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Abstract Delay coordinates are a widely used technique to pass from observations of a dynamical system to a representation of the dynamical system as an embedding in Euclidean space. Current proofs show that delay coordinates of a given dynamical system result in embeddings generically with respect to the observation function (Sauer et al., 1991). Motivated by applications of the embedding theory, we consider flow along a single periodic orbit where the observation function is fixed but the dynamics is perturbed. For an observation function that is fixed (as a nonzero linear combination of coordinates) and for the special case of periodic solutions, we prove that delay coordinates result in an embedding generically over the space of vector fields in the C r − 1 topology with r ≥ 2. Highlights • Beginning steps in embedology with a fixed observation function. • Proof of genericity of embedding of periodic solutions in R 3. • Neuroscience citations and the Lorenz example are used to show how periodic solutions arise in applications. [ABSTRACT FROM AUTHOR]

Abstract The first terms of the Wright–Fisher (WF) site frequency spectrum that follow the coalescent approximation are determined precisely, with a view to understanding the accuracy of the coalescent approximation for large samples. The perturbing terms show that the probability of a single mutant in the sample (singleton probability) is elevated in WF but the rest of the frequency spectrum is lowered. A part of the perturbation can be attributed to a mismatch in rates of merger between WF and the coalescent. The rest of it can be attributed to the difference in the way WF and the coalescent partition children between parents. In particular, the number of children of a parent is approximately Poisson under WF and approximately geometric under the coalescent. Whereas the mismatch in rates raises the probability of singletons under WF, its offspring distribution being approximately Poisson lowers it. The two effects are of opposite sense everywhere except at the tail of the frequency spectrum. The WF frequency spectrum begins to depart from that of the coalescent only for sample sizes that are comparable to the population size. These conclusions are confirmed by a separate analysis that assumes the sample size n to be equal to the population size N. Partly thanks to the canceling effects, the total variation distance of WF minus coalescent is 0. 12 ∕ log N for a population sized sample with n = N , which is only 1% for N = 2 × 1 0 4. The coalescent remains a good approximation for the site frequency spectrum of-large samples. [ABSTRACT FROM AUTHOR]

The Kingman coalescent is a commonly used model in genetics, which is often justified with reference to the Wright-Fisher (WF) model. Current proofs of convergence of WF and other models to the Kingman coalescent assume a constant sample size. However, sample sizes have become quite large in human genetics. Therefore, we develop a convergence theory that allows the sample size to increase with population size. If the haploid population size is N and the sample size is N 1 ∕ 3 − ϵ , ϵ > 0 , we prove that Wright-Fisher genealogies involve at most a single binary merger in each generation with probability converging to 1 in the limit of large N . Single binary merger or no merger in each generation of the genealogy implies that the Kingman partition distribution is obtained exactly. If the sample size is N 1 ∕ 2 − ϵ , Wright-Fisher genealogies may involve simultaneous binary mergers in a single generation but do not involve triple mergers in the large N limit. The asymptotic theory is verified using numerical calculations. Variable population sizes are handled algorithmically. It is found that even distant bottlenecks can increase the probability of triple mergers as well as simultaneous binary mergers in WF genealogies. [ABSTRACT FROM AUTHOR]

Spectral integration was deployed by Orszag and co-workers (1977, 1980, 1981) to obtain stable and efficient solvers for the incompressible Navier–Stokes equation in rectangular geometries. Since then several variations of spectral integration have appeared in the literature. In this article, we derive yet more versions of spectral integration. These new versions of spectral integration rely exclusively on banded matrices as opposed to banded matrices bordered with dense rows. In addition, we derive a factored form of spectral integration which relies only on bi- and tri-diagonal matrices. Key properties, such as the accuracy of spectral integration even when Green’s functions are not resolved by the underlying grid and the accuracy of spectral integration in spite of ill-conditioning of underlying linear systems are investigated. Timed comparisons show that reducing spectral integration to bi- and tri-diagonal systems leads to significant speed-ups. [ABSTRACT FROM AUTHOR]

