Peter Bates, Michigan State
Approximately Invariant Manifolds and Global Dynamics of Spike States
We investigate the existence of a true invariant manifold given an approximately invariant manifold for an infinite-dimensional dynamical system. We prove that if the given manifold is approximately invariant and approximately normally hyperbolic, then the dynamical system has a true invariant manifold nearby. We apply this result to reveal the global dynamics of boundary spike states for the generalized Allen-Cahn equation.
Arno Berger, University of Alberta
Digits and dynamics: from finite data to infinite dimensions
Understanding the distribution of digits and mantissae in numerical data,as generated by a dynamical system or otherwise, is a challenge that can be tackled in many different ways. This talk will discuss recent work utilising quantization, sharp distortion estimates, and dynamical systems techniques. While mostly of a finite (dimensional) nature, the results presented naturally lead to several intriguing questions concerning infinite dimensional systems.
Pietro-Luciano Buono, University of Ontario Institute
of Technology
Realization of critical eigenvalues for linear scalar and symmetric delay-differential
equations.
This talk discusses the realization of critical eigenvalues for linear delay-differential equations depending on n delays. The main result is as follows: for a set of n rationally independent positive numbers, there exists a scalar linear DDE depending on n delays such that the spectrum of the bounded linear operator has n eigenvalues on the imaginary axis with imaginary parts given by the chosen rationally independent numbers. I will also discuss a generalization to symmetric DDEs and in particular to delay-coupled rings of DDEs with dihedral symmetry.
Sue Ann Campbell, University of Waterloo
Delay Induced Canards
We consider a model for regenerative chatter in a drilling process. The model
is a nonlinear delay differential equation where the delay arises from the fact
that the cutting tool passes over the metal surface repeatedly. For any fixed
value of the delay, a large enough increase in the width of the chip being cut
results a Hopf bifurcation from the steady state, which is the origin of the
chatter vibration. We show that for zero delay the Hopf bifurcation is degenerate
and that for small delays this leads to a canard explosion. That is, as the
chip width is increased beyond the Hopf bifurcation value, there is a rapid
transition from a small amplitude limit cycle to a large relaxation cycle. Our
analysis relies on perturbation techniques and a small delay approximation of
the DDE model. We use numerical simulations and numerical continuation to support
our analysis and to determine when the small delay approximation fails. We discuss
how our results may apply to other systems
with time delays. This is joint work with Emily Stone and Thomas Erneux.
Alexandre N. Carvalho,Universidade de São
Paulo
Continuity of attractors and of its characterization
In this lecture we present some of our recent results on continuity of attractors and of its characterization under autonomous or non-autonomous perturbations. We introduce the class of \emph{gradient-like semigroups} which contains the class of \emph{gradient semigroups} (those with a Liapunov function) and prove that a perturbation of a gradient-like semigroup is again a gradient-like semigroup. The notion of gradient-like semigroups can be extended to non-autonomous evolution processes and we prove that a non-autonomous perturbation of a gradient-like semigroup is a gradient-like non-autonomous evolution process.
The results presented here are part of a joint work with J. A. Langa of the Universidad de Seville, Spain.
Yuming Chen, Wilfrid Laurier University
The global attractor of a delayed differential system
\noindent We consider a delayed differential system which describes the dynamics of a network of two identical neurons with delayed output. The technical tool is the discrete Lyapunov functional developed by Mallet-Paret and Sell. First, under some technical assumptions, the existence, nonexistence and uniqueness of periodic solutions in the level sets of the Lyapunov functional are established. Then the global attractor of the system is shown to be the finite union of the unstable sets of stationary points and periodic orbits.
Chris Cosner, University of Miami
Beyond diffusion: conditional dispersal in ecological models
Reaction-diffusion models have been widely used to describe the dynamics of
dispersing populations. However, many organisms disperse in ways that depend
on environmental conditions or the densities of other populations. Those can
include advection along environmental gradients and nonlinear diffusion, among
other possibilities. In this talk I will describe some models involving conditional
dispersal and discuss its effects and evolution. The presence of conditional
dispersal can
strongly influence the equilibria of population models, for example by causing
the population to concentrate at local maxima of resource density. The analysis
of the evolutionary aspects of dispersal is based on a study of models for two
competing populations that are ecologically identical except for their dispersal
strategies. The models consist of Lotka-Volterra competition systems with some
spatially varying coefficients and with diffusion, nonlinear diffusion, and/or
advection terms that reflect the dispersal strategies of the competing populations.
