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Centered around dynamics, DCDSB is an interdisciplinary journal focusing on the interactions between mathematical modeling, analysis and scientific computations. The mission of the Journal is to bridge mathematics and sciences by publishing research papers that augment the fundamental ways we interpret, model and predict scientific phenomena. The Journal covers a broad range of areas including chemical, engineering, physical and life sciences. A more detailed indication is given by the subject interests of the members of the Editorial Board.
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TOP 10 Most Read Articles in DCDSB, January 2015
1 
Approximation of attractors of nonautonomous dynamical systems
Volume 5, Number 2, Pages: 215  238, 2005
Bernd Aulbach,
Martin Rasmussen
and Stefan Siegmund
Abstract
Full Text
Related Articles
This paper is devoted to the numerical approximation of attractors. For general nonautonomous dynamical systems we first introduce a new
type of attractor which includes some classes of noncompact attractors such as
unbounded unstable manifolds. We then adapt two cell mapping algorithms
to the nonautonomous setting and use the computer program GAIO for the
analysis of an explicit example, a twodimensional system of nonautonomous
difference equations. Finally we present numerical data which indicate a bifurcation of nonautonomous attractors in the Duffingvan der Pol oscillator.

2 
Modelling the dynamics of endemic malaria in growing populations
Volume 4, Number 4, Pages: 1173  1202, 2004
G.A. Ngwa
Abstract
Full Text
Related Articles
A mathematical model for endemic malaria involving variable human
and mosquito populations is analysed. A threshold parameter $R_0$ exists
and the disease can persist if and only if $R_0$ exceeds $1$. $R_0$ is seen to be a generalisation
of the basic reproduction ratio associated with the RossMacdonald
model for malaria transmission. The disease free equilibrium always exist and
is globally stable when $R_0$ is below $1$. A perturbation analysis is used to approximate
the endemic equilibrium in the important case where the disease
related death rate is nonzero, small but significant. A diffusion approximation
is used to approximate the quasistationary distribution of the associated stochastic
model. Numerical simulations show that when $R_0$ is distinctly greater
than $1$, the endemic deterministic equilibrium is globally stable. Furthermore,
in quasistationarity, the stochastic process undergoes oscillations about
a mean population whose size can be approximated by the stable endemic
deterministic equilibrium.

3 
A spectral formulation of the Nekhoroshev theorem and its relevance for numerical and experimental data analysis
Volume 1, Number 1, Pages: 1  28, 2001
Massimiliano Guzzo
and Giancarlo Benettin
Abstract
Full Text
Related Articles
In this paper we provide an analytical characterization of the Fourier spectrum of
the solutions of quasiintegrable Hamiltonian systems, which completes the Nekhoroshev theorem
and looks particularly suitable to describe resonant motions. We also discuss the application of
the result to the analysis of numerical and experimental data. The comparison of the rigorous
theoretical estimates with numerical results shows a quite good agreement. It turns out that an
observation of the spectrum for a relatively short time scale (of order $1/\sqrt{\varepsilon}$, where $\varepsilon$ is a natural
perturbative parameter) can provide information on the behavior of the system for the much larger
Nekhoroshev times.

4 
Analysis of a chemostat model for bacteria and virulent bacteriophage
Volume 2, Number 4, Pages: 495  520, 2002
Edoardo Beretta,
Fortunata Solimano
and Yanbin Tang
Abstract
Full Text
Related Articles
The purpose of this paper is to study the mathematical properties of the
solutions of a model for bacteria and virulent bacteriophage system in a
chemostat. A general model was first proposed by Levin, Stewart and Chao
[13] and then, a specific one, by Lenski and Levin [12]. The numerical
simulations come from the experimental data referred in [12,13]. In our
Knowledge the analysis presented herefollowing is the first mathematical
attempt to analyse the model of bacteria and virulent bacteriophage and
presents two fresh frontiers: 1) the modeling of delay (latency period)
incorporating the realistic through time death rate in linear stability
analysis brings to characteristic equations with delay dependent parameters
for which only recently Beretta and Kuang [5] provided a geometric stability
switch criterion which application is presented along the paper; 2) the
modelling of the dynamics through three full delay stages can be reduced to
two using the integral representation for the density of infected
bacteria. The basic properties of the model which are investigated are the
existence of equilibria, positive invariance and boundedness of solutions
and permanence results. Second, using the geometric stability switch
criterion in the delay differential system with delay dependent parameters,
we present the local asymptotic stability of the equilibria by analyzing the
corresponding characteristic equation which coefficients depend on the time
delay (the latency period). Numerical simulations are
presented to illustrate the results of local stability. Then, we study the
global asymptotic stability of the boundary equilibria via Liapunov
functional method. Finally, we give a discussion about the model.