Let z1, z2, . . . , zN be a sequence of distinct grid points. A finite difference formula approximates the m-th derivative f(m)(0) as Σwkf (zk), with wk being the weight at zk. We derive an algorithm for finding the weights wk which uses fewer arithmetic operations and less memory than the algorithm in current use (Fornberg, Mathematics of Computation, vol. 51 (1988), pp. 699-706). The algorithm we derive uses fewer arithmetic operations by a factor of (5m + 5)/4 in the limit of large N. The optimized C++ implementation we describe is a hundred to five hundred times faster than MATLAB. The method of Fornberg is faster by a factor of five in MATLAB, however, and thus remains the most attractive option for MATLAB users. The algorithm generalizes easily to the calculation of spectral differentiation matrices, or equivalently, finite difference weights at several different points with a fixed grid. Unlike the algorithm in current use for the calculation of spectral differentiation matrices, the algorithm we derive suffers from no numerical instability. The order of accuracy of the finite difference formula for f(m)(0) with grid points hzk, 1 ⩽ k ⩽ N, is typically O (hN-m). However, the most commonly used finite difference formulas have an order of accuracy that is higher than normal. For instance, the centered difference approximation (f(h)-2f(0)+f(-h)) /h2 to f''(0) has an order of accuracy equal to 2 not 1. Even unsymmetric finite difference formulas can exhibit such superconvergence or boosted order of accuracy, as shown by the explicit algebraic condition that we derive. If the grid points are real, we prove a basic result stating that the order of accuracy can never be boosted by more than 1. [ABSTRACT FROM AUTHOR]

Abstract: Numerical solvers of the incompressible Navier–Stokes equations have reproduced turbulence phenomena such as the law of the wall, the dependence of turbulence intensities on the Reynolds number, and experimentally observed properties of turbulence energy production. In this article, we begin a sequence of investigations whose eventual aim is to derive and implement numerical solvers that can reach higher Reynolds numbers than is currently possible. Every time step of a Navier–Stokes solver in effect solves a linear boundary value problem. The use of Green’s functions leads to numerical solvers which are highly accurate in resolving the boundary layer, which is a source of delicate but exceedingly important physical effects at high Reynolds numbers. The use of Green’s functions brings with it a need for careful quadrature rules and a reconsideration of time steppers. We derive and implement Green’s function based solvers for the channel flow and plane Couette flow geometries. The solvers are validated by reproducing turbulence phenomena in good agreement with earlier simulations and experiment. [Copyright &y& Elsevier]

Let z1, ..., ZK be distinct grid points. If fk, 0 is the prescribed value of a function at the grid point zk and fk, r the prescribed value of the rth derivative, for 1 < r < nk - 1, the Hermite interpolant is the unique polynomial of degree N - 1 (N = n1 + ⋯ + nK) which interpolates the prescribed function values and function derivatives. We obtain another derivation of a method for Hermite interpolation recently proposed by Butcher et al. [Numer. Algorithms, 56 (2011), pp. 319-347]. One advantage of our derivation is that it leads to an efficient method for updating the barycentric weights. If an additional derivative is prescribed at one of the interpolation points, we show how to update the barycentric coefficients using only O(N) operations. Even in the context of confluent Newton series, a comparably efficient and general method to update the coefficients appears not to be known. If the method is properly implemented, it computes the barycentric weights with fewer operations than other methods and has very good numerical stability even when derivatives of high order are involved. We give a partial explanation of its numerical stability. [ABSTRACT FROM AUTHOR]

Suppose we are given a time series or a signal x(t) for 0 ≤ t ≤ T. We consider the problem of predicting the signal in the interval T