The evolutionary stability of dispersal strategies can be determined by analyzing
the stability of single-species equilibria in such models. In the case of simple
diffusion in spatially varying environments it has been known for some time
that the slower diffuser will exclude the faster diffuser, but conditional dispersal
can change that. In some cases a population whose dispersal strategy involves
advection along environmental gradients has the advantage or can coexist with
a population that simply diffuses. As is often the case in reaction-diffusion
theory, many of the results depend on the analysis of
eigenvalue problems for linearized models.
Walter Craig, McMaster University
Lagrangian and resonant tori for Hamiltonian PDEs
This talk discusses a version of the nonlinear Schroedinger equation posed on a lattice, which is an infinite-dimensional Hamiltonian dynamical system. Using the approach of classical KAM theory, we construct invariant tori of full dimension. Our approach is in fact related to constructions of lower dimensional tori, in that we use a normal form that controls the tangential frequencies of the torus in question, its normal frequencies, and its linear stability. We augment this with higher order Melnikov-like nonresonance conditions so that we additionally control the curvature of the action-frequency map. We also give a picture of the situation in the case of resonant tori. This represents joint work with J. Geng.
Susan Friedlander, University of Southern
California
Kolmogorov's Turbulence, Onsager's Conjecture and a Dyadic Model for the
Fluid Equations.
Kolmogorov predicted that the energy cascade mechanism in 3 dimensional turbulence
produces a striking phenomenon, namely positive energy dissipation in the limit
of vanishing viscosity. However, to date, there is no rigorous proof of this
phenomenon based on the Navier-Stokes equations.
We will discuss an infinite, nonlinearly coupled system of ODE that is a so
called "dyadic model" for the fluid equations. We prove that Kolmogorov's
dissipation anomaly exists for the dyadic model. Furthermore, the limiting value
of the energy dissipation is exactly the "turbulent" dissipation produced
by rough solutions of the inviscid model which are consistent with Onsager's
conjecture.
This is joint work with Alexey Cheskidov.
Hongjun Gao, Nanjing Normal University
Random Attractor for the 3D viscous stochastic primitive equations with additive
noise.
In this article, we obtain the existence and uniqueness of strong solutions to 3D viscous stochastic primitive equations (PEs) and the random attractor for 3D viscous PEs with additive white noise.
Martin Golubitsky, Mathematical Biosciences
Institute, Ohio State University
Feed-forward networks near Hopf bifurcation
Synchrony-breaking Hopf bifurcations in a small three-node feed-forward network
lead generically to periodic solutions whose amplitudes in the third node have
a surprising 1/6 power growth rate. Moreover, when this network is tuned near
such a Hopf bifurcation, it can act as an efficient frequency filter/ amplifier.
I will describe the general theory; recent experiments of McCullen and Mullin
on coupled electrical circuits that confirm this structure; and related work
with Claire Postletwaite, LieJune Shaiu, and Yanyan Zhang on periodic forcing
of systems near Hopf bifurcation and its application to auditory receptor cells
on the basilar
membrane in the cochlea.
Stephen Gourley, University of Surrey
A Nonlocal Reaction-Diffusion Model for Cellular Adhesion
Adhesion of cells to one another and their environment is an important regulator
of many biological processes but it is difficult to incorporate into continuum
mathematical models. An integro-partial differential equation model for cell
behaviour will be presented, in which the integral represents sensing by cells
of their local
environment. Aggregation patterns are investigated in a model incorporating
cell-cell adhesion, random cell movement, and cell proliferation. The model
is also extended to give a new representation of cancer growth, whose solutions
reflect the balance between cell-cell and cell-matrix adhesion in regulating
cancer invasion.
Solutions for cell density need to lie between zero and a positive density corresponding
to close cell packing. A number of conditions will be presented, each of which
is sufficient for the required boundedness. It can be demonstrated numerically
that cell density increases above the upper bound for some parameter sets not
satisfying these conditions. Conditions will also be presented which are sufficient
for global convergence to the uniform steady state.
