Erik S. Van Vleck

Experimental demonstration of chaotic instability in biological nitrification

Biological nitrification (that is, NH3 → NO2− → NO3−) is a key reaction in the global nitrogen cycle (N-cycle); however, it is also known anecdotally to be unpredictable and sometimes fails inexplicably. Understanding the basis of unpredictability in nitrification is critical because the loss or impairment of this function might influence the balance of nitrogen in the environment and also has biotechnological implications. One explanation for unpredictability is the presence of chaotic behavior; however, proving such behavior from experimental data is not trivial, especially in a complex microbial community. Here, we show that chaotic behavior is central to stability in nitrification because of a fragile mutualistic relationship between ammonia-oxidizing bacteria (AOB) and nitrite-oxidizing bacteria (NOB), the two major guilds in nitrification. Three parallel chemostats containing mixed microbial communities were fed complex media for 207 days, and nitrification performance, and abundances of AOB, NOB, total bacteria and protozoa were quantified over time. Lyapunov exponent calculations, supported by surrogate data and other tests, showed that all guilds were sensitive to initial conditions, suggesting broad chaotic behavior. However, NOB were most unstable among guilds and displayed a different general pattern of instability. Further, NOB variability was maximized when AOB were most unstable, which resulted in erratic nitrification including significant NO2− accumulation. We conclude that nitrification is prone to chaotic behavior because of a fragile AOB–NOB mutualism, which must be considered in all systems that depend on this critical reaction.

On the Compuation of Lyapunov Exponents for Continuous Dynamical Systems

In this paper, we consider discrete and continuous QR algorithms for computing all of the Lyapunov exponents of a regular dynamical system. We begin by reviewing theoretical results for regular systems and present general perturbation results for Lyapunov exponents. We then present the algorithms, give an error analysis of them, and describe their implementation. Finally, we give several numerical examples and some conclusions.

Unitary integrators and applications to continuous orthonormalization techniques

In this paper the issue of integrating matrix differential systems whose solutions are unitary matrices is addressed. Such systems have skew-Hermitian coefficient matrices in the linear case and a ...

Computation of a few Lyapunov exponents for continuous and discrete dynamical systems

In this paper, we consider QR-based method for computing the first few Lyapunov exponents of continuous and discrete dynamical systems. Algorithmic developments are discussed. Implementation details, error estimation and testing are also given.

DYNAMICS OF LATTICE DIFFERENTIAL EQUATIONS

In this paper recent work on the dynamics of lattice differential equations is surveyed. In particular, results on propagation failure and lattice induced anisotropy for traveling wave or plane wav...

Lyapunov Spectral Intervals: Theory and Computation

Different definitions of spectra have been proposed over the years to characterize the asymptotic behavior of nonautonomous linear systems. Here, we consider the spectrum based on exponential dichotomy of Sacker and Sell [J. Differential Equations, 7 (1978), pp. 320--358] and the spectrum defined in terms of upper and lower Lyapunov exponents. A main goal of ours is to understand to what extent these spectra are computable. By using an orthogonal change of variables transforming the system to upper triangular form, and the assumption of integral separation for the diagonal of the new triangular system, we justify how popular numerical methods, the so-called continuous QR and SVD approaches, can be used to approximate these spectra. We further discuss how to verify the property of integral separation, and hence how to a posteriori infer stability of the attained spectral information. Finally, we discuss the algorithms we have used to approximate the Lyapunov and Sacker--Sell spectra and present some numerical results.

Computation of Mixed Type Functional Differential Boundary Value Problems

We study boundary value differential-difference equations where the difference terms may contain both advances and delays. Special attention is paid to connecting orbits, in particular to the model- ing of the tails after truncation to a finite interval, and we reformulate these problems as functional differential equations over a bounded domain. Connecting orbits are computed for several such prob- lems including discrete Nagumo equations, an Ising model, and Frenkel-Kontorova type equations. We describe the collocation boundary value problem code used to compute these solutions, and the numerical analysis issues which arise, including linear algebra, boundary functions and conditions, and convergence theory for the collocation approximation on finite intervals. 1. Introduction. Nonlinear spatially discrete diffusion equations occur as models in many areas of science and engineering. When the underlying mathematical models contain differ- ence terms or delays as well as derivative terms, the resulting differential-difference equations present challenging analytical and computational problems. We demonstrate how functional differential boundary value problems with advances and delays arise from such models and describe a general approach for the numerical computation of solutions. Solutions are approx- imated for several such problems, and the numerical issues arising in their computation are

A Variant of Newton's Method for the Computation of Traveling Waves of Bistable Differential-Difference Equations

We consider a variant of Newtons method for solving nonlinear differential-difference equations arising from the traveling wave equations of a large class of nonlinear evolution equations. Building on the Fredholm theory recently developed by Mallet-Paret we prove convergence of the method. The utility of the method is demonstrated with a series of examples.

Traveling Wave Solutions for Bistable Differential-Difference Equations with Periodic Diffusion

We consider traveling wave solutions to spatially discrete reaction-diffusion equations with nonlocal variable diffusion and bistable nonlinearities. To find the traveling wave solutions we introduce an ansatz in which the wave speed depends on the underlying lattice as well as on time. For the case of spatially periodic diffusion we obtain analytic solutions for the traveling wave problem using a piecewise linear nonlinearity. The formula for the wave forms is implicitly defined in the general periodic case and we provide an explicit formula for the case of period two diffusion. We present numerical studies for time t=0 fixed and for the time evolution of the traveling waves. When t=0 we study the cases of homogeneous, period two, and period four diffusion coefficients using a cubic nonlinearity, and uncover, numerically, a period doubling bifurcation in the wave speed versus detuning parameter relation. For the time evolution case we also discover a detuning parameter dependent bifurcation in observed p...

Spatially Discrete FitzHugh--Nagumo Equations

We consider pulse and front solutions to a spatially discrete FitzHugh--Nagumo equation that contains terms to represent both depolarization and hyperpolarization of the nerve axon. We demonstrate a technique for deriving candidate solutions for the McKean nonlinearity and present and apply solvability conditions necessary for existence. Our equation contains both spatially continuous and discrete diffusion terms.