5 
A hierarchy of cancer models and their mathematical challenges
Volume 4, Number 1, Pages: 147  159, 2003
Avner Friedman
Abstract
Full Text
Related Articles
A variety of PDE models for tumor growth have been developed
in the last three decades. These models are based on mass conservation laws and on reactiondiffusion processes within the tumor.

6 
Bifurcations of an SIRS epidemic model with nonlinear
incidence rate
Volume 15, Number 1, Pages: 93  112, 2010
Zhixing Hu,
Ping Bi,
Wanbiao Ma
and Shigui Ruan
Abstract
References
Full Text
Related Articles
The main purpose of this paper is to explore the dynamics of an
epidemic model with a general nonlinear incidence
$\beta SI^p/(1+\alpha I^q)$. The existence and stability
of multiple endemic equilibria of the epidemic model are analyzed.
Local bifurcation theory is applied to explore the rich dynamical
behavior of the model. Normal forms of the model are derived for
different types of bifurcations, including Hopf and BogdanovTakens
bifurcations. Concretely speaking, the first Lyapunov coefficient is
computed to determine various types of Hopf bifurcations. Next,
with the help of the BogdanovTakens normal form, a family of
homoclinic orbits is arising when a Hopf and a saddlenode
bifurcation merge. Finally, some numerical results and simulations
are presented to illustrate these theoretical results.

7 
A collocation method for the numerical
Fourier analysis of quasiperiodic functions.
I: Numerical tests and examples
Volume 14, Number 1, Pages: 41  74, 2010
Gerard Gómez,
Josep–Maria Mondelo
and Carles Simó
Abstract
Full Text
Related Articles
The purpose of this paper is to develop a numerical procedure for the
determination of frequencies and amplitudes of a quasiperiodic
function, starting from equallyspaced samples of it on a finite time
interval. It is based on a collocation method in frequency domain.
Strategies for the choice of the collocation harmonics are discussed,
in order to ensure good conditioning of the resulting system of
equations. The accuracy and robustness of the procedure is checked
with several examples. The paper is ended with two applications of its
use as a dynamical indicator. The theoretical support for the method
presented here is given in a companion paper [21].

8 
On the Hamiltonian dynamics and geometry of the Rabinovich system
Volume 15, Number 3, Pages: 789  823, 2011
Răzvan M. Tudoran
and Anania Gîrban
Abstract
References
Full Text
Related Articles
In this paper, we describe some relevant dynamical and geometrical properties of the Rabinovich system from the Poisson geometry and the dynamics point of view. Starting with a LiePoisson realization of the Rabinovich system, we determine and then completely analyze the Lyapunov stability of all the equilibrium states of the system, study the existence of periodic orbits, and then using a new geometrical approach, we provide the complete dynamical behavior of the Rabinovich system, in terms of the geometric semialgebraic properties of a twodimensional geometric figure, associated with the problem. Moreover, in tight connection with the dynamical behavior, by using this approach, we also recover all the dynamical objects of the system (e.g. equilibrium states, periodic orbits, homoclinic and heteroclinic connections). Next, we integrate the Rabinovich system by Jacoby elliptic functions, and give some Lax formulations of the system. The last part of the article discusses some numerics associated with the Poisson geometrical structure of the Rabinovich system.

9 
On the propagation of tsunami waves, with emphasis on the tsunami of 2004
Volume 12, Number 3, Pages: 525  537, 2009
Adrian Constantin
Abstract
Full Text
Related Articles
In this review paper we discuss the range of validity of nonlinear dispersive
integrable equations for the modelling of the propagation of tsunami waves. For the 2004
tsunami the available measurements and the geophyiscal scales involved
rule out a connection between integrable nonlinear wave equations and tsunami dynamics.

10 
Modelling the effect of imperfect vaccines on disease epidemiology
Volume 4, Number 4, Pages: 999  1012, 2004
S.M. Moghadas
Abstract
Full Text
Related Articles
We develop a mathematical model to monitor the effect of imperfect
vaccines on the transmission dynamics of infectious diseases. It is
assumed that the vaccine efficacy is not $100\%$ and may wane with
time. The model will be analyzed using a new technique based on some
results related to the Poincaré index of a piecewise smooth Jordan
curve defined as the boundary of a positively invariant region for
the model. Using global analysis of the model, it is shown that
reducing the basic reproductive number ($\mathcal{R}_0$) to values less than
one no longer guarantees disease eradication. This analysis is
extended to determine the threshold level of vaccination coverage
that guarantees disease eradication.

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