Jack K. Hale, Georgia Institute of Technology
Perturbing Periodic Orbits in Infinite Dimensions
For an autonomous ODE with the flow possessing a compact invariant set which is a smooth manifold without boundary, there is a vast literature on the effects of perturbations, both autonomous and nonautonmous. Many of the efforts break the problem down into two parts. The first is the persistence of the manifold in the base space for autonomous perturbations and the base space times the reals in the nonautonomous case. The second step is to study the flow on the perturbed manifold.
In the infinite dimensional case, much is known about RFDE with finite delay
and parabolic systems. Otherwise, there are only a few important contributions
in other
cases; in particular, PDE which do not have a smoothing property in time. There
are many obstables to applying the methods of ODE. One important reason is that
the evolutionary equations involve unbouded operators. Also, for PDE, there
are so many important perturbation parameters, some of which are regular and
some irregular.
Raugel and I have been attempting to develop methods which will be sophisticated enough to obtain results similar to the ones for ODE and allow general types of perturbations. Since the problems are so complicated, we have chosen the basic invariant set to be a periodic orbit. We discuss this case in some detail.
Wenzhang Huang, University of Alabama in Huntsville
The Minimum Wave Speed of Traveling Waves for a Competition Model
Consider a reaction-diffusion system that serves as a 2-species Lotka-Volterra competition model with each species having logistic growth in the absence of the other. Suppose that the corresponding reaction system has one unstable boundary equilibrium $E_1$ and a stable boundary equilibrium $E_2$. Then it is well known that there exists a positive number $c_*$, called the minimum wave speed, such that, for each $c\ge c_*$, the reaction-diffusion system has a positive traveling wave solution of wave speed $c$ connecting $E_1$ and $E_2$, and the system has no nonnegative traveling wave with wave speed less than $c_*$ that connects $E_1$ and $E_2$. Although much research work has been done to give an estimate of $c_*$, the important problem on finding an algebraic or analytic expression for the minimum wave speed $c_*$ remains open. In this talk we will introduce a new approach that enable us to determine precisely the minimum wave speed algebraically.
Michael Jolly, University of Indiana
Estimates on enstrophy, palinstrophy, and invariant measures for 2-D turbulence
We construct semi-integral curves which bound the projection of the global
attractor of the 2-D Navier-Stokes equations in the plane spanned by enstrophy
and
palinstrophy. Of particular interest are certain regions of the plane where
palinstrophy dominates enstrophy. Previous work shows that if solutions on the
global attractor spend a significant amount of time in such a region, then there
is a cascade of enstrophy to smaller length scales, one of the main features
of 2-D turbulence theory. The semi-integral curves divide the plane into regions
in which a range for the direction of the flow is determined. This allows us
to estimate the average time it takes for an intermittent solution to burst
into a region of large palinstrophy. We also show that the time average of palinstrophy
achieves its maximal value only at statistical steady states where the nonlinear
term is zero.
Barbara Keyfitz, Fields Institute
The Trouble with Conservation Laws
I address difficulties in conservation laws from the point of view of infinite dimensional dynamical systems and evolution equations. Unlike systems that behave somewhat like finite-dimensional dynamical systems, conservation laws are not amenable to simple finite dimensional approximations. Only in rather non-intuitive spaces can one find compact approximations. Even the "obvious" approximation (by parabolic systems imitating physical viscosity) was shown to converge only recently, and only after application of some very deep arguments. Furthermore, all analysis developed so far works only for systems in a single space variable. Again, conservation laws differ in this respect from many well-known evolution equations.
Over the last dozen years, several groups have begun to approach multidimensional systems by looking at self-similar reductions. The reduced system changes type in an interesting way. This talk will review work of Morawetz, Popivanov and others on linear change of type and its relation to evolution systems. Our objective is to find well-posed problems for the reduced (quasilinear) system and, ultimately, to use this as a tool to understand the dynamics of multidimensional systems.
Tibor Krisztin, University of Szeged
Morse decomposition for differential equations with state-dependent delay
We consider a class of functional differential equations representing equations
with state-dependent delay.
It is shown that, under certain technical conditions, the global attractor of
the solution semiflow has a Morse decomposition.
