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We investigate memory dependent asymptotic growth in scalar Volterra equations with sublinear nonlinearity. To obtain precise results we utilise the powerful theory of regular variation extensively. By computing the growth rate in terms of a related ordinary differential equation we show that when the memory effect is so strong that the kernel tends to infinity, the growth rate of solutions depends explicitly on the memory of the system. Finally, we employ a fixed point argument to determine analogous results for a perturbed Volterra equation and show that, for a sufficiently large perturbation, the solution tracks the perturbation asymptotically, even when the forcing term is potentially highly non-monotone., Comment: 29 pages, 1 figure

This article revisits the approximation problem of systems of nonlinear delay differential equations (DDEs) by a set of ordinary differential equations (ODEs). We work in Hilbert spaces endowed with a natural inner product including a point mass, and introduce polynomials orthogonal with respect to such an inner product that live in the domain of the linear operator associated with the underlying DDE. These polynomials are then used to design a general Galerkin scheme for which we derive rigorous convergence results and show that it can be numerically implemented via simple analytic formulas. The scheme so obtained is applied to three nonlinear DDEs, two autonomous and one forced: (i) a simple DDE with distributed delays whose solutions recall Brownian motion; (ii) a DDE with a discrete delay that exhibits bimodal and chaotic dynamics; and (iii) a periodically forced DDE with two discrete delays arising in climate dynamics. In all three cases, the Galerkin scheme introduced in this article provides a good approximation by low-dimensional ODE systems of the DDE's strange attractor, as well as of the statistical features that characterize its nonlinear dynamics., 6 figures and 1 table

Symmetry preserving difference schemes approximating second and third order ordinary differential equations are presented. They have the same three or four-dimensional symmetry groups as the original differential equations. The new difference schemes are tested as numerical methods. The obtained numerical solutions are shown to be much more accurate than those obtained by standard methods without an increase in cost. For an example involving a solution with a singularity in the integration region the symmetry preserving scheme, contrary to standard ones, provides solutions valid beyond the singular point., Comment: 26 pages 7 figures

In this article we consider rough differential equations (RDEs) driven by non-geometric rough paths, using the concept of branched rough paths introduced in Gubinelli (2004). We first show that branched rough paths can equivalently be defined as $\gamma$-H\"older continuous paths in some Lie group, akin to geometric rough paths. We then show that every branched rough path can be encoded in a geometric rough path. More precisely, for every branched rough path $\mathbf{X}$ lying above a path $X$, there exists a geometric rough path $\bar{\mathbf{X}}$ lying above an extended path $\bar X$, such that $\bar{\mathbf{X}}$ contains all the information of $\mathbf{X}$. As a corollary of this result, we show that every RDE driven by a non-geometric rough path $\mathbf{X}$ can be rewritten as an extended RDE driven by a geometric rough path $\bar{\mathbf{X}}$. One could think of this as a generalisation of the It\^o-Stratonovich correction formula., Comment: Fixed abstract for display with MathJax

Sobolev descent has long been established as an efficient method for numerically solving boundary value problems, ordinary differential equations and partial differential equations in a small number of iterations. We demonstrate that for any linear ordinary differential equation with initial value conditions sufficient to assure a unique solution, there exists a Hilbert space in which gradient descent will converge to the solution in one iteration. We provide two elementary examples, one initial value problem and one boundary value problem, demonstrating the effectiveness of the theory in numerical settings. As there are ample efficient numerical methods for solving such problems, the significance of the paper is in the approach and the question it raises. Namely, do such spaces exist for wider classes of differential equations?

We consider inverse or parameter estimation problems for general nonlinear nonautonomous dynamical systems with delays. The parameters may be from a Euclidean set as usual, may be time dependent coefficients or may be probability distributions across a population as arise in aggregate data problems. Theoretical convergence results for finite dimensional approximations to the systems are given. Several examples are used to illustrate the ideas and computational results that demonstrate efficacy of the approximations are presented.

By applying hybrid functions of general block-pulse functions and the second Chebyshev polynomials,integrodifferential systems are converted into a system of algebraic equations. The approximate solutions of integrodifferential systems are derived. The numerical examples illustrate that the algorithms are valid., Comment: 10 pages, 0 figures

We survey properties of the Ankeny-Onishi sieve function and establish inequalities for $j_\kappa(\kappa)$ and for $1-j_\kappa(u)$ for $u \to \infty$.

The monodromy operator of a linear delay differential equation with periodic coefficients is formulated as an integral operator. The kernel of this operator includes a factor formed from the fundamental solution of the linear delay differential equation. Although the properties of the fundamental solutions are known, in general there is no closed form for the fundamental solution. This paper describes a collocation procedure to approximate the fundamental solution before the integral operator is discretized. Using arguments on collectively compact operators, the eigenvalues of the discretized monodromy operator are shown to converge to the eigenvalues of the monodromy operator in integral form. The eigenvalues of the monodromy operator tell the stability of the linear delay differential equation. An application to several cases of the Van der Pol oscillator with delay will be given. AMS (MOS) Subject Classification. 34K11, 34K13, 34K28, 45P05, 47A75, 47B38, 47G10.

This work develops an epsilon-uniform finite element method for singularly perturbed boundary value problems. A surprising and remarkable observation is illustrated: By moving one node arbitrarily in between its adjacent nodes, the new finite element solution always intersect with original one at fixed point. Using this fact, an effective epsilon-uniform approximation out of boundary is proposed by adding one point only in the grid that contains boundary layer. The thickness of boundary layer is not necessary to be known from priori estimation. Numerical results are carried out and compared to Shishkin mesh for demonstration purpose., Comment: 18 pages, 7 figures

We study a scheme from citeFV99, which allows to approximate periodic traveling waves in the nonlinear atomic chain with nearest neighbour interactions. We prove a compactness result for this scheme, and derive some generalizations. Moreover, we discuss the thermodynamic properties of traveling waves.

We study the dynamics of two mutually coupled oscillators with a time delayed coupling. Due to the delay, the allowed frequencies of the oscillators are shown to be discretized. The phenomenon is observed in the case when the delay is much larger than the characteristic period of the solitary uncoupled oscillator.

In this paper, we develop a strong Milstein approximation scheme for solving stochastic delay differential equations (SDDEs). The scheme has convergence order 1. In order to establish the scheme, we prove an infinite-dimensional Itô formula for "tame'' functions acting on the segment process of the solution of an SDDE. It is interesting to note that the presence of the memory in the SDDE requires the use of the Malliavin calculus and the anticipating stochastic analysis of Nualart and Pardoux. Given the nonanticipating nature of the SDDE, the use of anticipating calculus methods in the context of strong approximation schemes appears to be novel.