Tasso J. Kaper

Asymptotic analysis of two reduction methods for systems of chemical reactions

Abstract This paper concerns two methods for reducing large systems of chemical kinetics equations, namely, the method of intrinsic low-dimensional manifolds (ILDMs) due to Maas and Pope [Combust. Flame 88 (1992) 239] and an iterative method due to Fraser [J. Chem. Phys. 88 (1988) 4732] and further developed by Roussel and Fraser [J. Chem. Phys. 93 (1990) 1072]. Both methods exploit the separation of fast and slow reaction time scales to find low-dimensional manifolds in the space of species concentrations where the long-term dynamics are played out. The asymptotic expansions of these manifolds (e↓0, where e measures the ratio of the reaction time scales) are compared with the asymptotic expansion of M e , the slow manifold given by geometric singular perturbation theory. It is shown that the expansions of the ILDM and M e agree up to and including terms of O (e) ; the former has an error at O (e 2 ) that is proportional to the local curvature of M 0 . The error vanishes if and only if the curvature is zero everywhere. The iterative method generates, term by term, the asymptotic expansion of M e . Starting from M 0 , the ith application of the algorithm yields the correct expansion coefficient at O (e i ) , while leaving the lower-order coefficients invariant. Thus, after l applications, the expansion is accurate up to and including the terms of O (e l ) . The analytical results are illustrated on a planar system from enzyme kinetics (Michaelis–Menten–Henri) and a model planar system due to Davis and Skodje.

Stability analysis of singular patterns in the 1D Gray-Scott model: a matched asymptotics approach

Abstract In this work, we analyze the linear stability of singular homoclinic stationary solutions and spatially periodic stationary solutions in the one-dimensional Gray-Scott model. This stability analysis has several implications for understanding the recently discovered phenomena of self-replicating pulses. For each solution constructed in A. Doelman et al. [Nonlinearity 10 (1997) 523–563], we analytically find a large open region in the space of the two scaled parameters in which it is stable. Specifically, for each value of the scaled inhibitor feed rate, there exists an interval, whose length and location depend on the solution type, of values of the activator (autocatalyst) decay rate for which the solution is stable. The upper boundary of each interval corresponds to a subcritical Hopf bifurcation point, and the lower boundary is explicitly determined by finding the parameter value where the solution ‘disappears’, i.e., below which it no longer exists as a solution of the steady state system. Explicit asymptotic formulae show that the one-pulse homoclinic solution gains stability first as the second parameter is decreased, and then successively, the spatially periodic solutions (with decreasing period) become stable. Moreover, the stability intervals for different solutions overlap. These stability results are derived via the reduction of a fourth-order slow-fast eigenvalue problem to a second-order nonlocal eigenvalue problem (NLEP). Explicit determination of these stability intervals plays a central role in understanding pulse self-replication. Numerical simulations confirm that the spatially periodic stationary solutions are attractors in the pulse-splitting regime; and, moreover, whenever, for a given solution, the value of the activator decay rate was taken to lie in the regime below that solution s stability interval, initial data close to that solution were observed to evolve toward a different spatially periodic stationary solution, one whose stability interval inclucded the parameter value. The main analytical technique used is that of matched asymptotic expansions.

Analysis of the Computational Singular Perturbation Reduction Method for Chemical Kinetics

Abstract This article is concerned with the asymptotic accuracy of the Computational Singular Perturbation (CSP) method developed by Lam and Goussis [The CSP method for simplifying kinetics, Int. J. Chem. Kin. 26 (1994) 461–486] to reduce the dimensionality of a system of chemical kinetics equations. The method, which is generally applicable to multiple-time scale problems arising in a broad array of scientific disciplines, exploits the presence of disparate time scales to model the dynamics by an evolution equation on a lower-dimensional slow manifold. In this article it is shown that the successive applications of the CSP algorithm generate, order by order, the asymptotic expansion of a slow manifold. The results are illustrated on the Michaelis–Menten–Henri equations of enzyme kinetics.

Tracking invariant manifolds up to exponentially small errors

This work establishes a new tool for proving the existence of multiple-pulse homoclinic orbits in perturbed Hamiltonian systems and general multidimensional singular-perturbation problems. The center-stable and center-unstable manifolds of slow manifolds in these problems intersect transversely at angles that are of the same order as the asymptotically small parameter in the problem, which can be either an amplitude or a frequency. To deal with the difficulties associated with small angles of intersection, we develop the exchange lemma with exponentially small error (ELESE), which is the main technical result of this work. This lemma enables highly accurate tracking of invariant manifolds while orbits on them spend long intervals of time near slow manifolds.

Alpha-Frequency Rhythms Desynchronize over Long Cortical Distances: A Modeling Study

Neocortical networks of excitatory and inhibitory neurons can display alpha(α)-frequency rhythms when an animal is in a resting or unfocused state. Unlike some γ- and β-frequency rhythms, experimental observations in cats have shown that these α-frequency rhythms need not synchronize over long cortical distances. Here, we develop a network model of synaptically coupled excitatory and inhibitory cells to study this asynchrony. The cells of the local circuit are modeled on the neurons found in layer V of the neocortex where α-frequency rhythms are thought to originate. Cortical distance is represented by a pair of local circuits coupled with a delay in synaptic propagation. Mathematical analysis of this model reveals that the h and T currents present in layer V pyramidal (excitatory) cells not only produce and regulate the α-frequency rhythm but also lead to the occurrence of spatial asynchrony. In particular, these inward currents cause excitation and inhibition to have nonintuitive effects in the network, with excitation delaying and inhibition advancing the firing time of cells; these reversed effects create the asynchrony. Moreover, increased excitatory to excitatory connections can lead to further desynchronization. However, the local rhythms have the property that, in the absence of excitatory to excitatory connections, if the participating cells are brought close to synchrony (for example, by common input), they will remain close to synchrony for a substantial time.