The result can be applied for equations with threshold delays, signal transmission
delays, and delay functions depending
on the present state of the system. The proof is analogous to that of Mallet-Paret
given for equations with constant delay,
however, nontrivial modifications are necessary. A discrete Lyapunov functional,
which is a version of that of
Mallet-Paret and Sell, counts sign changes on intervals of the form $[t-r(t),t]$
where $r$ may depend on the state variable.
A crucial property in the proof of the boundedness of the discrete Lyapunov
functional on the global attractor is that,
for globally defined solutions, the map $t\mapsto t-r(t)$ is monotone increasing.
Bernhard Lani-Wayda, JLU Gießen
Attractors for delay equations with monotone and non-monotone feedback.
It is known that, for monotone $f$ with negative feedback ($ \text{ sign} (f(x))
= - \text{ sign} (x) $), the infinite-dimensional
dynamical system generated by the delay equation $ \dot x(t) = f(x(t-1)$ possesses
a two-dimensional invariant manifold $W$ with Poincar\'{e}-Bendixson-like dynamics,
which attracts all slowly oscillating solutions.
We consider parametrized families of nonlinearities starting from such $f$, but then changing to non-monotone shape, and how the attractor changes with the nonlinearity. In particular, it looses the disk-like structure.
Marta Lewicka, University of Minnesota
Derivation of shell theories from 3d nonlinear elasticity.
A longstanding problem in the mathematical theory of elasticity is to predict theories of lower-dimensional objects (such as rods, plates or shells), subject to mechanical deformations, starting from the 3d nonlinear theory. For plates, a recent effort has lead to rigorous justification of a hierarchy of such theories (membrane, Kirchhoff, von K\'arm\'an). For shells, despite extensive use of their ad-hoc generalizations present in the engineering applications, much less is known from the mathematical point of view.
In this talk, I will discuss the limiting behaviour (using the notion of Gamma-limit) of the 3d nonlinear elasticity for thin shells around an arbitrary smooth 2d mid-surface S. We prove that the minimizers of the 3d elastic energy converge, after suitable rescaling, to minimizers of a hierarchy of shell models.
The limiting functionals (which for plates yield respectively the von Karman,
linear, or linearized Kirchhoff theories) are intrinsically linked with the
geometry of S. They are defined on the space of infinitesimal isometries of
S (which replaces the 'out-of-plane-displacements' of plates), and the space
of finite strains (which replaces strains of the `in-plane-displacements'),
thus clarifying the effects of rigidity of S on the derived theories. The different
limiting theories correspond to
different magnitudes of the applied forces, in terms of the shell thickness.
This is joint work with M.G. Mora and R. Pakzad.
Xing Liang, University of Science and Technology
of China and University of Tokyo
A variational problem associated with the minimal speed of travelling waves
for spatially periodic reaction-diffusion equations.
We consider the equation $u_t=u_{xx}+b(x)u(1-u),$ $x\in\mathbb R,$ where $b(x)$
is a nonnegative measure on $\mathbb R$ that is periodic in $x.$ In the case
where $b(x)$ is a smooth periodic function, it is known that there exists a
travelling wave with speed $c$ for any $c\geq c^*(b),$ where $c^*(b)$ is a certain
positive number depending on $b.$ Such a travelling wave is often called a \lq\lq
pulsating travelling wave" or a \lq\lq periodic travelling wave",
and $c^*(b)$ is called the \lq\lq minimal speed". In this paper, we first
extend this theory by showing the existence of the minimal speed $c^*(b)$ for
any nonnegative measure $b$ with period $L.$ Next we study the question of maximizing
$c^*(b)$ under the constraint $\int_0^Lb(x)dx=\alpha L,$ where $\alpha$ is an
arbitrarily given constant. This question is closely related to the problem
studied by mathematical ecologists in late 1980's but its answer has not been
known. We answer this question by proving that the maximum is attained by periodically
arrayed Dirac's delta functions $\alpha L\sum_{k\in\mathbb Z}\delta(x+kL).$
This is a joint work with Prof. H.Matano and Dr. X. Lin .
Xiao-Biao Lin, University of North Carolina
Traveling Wave Solutions of a Model of Liquid/Vapor Phase Transition
We will discuss traveling wave solutions for dynamical flows involving liquid/vapor
phase transition. The model is a coupled system of viscous conservation laws
and a reaction-diffusion equation. Sufficient and necessary conditions for the
existence of four tyeps of traveling waves will be given:
(1) Liquefaction waves; (2) Evaporation waves; (3) Collapsing waves;
(4) Explosion waves.
