Spot Self-Replication and Dynamics for the Schnakenburg Model in a Two-Dimensional Domain

The dynamical behavior of multi-spot solutions in a two-dimensional domain Ω is analyzed for the two-component Schnakenburg reaction–diffusion model in the singularly perturbed limit of small diffusivity ε for one of the two components. In the limit ε→0, a quasi-equilibrium spot pattern in the region away from the spots is constructed by representing each localized spot as a logarithmic singularity of unknown strength S _j for j=1,…,K at unknown spot locations x _j ∈Ω for j=1,…,K. A formal asymptotic analysis, which has the effect of summing infinite logarithmic series in powers of −1/log ε, is then used to derive an ODE differential algebraic system (DAE) for the collective coordinates S _j and x _j for j=1,…,K, which characterizes the slow dynamics of a spot pattern. This DAE system involves the Neumann Green’s function for the Laplacian. By numerically examining the stability thresholds for a single spot solution, a specific criterion in terms of the source strengths S _j , for j=1,…,K, is then formulated to theoretically predict the initiation of a spot-splitting event. The analytical theory is illustrated for spot patterns in the unit disk and the unit square, and is compared with full numerical results computed directly from the Schnakenburg model.      

Periodic solutions for planar autonomous systems with nonsmooth periodic perturbations

In this paper we consider a class of planar autonomous systems having an isolated limit cycle x₀ of smallest period T>0 such that the associated linearized system around it has only one characteristic multiplier with absolute value 1. We consider two functions, defined by means of the eigenfunctions of the adjoint of the linearized system, and we formulate conditions in terms of them in order to have the existence of two geometrically distinct families of T-periodic solutions of the autonomous system when it is perturbed by nonsmooth T-periodic nonlinear terms of small amplitude. We also show the convergence of these periodic solutions to x₀ as the perturbation disappears and we provide an estimation of the rate of convergence. The employed methods are mainly based on the theory of topological degree and its properties that allow less regularity on the data than that required by the approach, commonly employed in the existing literature on this subject, based on various versions of the implicit function theorem.     

Asymptotics for steady‐state voltage potentials in a bidimensional highly contrasted medium with thin layer

We study the behaviour of steady‐state voltage potentials in two kinds of bidimensional media composed of material of complex permittivity equal to 1 (respectively, α) surrounded by a thin membrane of thickness h and of complex permittivity α (respectively, 1). We provide in both cases a rigorous derivation of the asymptotic expansion of steady‐state voltage potentials at any order as h tends to zero, when Neumann boundary condition is imposed on the exterior boundary of the thin layer. Our complex parameter α is bounded but may be very small compared to 1, hence our results describe the asymptotics of steady‐state voltage potentials in all heterogeneous and highly heterogeneous media with thin layer. The asymptotic terms of the potential in the membrane are given explicitly in local coordinates in terms of the boundary data and of the curvature of the domain, while these of the inner potential are the solutions to the so‐called dielectric formulation with appropriate boundary conditions. The error estimates are given explicitly in terms of h and α with appropriate Sobolev norm of the boundary data. We show that the two situations described above lead to completely different asymptotic behaviours of the potentials. Copyright © 2007 John Wiley & Sons, Ltd.     

Averaging on slow and fast cycles of a three time scale system

Pontryagin–Rodygin?s Theorem for slow and fast systems describes the slow drift during the rolling up of the trajectories around the cycles of the fast dynamics. This drift is approximated by the averaging on the cycles. The calculation of this average is generally a difficult task since it requires the knowledge of the closed orbits and their periods. We present two paradigms of three time scale systems where we can overcome this limitation. It is the case of systems the fast dynamics of which have cycles with relaxation presenting or not a canard phenomenon. We can not apply Pontryagin–Rodygin?s Theorem to these systems because their fast equation is itself singularly perturbed. We also investigate the extension of the results to unbounded time intervals. The results are stated classically and proved within the framework of nonstandard analysis.     

