doi: 10.3934/dcdsb.2018163 +[Abstract](315) +[HTML](163) +[PDF](4828.36KB)

In this article we present, for the first time, domain-growth induced pattern formation for reaction-diffusion systems with linear cross-diffusion on evolving domains and surfaces. Our major contribution is that by selecting parameter values from spaces induced by domain and surface evolution, patterns emerge only when domain growth is present. Such patterns do not exist in the absence of domain and surface evolution. In order to compute these domain-induced parameter spaces, linear stability theory is employed to establish the necessary conditions for domain-growth induced cross-diffusion-driven instability for reaction-diffusion systems with linear cross-diffusion. Model reaction-kinetic parameter values are then identified from parameter spaces induced by domain-growth only; these exist outside the classical standard Turing space on stationary domains and surfaces. To exhibit these patterns we employ the finite element method for solving reaction-diffusion systems with cross-diffusion on continuously evolving domains and surfaces.

Partitioned second order method for magnetohydrodynamics in Elsässer variables
Yong Li and Catalin Trenchea
2018, 23(7): 2803-2823 doi: 10.3934/dcdsb.2018106 +[Abstract](396) +[HTML](346) +[PDF](428.11KB)

Magnetohydrodynamics (MHD) studies the dynamics of electrically conducting fluids, involving Navier-Stokes equations coupled with Maxwell equations via Lorentz force and Ohm's law. Monolithic methods, which solve fully coupled MHD systems, are computationally expensive. Partitioned methods, on the other hand, decouple the full system and solve subproblems in parallel, and thus reduce the computational cost.

This paper is devoted to the design and analysis of a partitioned method for the MHD system in the Elsässer variables. The stability analysis shows that for magnetic Prandtl number of order unity, the method is unconditionally stable. We prove the error estimates and present computational tests that support the theory.

Polynomial interpolation and a priori bootstrap for computer-assisted proofs in nonlinear ODEs
Maxime Breden and Jean-Philippe Lessard
2018, 23(7): 2825-2858 doi: 10.3934/dcdsb.2018164 +[Abstract](337) +[HTML](159) +[PDF](2931.51KB)

In this work, we introduce a method based on piecewise polynomial interpolation to enclose rigorously solutions of nonlinear ODEs. Using a technique which we call a priori bootstrap, we transform the problem of solving the ODE into one of looking for a fixed point of a high order smoothing Picard-like operator. We then develop a rigorous computational method based on a Newton-Kantorovich type argument (the radii polynomial approach) to prove existence of a fixed point of the Picard-like operator. We present all necessary estimates in full generality and for any nonlinearities. With our approach, we study two systems of nonlinear equations: the Lorenz system and the ABC flow. For the Lorenz system, we solve Cauchy problems and prove existence of periodic and connecting orbits at the classical parameters, and for ABC flows, we prove existence of ballistic spiral orbits.

Underlying one-step methods and nonautonomous stability of general linear methods
Andrew J. Steyer and Erik S. Van Vleck
2018, 23(7): 2859-2877 doi: 10.3934/dcdsb.2018108 +[Abstract](631) +[HTML](333) +[PDF](473.14KB)

We generalize the theory of underlying one-step methods to strictly stable general linear methods (GLMs) solving nonautonomous ordinary differential equations (ODEs) that satisfy a global Lipschitz condition. We combine this theory with the Lyapunov and Sacker-Sell spectral stability theory for one-step methods developed in [34,35,36] to analyze the stability of a strictly stable GLM solving a nonautonomous linear ODE. These results are applied to develop a stability diagnostic for the solution of nonautonomous linear ODEs by strictly stable GLMs.

Conditioning and relative error propagation in linear autonomous ordinary differential equations
Stefano Maset
2018, 23(7): 2879-2909 doi: 10.3934/dcdsb.2018165 +[Abstract](214) +[HTML](154) +[PDF](1044.93KB)

In this paper, we study the relative error propagation in the solution of linear autonomous ordinary differential equations with respect to perturbations in the initial value. We also consider equations with a constant forcing term and a nonzero equilibrium. The study is carried out for equations defined by normal matrices.

Computational techniques to locate crossing/sliding regions and their sets of attraction in non-smooth dynamical systems
Alessandro Colombo, Nicoletta Del Buono, Luciano Lopez and Alessandro Pugliese
2018, 23(7): 2911-2934 doi: 10.3934/dcdsb.2018166 +[Abstract](328) +[HTML](153) +[PDF](3759.2KB)

Several models in the applied sciences are characterized by instantaneous changes in the solutions or discontinuities in the vector field. Knowledge of the geometry of interaction of the flow with the discontinuities can give new insights on the behaviour of such systems. Here, we focus on the class of the piecewise smooth systems of Filippov type. We describe some numerical techniques to locate crossing and sliding regions on the discontinuity surface, to compute the sets of attraction of these regions together with the mathematical form of the separatrices of such sets. Some numerical tests will illustrate our approach.

Smooth to discontinuous systems: A geometric and numerical method for slow-fast dynamics
Luca Dieci and Cinzia Elia
2018, 23(7): 2935-2950 doi: 10.3934/dcdsb.2018112 +[Abstract](568) +[HTML](326) +[PDF](6323.89KB)

We consider a smooth planar system having slow-fast motion, where the slow motion takes place near a curve γ. We explore the idea of replacing the original smooth system with a system with discontinuous right-hand side (DRHS system for short), whereby the DRHS system coincides with the smooth one away from a neighborhood of γ. After this reformulation, in the region of phase-space where γ is attracting for the DRHS system, we will obtain sliding motion on γ and numerical methods apt at integrating for sliding motion can be applied. Moreover, we further bypass resolving the sliding motion and monitor entries (transversal) and exits (tangential) on the curve γ, a fact that can be done independently of resolving for the motion itself. The end result is a method free from the need to adopt stiff integrators or to worry about resolving sliding motion for the DRHS system. We illustrate the performance of our method on a few problems, highlighting the feasibility of using simple explicit Runge-Kutta schemes, and that we obtain much the same orbits of the original smooth system.

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