This is joint work with Haitao Fan, Georgetown University.
Weishi Liu, University of Kansas
Effects of some turning points on global dynamics
In this talk, we will consider singularly perturbed systems with turning points.For
a class of turning points, a new structure is revealed that plays a critical
role in the organization of global dynamics. A concrete example will also be
discussed for an illustration of the abstract result.
Michael C. Mackey, McGill University
Temporal Dynamics in the Tryptophan and Lactose Operons
This talk will focus on the temporal dynamics of the lactose and tryptophan operons, and the ability of apparently realistic mathematical models to capture these dynamics. The dynamics in question range from stable steady states through bistability and oscillatory expression of gene products.
Pierre Magal, University Le Havre
Semilinear Non-densely Defined Cauchy Problems: Center Manifold Theorem and
Applications
Several types of differential equations, such as delay differential equations, age-structure models in population dynamics, evolution equations with nonlinear boundary conditions, can be written as semi-linear Cauchy problems with an operator which is not densely defined. The goal of this presentation is first to present several examples, and then to turn to a center manifold theory for semi-linear Cauchy problems with non-dense domain. Using Liapunov-Perron method and following the techniques of Vanderbauwhede and Iooss in treating infinite dimensional systems, we study the existence and smoothness of center manifolds for semi-linear Cauchy problems with non-dense domain. We will conclude this presentation with several example of PDE where one can investigate Hopf bifurcation using this center-manifold theorem.
Hiroshi Matano, Tokyo
A braid-group method for blow-up in nonlinear heat equations.
In this talk I will present intriguing applications of the braid group theory
to the study of blow-up in a nonlinear heat
equation $u_t = \Delta u + u^p$, where $p$ is supercritical in the Sobolev sense.
One of the goals is to classify the type II
blow-up rates by analyzing the topological properties of certain braids. I will
also discuss other applications of this method.
Connell McCluskey, Wilfrid Laurier University
A Global Result for a Disease Model with Infinite Delay
A recent paper (MBE 2008, 5:389-402) by G. Rost and J. Wu presented an SEIR
disease model using infinite delay to account for varying infectivity. They
gave a thorough analysis leaving out only the elusive global stablity of the
endemic equilibrium. A solution to that problem will be given, making use of
a Lyapunov functional. The functional includes a term that integrates over all
previous states of the system.
Konstantin Mischaikow, Rutgers University
Building a Database for Global Dynamics of Parameterized Nonlinear Systems
James Muldowney, University of Alberta
Bendixson conditions for differential equations in Banach Spaces
It will be shown that a flow which diminishes a measure of 2-dimensional surface
area cannot contain non-constant periodic orbits. Concrete conditions that preclude
the existence of periodic solutions for a parabolic PDE will be given.
Roger Nussbaum, Rutgers University
Nonlinear Differential-Delay Equations with State Dependent Time Lag(s)
This lecture will mostly concentrate on the differential delay equation
(1) ax'(t)= f(x(t),x(t-r)),
where r:=r(x(t)) and f and r are given functions. A central question is what can be said about the limiting shape of slowly oscillating periodic solutions of eq.(1) as a--0. We shall briefly describe some of the tools which have been used to study such equations, and we shall illustrate our results by discussing some simple-looking equations which already exhibit most of the essential difficulties:
(2) ax'(t)= -x(t) -kx(t-r), a0, k1, r=1+x(t) or r=1-(x(t))^2.
All of the results which we shall discuss represent joint work with John Mallet-Paret.
Carmen Núñez, University of Valladolid
Global attractivity in monotone concave differential equations with infinite
delay
We study the dynamical behavior of the trajectories defined by a recurrent family of monotone functional differential equations with infinite delay and concave nonlinearities. We analyze different sceneries which require the existence of a lower solution and of a bounded trajectory ordered in an appropriate way, for which we prove the existence of a globally asymptotically stable minimal set given by a 1\nbd-cover of the base flow. We apply these results to the description of the long term dynamics of a nonautonomous model representing a stage-structured population growth without irreducibility assumptions on the coefficient matrices. This work is made in collaboration with Rafael Obaya and Ana M. Sanz.
