ISSN:

1531-3492

eISSN:

1553-524X

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## Discrete & Continuous Dynamical Systems - B

January 2008 , Volume 9 , Issue 1

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2008, 9(1): 1-10
doi: 10.3934/dcdsb.2008.9.1

*+*[Abstract](1801)*+*[PDF](255.6KB)**Abstract:**

We propose and analyze two different Bayesian online algorithms for learning in discrete Hidden Markov Models and compare their performance with the already known Baldi-Chauvin Algorithm. Using the Kullback-Leibler divergence as a measure of generalization we draw learning curves in simplified situations for these algorithms and compare their performances.

2008, 9(1): 11-36
doi: 10.3934/dcdsb.2008.9.11

*+*[Abstract](1531)*+*[PDF](684.3KB)**Abstract:**

We consider focusing nonlinear Schrödinger equations (NLS), in the $L^2$-critical and supercritical cases. We present a systematic numerical investigation of the dependence of the blow-up time on properties of the data or on the (parameters of the) equation in three cases: dependence on the strength of the nonlinearity in the equation when the initial data is fixed; dependence on the strength of a damping term in the equation when the initial data is fixed; and dependence upon the strength of a quadratic oscillation in the initial data when the equation and the initial profile are fixed. For some cases, analytic results are available and presented. In most situations our numerical counterexamples show that monotonicity in the evolution of the blow-up time does not occur. In addition they show that in certain regimes the blow-up time is very sensitive to the different parameters that we modulate.

Our numerical solutions are very reliable since not only we test independence on the precise setting of the numerical problem (size of the periodic domain, discretization etc.) but we compare the same simulations with two different methods in two independent codes: a spectral time splitting code and a relaxation method, with results identical at the order of precision.

2008, 9(1): 37-46
doi: 10.3934/dcdsb.2008.9.37

*+*[Abstract](1323)*+*[PDF](146.4KB)**Abstract:**

Aging is an abundant property of materials, populations, and networks. We consider some classes of cellular automata (Deterministic Walks in Random Environments) where the process of aging is described by a time dependent function, called a rigidity of the environment. Asymptotic laws for the dynamics of perturbations propagating in such environments with aging are obtained.

2008, 9(1): 47-64
doi: 10.3934/dcdsb.2008.9.47

*+*[Abstract](1681)*+*[PDF](206.8KB)**Abstract:**

Stochastic differential equations with Poisson driven jumps of random magnitude are popular as models in mathematical finance. Strong, or pathwise, simulation of these models is required in various settings and long time stability is desirable to control error growth. Here, we examine strong convergence and mean-square stability of a class of implicit numerical methods, proving both positive and negative results. The analysis is backed up with numerical experiments.

2008, 9(1): 65-73
doi: 10.3934/dcdsb.2008.9.65

*+*[Abstract](1964)*+*[PDF](156.8KB)**Abstract:**

In this paper, we consider a nonlocal parabolic initial value problem that models a single species which is diffusing, aggregating, reproducing and competing for space and resources. We establish a comparison principle and construct monotone sequences to show the existence and uniqueness of the solution to the problem. We also analyze the long-time behavior of the solution.

2008, 9(1): 75-82
doi: 10.3934/dcdsb.2008.9.75

*+*[Abstract](1549)*+*[PDF](368.0KB)**Abstract:**

The dynamics of the transformation $F: (x,y)\rightarrow (x(4-x-y),xy)$ defined on the plane triangle $\Delta$ of vertices $(0,0)$, $(0,4)$ and $(4,0)$ plays an important role in the behaviour of the Lotka--Volterra map. In 1993, A. N. SharkovskiĬ (Proc. Oberwolfach 20/1993) stated some problems on it, in particular a question about the trasitivity of $F$ was posed. The main aim of this paper is to prove that for every non--empty open set $\mathcal{U} \subset \Delta$ there is an integer $n_{0}$ such that for each $n>n_{0}$ it is $F^{n}(\mathcal{U}) \supseteq \Delta \setminus P_{\varepsilon}$, where $P_{\varepsilon} = \{ (x,y) \in D : y<\beta, \mbox{ $where$ F(t,\varepsilon)=(\alpha,\beta) \mbox{ and } t \in[0,2] \}$ and $\varepsilon \rightarrow 0$ for $n \rightarrow \infty$. Consequently, we show that the map $F$ is transitive, it is not topologically exact and it is almost topologically exact. Additionally, we prove that the union of all preimages of the point $(1,2)$ is a dense subset of $\Delta$.