含非线性扰动的变时滞中立系统时滞相关稳定性新条件

研究一类含非线性扰动和混合时变时滞的中立型系统时滞相关鲁棒稳定性问题.通过构造包含三重积分项的Lyapunov-Krasovskii的泛函,结合积分不等式和自由权矩阵技术,建立了基于线性矩阵不等式(LMI)形式的离散时滞和中立时滞均相关的稳定性新判据.判据扩大了系统稳定所允许的最大时滞上界范围,具有更低的保守性.利用MATLAB的LMI工具箱进行了数值仿真,仿真算例验证了所提出的稳定性判据的有效性.     

Singularly Perturbed Linear and Semilinear Hyperbolic Systems: Kinetic Theory Approach to Some Folk’s Theorems

The fact that a solution of a wave equation with strong damping can be approximated by a solution of an appropriate diffusion equation has been common knowledge to physicists and mathematicians for a few decades. In numerous branches of science one can find attempts to prove this results by various ad hoc methods. However, it seems to be difficult to trace a rigorous and exhaustive treatment of the subject. By using a recently developed asymptotic procedure, first applied in kinetic theory, we present a systematic approach to a wide spectrum of singularly perturbed hyperbolic problems and we also hope that our method will provide new insight into the field.     

Aspects of Scalar Brane-World Cosmological Perturbations

We summarize an approach to deal with scalar brane-world cosmological perturbations based on Mukohyama's master equation. We also give its relation to one based on perturbing the effective Einstein's equations on the brane (involving the Weyl fluid).     

Decomposition of Vocal Cycle Length Perturbations into Vocal Jitter and Vocal Microtremor, and Comparison of Their Size in Normophonic Speakers

A statistical method that enables raw vocal cycle length perturbations to be decomposed into perturbations ascribed to vocal jitter and vocal tremor is presented, together with a comparison of the size of jitter and tremor. The method is based on a time series model that splits the vocal cycle length perturbations into uncorrelated cycle-to-cycle perturbations ascribed to vocal jitter and supra-cycle perturbations ascribed to vocal tremor. The corpus was composed of 114 vocal cycle length time series for sustained vowels [a], [i], and [u] produced by 22 male and 16 female normophonic speakers. The results were the following. First, 100 out of 114 time series were decomposed successfully by means of the time series model. Second, vocal perturbations ascribed to tremor were significantly larger than perturbations ascribed to jitter. Third, the correlation between vocal jitter and vocal tremor was moderate, but statistically significant. Fourth, small but statistically significant differences were observed among the three vowel timbres in the relative jitter and the arithmetic difference of jitter and tremor. Fifth, the differences between male and female speakers were not statistically significant in the relative raw perturbations, the relative jitter, or the modulation level owing to tremor.     

Newtonian versus relativistic nonlinear cosmology

Both for the background world model and its linear perturbations Newtonian cosmology coincides with the zero-pressure limits of relativistic cosmology. However, such successes in Newtonian cosmology are not purely based on Newton's gravity, but are rather guided ones by previously known results in Einstein's theory. The action-at-a-distance nature of Newton's gravity requires further verification from Einstein's theory for its use in the large-scale nonlinear regimes. We study the domain of validity of the Newtonian cosmology by investigating weakly nonlinear regimes in relativistic cosmology assuming a zero-pressure and irrotational fluid. We show that, first, if we ignore the coupling with gravitational waves the Newtonian cosmology is exactly valid even to the second order in perturbation. Second, the pure relativistic correction terms start appearing from the third order. Third, the correction terms are independent of the horizon scale and are quite small in the large-scale near the horizon. These conclusions are based on our special (and proper) choice of variables and gauge conditions. In a complementary situation where the system is weakly relativistic but fully nonlinear (thus, far inside the horizon) we can employ the post-Newtonian approximation. We also show that in the large-scale structures the post-Newtonian effects are quite small. As a consequence, now we can rely on the Newtonian gravity in analyzing the evolution of nonlinear large-scale structures even near the horizon volume.