Rafael Obaya, Universidad de Valladolid
Exponential ordering for scalar neutral functional differential equations
with infinite delay.
We study some properties of the exponential ordering for scalar nonautonomous
families of functional differential equations and neutal functional differential
equations with stable D-operator. We discuss some properties which imply that
the omega limit set of a relatively compact trajectory is a copy of the base.
We apply these
results in the study of some kind of compartmental systems.
Sérgio Oliva, Universidade de São
Paulo
Analytical Methods for Approximation Schemes in Partial Functional Differential
Equations
The goal of this paper is to present an approximation scheme for a reaction-diffusion
equation with finite delay, which has been used as a model to study the evolution
of a population with density distribution $u$, in such a way that the resulting
finite dimensional ordinary differential system contains the same asymptotic
dynamics as the reaction-diffusion equation.
Ken Palmer, National Taiwan University
Homoclinic Orbits in Singularly Perturbed Systems
We begin by considering three nonlinear oscillators studied by Cherry, Iglisch, and Kurland and Levi respectively. In all three systems transversal homoclinic or heteroclinic orbits arise after perturbation. It turns out that all three equations can be regarded as singularly perturbed (or slowly varying) systems where the unperturbed system has one or two normally hyperbolic centre manifolds. We study the general question of fast connecting orbits in such systems and derive a general bifurcation function, the zeros of which correspond to such connecting orbits. We also discuss the question of when such orbits connect equilibria on the centre manifolds.
Peter Polacik, University of Minnesota
Parabolic Liouville theorems and their applications.
Parabolic Liouville theorems state that if u is an entire solution of a specific
parabolic equation and u is contained in an admissible class of solutions, then
u ? 0.
As an admissible class one can take nonnegative solutions or radial solutions
withbounded zero number. We present available Liouville theorems and some of
their numerous applications.
Genevieve Raugel, Paris de Sud
Dynamics of some equations in fluid mechanics
In this talk, we mainly consider two systems, arising in fluid mechanics, namely the second grade fluid equations and the ``hyperbolic second order Navier-Stokes equations". We shall prove existence and uniqueness results for both systems and study their dynamical properties. Both systems depend on a parameter. When this parameter is small, these systems can be considered as non regular perturbations of the Navier-Stokes equations. We shall compare the dynamics of these perturbed systems with those of the Navier-Stokes equations, when the parameter goes to zero.
Shigui Ruan, University of Miami,
Center Manifolds for Semilinear Equations with Non-dense Domain and Applications
on Hopf Bifurcation in Age Structured Models
Age structured models arise naturally in population dynamics and epidemiology.
The existence of non-trivial periodic solutions in age structured models has
been a very interesting and difficult problem. It is believed that such periodic
solutions are induced by Hopf bifurcation, however there is no general Hopf
bifurcation theorem available for age structured models. One of the difficulties
is that, rewriting age structured models as a semilinear equation, the domain
of the linear
operator is not dense. In this talk, we first introduce the center manifold
theory for semilinear equations with non-dense domain. We then use the center
manifold theorem to establish a Hopf bifurcation theorem for age structured
models (based on joint work with Pierre Magal).
Arnd Scheel, University of Minnesota
How robust are Liesegang patterns?
TBA
Wenxian Shen, Auburn University
Spreading and Generalized Propagating Speeds of KPP Models in Time Varying
Environments
The current talk is concerned with the spreading and generalized propagating
speeds of KPP models in time recurrent environments, which including time periodic
and almost periodic environments as special cases. It first introduces the notions
of spreading speed intervals, generalized propagating speed intervals, and traveling
wave solutions. Some fundamental properties of spreading and generalized propagating
speeds are then presented. When the environment is unique ergodic and the so
called linear determinacy condition is satisfied, it is shown that the spreading
speed interval in any direction is a singleton (called the spreading speed),
which equals the classical spreading speed if the environment is actually periodic.
Moreover, in such case, a variational principle for the spreading speed is established
and it is shown that there is a front of speed c in a given direction if and
only if c is greater than or equal to the spreading speed in that direction.