2008, 9(1): 83-101
doi: 10.3934/dcdsb.2008.9.83

*+*[Abstract](1744)*+*[PDF](217.4KB)**Abstract:**

We study the coupled map lattice model of tree dispersion. Under quite general conditions on the nonlinearity of the local growth function and the dispersion (coupling) function, we show that when the maximal dispersal distance is finite and the spatial redistribution pattern remains unchanged in time, the moving front will always converge in the strongest sense to an asymptotic state: a traveling wave with finite length of the wavefront. We also show that when the climate becomes more favorable to growth or germination, the front at any nonzero density level will have a positive acceleration. An estimation of the magnitude of the acceleration is given.

2008, 9(1): 103-128
doi: 10.3934/dcdsb.2008.9.103

*+*[Abstract](1352)*+*[PDF](955.2KB)**Abstract:**

This work presents a hierarchy of mathematical models for describing the motion of phototactic bacteria, i.e., bacteria that move towards light. Based on experimental observations, we conjecture that the motion of the colony towards light depends on certain group dynamics. This group dynamics is assumed to be encoded as an individual property of each bacterium, which we refer to as ’excitation’. The excitation of each individual bacterium changes based on the excitation of the neighboring bacteria. Under these assumptions, we derive a stochastic model for describing the evolution in time of the location of bacteria, the excitation of individual bacteria, and a surface memory effect. A discretization of this model results in an interacting stochastic many-particle system. The third, and last model is a system of partial differential equations that is obtained as the continuum limit of the stochastic particle system. The main theoretical results establish the validity of the new system of PDEs as the limit dynamics of the multi-particle system.

2008, 9(1): 129-143
doi: 10.3934/dcdsb.2008.9.129

*+*[Abstract](1258)*+*[PDF](225.3KB)**Abstract:**

In this paper we consider an evolution problem which model the frictional skin effects in piezoelectricity. The model consists of the system of the hemivariational inequality of hyperbolic type for the displacement and the time dependent elliptic equation for the electric potential. In the hemivariational inequality the viscosity term is noncoercive and the friction forces are derived from a nonconvex superpotential through the generalized Clarke subdifferential. The existence of weak solutions is proved by embedding the problem into a class of second order evolution inclusions and by applying a parabolic regularization method.

2008, 9(1): 145-162
doi: 10.3934/dcdsb.2008.9.145

*+*[Abstract](1416)*+*[PDF](4932.0KB)**Abstract:**

Phase-and-frequency-locking phenomena among coupled biological oscillators are a topic of current interest, in particular to neuroscience. In the case of mono-directionally pulse-coupled oscillators, phase-locking is well understood, where the phenomenon is globally described by Arnold tongues. Here, we develop the tools that allow corresponding investigations to be made for more general pulse-coupled networks. For two bi-directionally coupled oscillators, we prove the existence of three-dimensional Arnold tongues that mediate from the mono- to the bi-directional coupling topology. Under this transformation, the coupling strength at which the onset of chaos is observed is invariant. The developed framework also allows us to compare information transfer in feedforward versus recurrent networks. We find that distinct laws govern the propagation of phase-locked spike-time information, indicating a qualitative difference between classical artificial vs. biological computation.

2008, 9(1): 163-182
doi: 10.3934/dcdsb.2008.9.163

*+*[Abstract](1366)*+*[PDF](220.5KB)**Abstract:**

In this paper, some patch recovery methods are proposed and analyzed for finite element approximation of elasticity problems using quadrilateral meshes. Under a mild mesh condition, superconvergence results are established for the recovered stress tensors. Consequently,

*a posteriori*error estimators based on the recovered stress tensors are asymptotically exact.

2008, 9(1): 183-198
doi: 10.3934/dcdsb.2008.9.183

*+*[Abstract](1390)*+*[PDF](512.2KB)**Abstract:**

A digital spiking neuron is used to generate spike-trains of variable spike-intervals. Multiple co-existing periodic spike-trains are observed, depending on initial states. By focusing on a simple parameter case, we clarify the number of co-existing periodic spike-trains and determine their periods theoretically. Using a spike-interval modulation, the spike-train is coded by a digital sequence. We clarify that the set of co-existing periodic spike-trains is in a one-to-one relation to a set of binary numbers. We finally discuss to what extent these theoretical results may provide the mathematical basis for technological applications.

2018 Impact Factor: 1.008

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