Coupled pulsation and translation of two gas bubbles in a liquid

We present and analyse a model for the spherical pulsations and translational motions of a pair of interacting gas bubbles in an incompressible liquid. The model is derived rigorously in the context of potential flow theory and contains all terms up to and including fourth order in the inverse separation distance between the bubbles. We use this model to study the cases of both weak and moderate applied acoustic forcing. For weak acoustic forcing, the radial pulsations of the bubbles are weakly coupled, which allows us to obtain a nonlinear time-averaged model for the relative distance between the bubbles. The two parameters of the time-averaged model classify four dierent dynamical regimes of relative translational motion, two of which correspond to the attraction and repulsion of classical secondary Bjerknes theory. Also predicted is a pattern in which the bubbles exhibit stable, time-periodic translational oscillations along the line connecting their centres, and another pattern in which there is an unstable separation distance such that bubble pairs can either attract or repel each other depending on whether their initial separation distance is smaller or larger than this value. Moreover, it is shown that the full governing equations possess the dynamics predicted by the time-averaged model. We also study the case of moderateamplitude acoustic forcing, in which the bubble pulsations are more strongly coupled to each other and bubble translation also aects the radial pulsations. Here, radial harmonics and nonlinear phase shifting play a signicant role, as bubble pairs near resonances are observed to translate in patterns opposite to those predicted by classical secondary Bjerknes theory. In this work, dynamical systems techniques and the method of averaging are the primary mathematical methods that are employed.

Introduction to Focus Issue: Mixed Mode Oscillations: Experiment, Computation, and Analysis

Mixed mode oscillations (MMOs) occur when a dynamical system switches between fast and slow motion and small and large amplitude. MMOs appear in a variety of systems in nature, and may be simple or complex. This focus issue presents a series of articles on theoretical, numerical, and experimental aspects of MMOs. The applications cover physical, chemical, and biological systems.

Multi-bump orbits homoclinic to resonance bands

We establish the existence of several classes of multi-bump orbits homoclinic to resonance bands for completely-integrable Hamiltonian systems subject to small-amplitude Hamiltonian or dissipative perturbations. Each bump is a fast excursion away from the resonance band, and the bumps are interspersed with slow segments near the resonance band. The homoclinic orbits, which include multi-bump Silnikov orbits, connect equilibria and periodic orbits in the resonance band. The main tools we use in the existence proofs are the exchange lemma with exponentially small error and the existence theory of orbits homoclinic to resonance bands which make only one fast excursion away from the resonance bands.

Fast and Slow Dynamics for the Computational Singular Perturbation Method

The computational singular perturbation (CSP) method ofLam and Goussis is an iterative method to reduce the dimensionality of systems of ordinary differential equations with multiple time scales. In [J. Nonlinear Sci., 14 (2004), pp. 59--91], the authors of this paper showed that each iteration of the CSP algorithm improves the approximation of the slow manifold by one order. In this paper, it is shown that the CSP method simultaneously approximates the tangent spaces to the fast fibers along which solutions relax to the slow manifold. Again, each iteration adds one order of accuracy. In some studies, the output of the CSP algorithm is postprocessed by linearly projecting initial data onto the slow manifold along these approximate tangent spaces. These projections, in turn, also become successively more accurate.

An analytical study of transport in Stokes flows exhibiting large-scale chaos in the eccentric journal bearing

In the present work, we apply new tools from the field of adiabatic dynamical systems theory to make quantitative predictions of various important mixing quantities in quasi-steady Stokes flows which possess slowly varying saddle stagnation points. Many of these quantities can be obtained before experiments or numerical simulations are performed using only knowledge of the streamlines in steady-state flows and the externally determined flow parameters. The location and size of the main region in which mixing occurs is determined to leading order by the slowly sweeping instantaneous stagnation streamlines. Tracer patches get stretched by an amount O (1/e) during each period, and the average striation thickness of the patch decreases by a factor of e in the same time. Also, the rate of stretching of material interfaces is bounded from below with an analytically obtained exponentially growing lower bound. Finally, the highly regular appearance of islands in quasi-steady Stokes’ flows is explained using an extension of the KAM theory. As an example to illustrate these results, we study the transport of passive scalars in a low Reynolds number flow in the two-dimensional eccentric journal bearing when the angular velocities of the cylinders are slowly modulated, continuously and periodically in time, with frequency e. In contrast to the flows usually studied with dynamical systems, these slowly varying systems are singular perturbation (apparently far from integrable) problems exhibiting large-scale chaos, in which the non-integrability is due to the slow, continuous O (1) modulation of the position of the saddle stagnation point and the two streamlines stagnating on it.