Stefan Siegmund, Dresden University of Technology
Differential Equations with Random Delay
We present a first step towards a general theory of differential equations incorporating unbounded random delays. The main technical tool relies on recent work of Zeng Lian and Kening Lu, which generalizes the Multiplicative Ergodic Theorem by V.I. Oseledets to Banach spaces.
Hal Smith, Arizona State University
Persistence Theory for Semidynamical Systems
Persistence, sometimes called Permanence, for dynamical systems arising in the biological sciences implies that a set of populations avoids extinction. The theory, developed beginning in the 1970s, aims to answer the question "Which species survive in the long run?" We survey some of the theoretical results of this theory and its applications. Special attention will be given to discrete-time systems.
Horst R. Thieme, Arizona State University
Differentiability of convolutions
If $T=\{T(t); t \ge 0\}$ is a strongly continuous family of bounded linear
operators between two Banach spaces $X$ and $Y$ and $f \in L^1(0,b,$ $X)$, the
convolution of $T$ with $f$ is defined by $(T *f)(t) = \int_0^t T(s)f(t-s)ds$.
It is shown that $T*f$ is continuously differentiable for all $f \in C(0,b,X)$
if and only if $T$ is of bounded semi-variation on $[0,b]$. Further $T*f$ is
continuously differentiable for all $f \in L^p(0,b, X)$ ($1 \le p < \infty$)
if and only if $T$ is of bounded semi-$p$-variation on $[0,b]$ and $T(0)=0$.
If $T$ is an integrated semigroup with generator $A$, these respective conditions
are necessary and sufficient for the Cauchy problem $u'= Au +f $, $u(0)=0$,
to have integral (or mild) solutions for all $f$ in the respective function
vector spaces. A converse is proved to a well-known result by Da Prato and Sinestrari:
the generator $A$ of an integrated semigroup is a Hille-Yosida operator if,
for some $b>0$, the
Cauchy problem has integral solutions for all $f\in L^1(0,b, X)$. Integrated
semigroups of bounded semi-$p$-variation are preserved under bounded additive
perturbations of their generators and under commutative sums of generators if
one of them generates a $C_0$-semigroup.
Stephen Schecter, North Carolina State University
Stability of fronts in gasless combustion
For gasless combustion in a one-dimensional solid, we show a type of nonlinear stability of the physical combustion front: if a perturbation of the front is small in both a spatially uniform norm and an exponentially weighted norm, then the perturbation stays small in the spatially uniform norm and decays in the exponentially weighted norm, provided the linearized operator has no eigenvalues in the right half-plane other than zero. Using the Evans function, we show that the zero eigenvalue must be simple. Factors that complicate the analysis are: (1) the linearized operator is not sectorial, and (2) the linearized operator only has good spectral properties when the weighted norm is used, but then the nonlinear term is not Lipschitz. The result is nevertheless physically natural.
Coauthors:
Anna Ghazaryan, University of North Carolina at Chapel Hill Yuri Latushkin,
University of Missouri Aparecido de Souza, Universidade Federal de Campina Grande
(Brazil)
Arnd Scheel, University of Minnesota
How robust are Liesegang patterns?
Liesegang patterns are quite common stationary patterns in reaction-diffusion
systems. They consist of precipitation spots that are spaced at geometrically
increasing distances from the boundary. We will show that such patterns are
untypical in generic reaction-diffusion systems, but robust in a class of systems
with an irreversible chemical reaction. The main result gives necessary and
sufficient conditions for the existence of Liesegang patterns. The proof involves
the analysis of a degenerate reversible homoclinic orbit and an invariant manifold
theorem for non-smooth Poincare maps.
Hans-Otto Walther, Universität Giessen
Algebraic-delay differential systems, state-dependent delay, and temporal
order of reactions
Systems of the form
x'(t) & = & g(r(t),x_t)\\
0 & = & \Delta(r(t),x_t)
generalize differential equations with delays $r(t)<0$ which are given implicitly
by the history $x_t$ of the state. We show that the associated initial value
problem generates a semiflow with differentiable solution operators on a Banach
manifold. The theory covers reaction delays, signal transmission delays, threshold
delays, and delays depending on the present state $x(t)$ only. As an application
we consider a model for the regulation of the density of white blood cells and
study monotonicity properties of the delayed argument function $\tau:t\mapsto
t+r(t)$. There are solutions $(r,x)$ with $\tau'(t)0$ and others with $\tau'(t)<0$.