The Lorenz attractor is one of the best-known examples of applied mathematics. However, much of what is known about it is a result of numerical calculations and not of mathematical analysis. As a step toward mathematical analysis, we allow the time variable in the three-dimensional Lorenz system to be complex, hoping that solutions that have resisted analysis on the real line will give up their secrets in the complex plane. Knowledge of singularities being fundamental to any investigation in the complex plane, we build upon earlier work and give a complete and consistent formal development of complex singularities of the Lorenz system using the psi series. The psi series contain two undetermined constants. In addition, the location of the singularity is undetermined as a consequence of the autonomous nature of the Lorenz system. We prove that the psi series converge, using a technique that is simpler and more powerful than that of Hille, thus implying a two-parameter family of singular solutions of the Lorenz system. We pose three questions, answers to which may bring us closer to understanding the connection of complex singularities to Lorenz dynamics. [ABSTRACT FROM AUTHOR]

We derive necessary conditions that traveling wave and other solutions of the Navier–Stokes equations must satisfy in the pipe, Couette, and channel flow geometries. Some conditions are exact and must hold for any traveling wave solution or periodic solution irrespective of the Reynolds number (Re). Other conditions are asymptotic in the limit Re→∞. For the pipe flow geometry, we give computations up to Re=100 000 showing the connection of our asymptotic conditions to critical layers that accompany vortex structures at high Re. [ABSTRACT FROM AUTHOR]

In a 1963 paper, Lorenz inferred that the Lorenz attractor must be an infinite complex of surfaces. We investigate this fractal property of the Lorenz attractor in two ways. Firstly, we obtain explicit plots of the fractal structure of the Lorenz attractor using symbolic dynamics and multiple precision computations of periodic orbits. The method we derive for multiple precision computation is based on iterative refinement and can compute even highly unstable periodic orbits with long symbol sequences with as many as 100 digits of accuracy. Ordinary numerical integrations are much too crude to show even the coarsest splitting of surfaces, and there appear to be no other explicit computations of the fractal structure in the extensive literature about the Lorenz attractor. Secondly, we apply a well known formula that gives the Hausdorff dimension of the Lorenz attractor in terms of the characteristic multipliers of its unstable periodic orbits. The formula converges impressively and the Hausdorff dimension of the Lorenz attractor appears to be 2.0627160. We use comparison with explicit computations of the fractal structure and discuss the accuracy of this formula and its applicability to the Lorenz equations. Additionally, we apply periodic orbit theory to the Lorenz attractor and exhibit its spectral determinant and compute its Lyapunov exponent. [Copyright &y& Elsevier]

The Wisdom-Holman algorithm is an effective method for numerically solving nearly integrable systems. It takes into account the exact solution of the integrable part. If the nearly integrable system is the solar system, for example, the Wisdom-Holman algorithm uses the solution consisting of Keplerian orbits obtained when the interplanetary interactions are ignored. The effectiveness of the algorithm lies in its ability to take long timesteps. We use the Duffing oscillator and Kepler's problem with forcing to deduce how long those timesteps can be. For nearly Keplerian orbits, the timesteps must be at least six per orbital period even when the orbital eccentricity is zero. High eccentricity of the Keplerian orbits constrains the algorithm and forces it to take shorter timesteps. The analysis is applied to the solar system and other problems. [ABSTRACT FROM AUTHOR]

Provides information on a study which derived a numerical algorithm based on the Lindstedt-Poincare technique for computing periodic orbits of dynamical systems. Proof of quadratic convergence; Details on the coupled Josephson junctions; Conclusion.

The Whitney embedding theorem is a basic result of differential topology and it is natural to ask for versions of the embedding theorem for time series data. Embedding theorems for dynamical time series data were claimed by Takens (1981) and proved in a different setting by Sauer, Yorke, and Casdagli (1991). Those results were stated assuming the observation function to be parametrized. A possibly more pertinent setting is to assume the dynamical system to be parametrized, with the observation function fixed, for example, as a projection to a certain coordinate. We prove an embedding theorem in such a setting. Our proof introduces a technique that relies on the notion of Lebesgue points. • Delay embeddings are prevalent with respect to perturbations of the dynamics. • A gap in the work on prevalence by Sauer, Yorke, and Casdagli (1991) is closed. • Discussion of the completeness of the proof of Takens' embedding theorem. [ABSTRACT FROM AUTHOR]