These other solutions correspond to feedback which reverses temporal order;
they are short-lived and less abundant. Transient behaviour with a sign change
of $\tau'$ is impossible.
Gail S. K. Wolkowicz, McMaster University
Comparison of Predator-Prey Models with Discrete Time Delay
We consider the dynamics of the classical predator-prey model and the predator-prey model in the chemostat when a discrete delay is introduced to model the time between the capture of the prey and its conversion to biomass. In both models we use Holling type I response functions so that no oscillatory behavior is possible in the associated system when there is no delay. With delay, the characteristic equation in both cases is a second order transcendental equation with parameters that depend on the delay. In both models, we prove that as the parameter modeling the delay is varied Hopf bifurcation can occur, and show numerically that the bifurcation results in asymptotically stable periodic orbits. However, we show that there appear to be differences in the possible sequences of bifurcations for the two models.
Yuan Yuan, Memorial University of Newfoundland
Pattern Formation in a Ring Network with Delay
We consider a ring network of three identical neurons with delayed feedback.
Regarding the coupling coefficients as bifurcation parameters, we obtain codimension
one bifurcation (including a Fold bifurcation and Hopf bifurcation) and codimension
two bifurcations (including Fold-Fold bifurcations, Fold-Hopf bifurcations and
Hopf-Hopf bifurcations). We also give concrete formulae for the normal form
coefficients derived via the center manifold reduction that
provide detailed information about the bifurcation and stability of various
bifurcated solutions. In particular, we obtain stable or unstable equilibria,
periodic solutions, and quasi-periodic solutions.
This is a joint work with Shangjiang Guo.
Chongchun Zeng, Georgia Tech
Unstable manifolds and $L^2$ nonlinear instability of the Euler equation
We consider a steady state $v_0$ of the Euler equation in a fixed bounded domain in $R^n$. Suppose the linearized equation has an exponential dichotomy with a finite dimensional unstable subspace. By rewriting the Euler equation as an ODE on an infinite dimensional manifold in $H^k$, $k>\frac n2 +1$, the unstable manifold of $v_0$ is constructed under certain conditions on the Lyapunov exponents of the vector field $v_0$. This in turn shows the nonlinear instability of $v_0$ in the sense that small $H^k$ perturbations can lead to $L^2$ derivation of the solutions.
Yuncheng You, University of South Florida
Global Dynamics of Cubic Autocatalytic Reaction-Diffusion Systems
Consider a class of cubic and mixed-order autocatalytic reaction-diffusion systems arising from kinetic modeling of chemical and biochemical reactions of two components or from biological pattern formations. We shall talk about the existence of a global attractor and an exponential attractor for the solution semiflow of these reaction-diffusion systems on three-dimensional bounded domains and related issues.
Xiaoqiang Zhao, Memorial University of Newfoundland
Spatial Dynamics of Abstract Monostable Evolution Systems With Applications
In this talk, I will report on the theory of spreading speeds and traveling
waves for abstract monotone semiflows with spatial structure. Under appropriate
assumptions, we show that the spreading speeds coincide with the minimal wave
speeds for monotone traveling waves in the positive and negative directions.
Then we use this theory to study the spatial dynamics of a reaction-diffusion-advection
population model, a parabolic equation in a periodic cylinder with the Dirichlet
boundary condition, a porous medium equation in a tube, and a lattice system
in a periodic habitat. If time permits, I will also mention our research on
some non-monotone evolution systems. This talk is based on my recent joint works
with X. Liang, S.-B. Hsu, J. Fang, and Y. Jin.
Xingfu Zou, University of Western Ontario
Impact of map dynamics on the dynamics of the associated delay reaction diffusion
equation with Neumann condition
We are concerned with the dynamics of a class of delay reaction diffusion
equation with a parameter $\mu$. By letting $\mu \rightarrow +\infty$,
such an equation is formerly reduced to an interval dynamical system.
With the help of the famous Sarkovskii's theorem, we obtain some new yet
simple sufficient conditions that assure the global stability of the delayed
reaction
diffusion equation with the parameter. We also give several examples to illustrate
our main results.
This is a joint work with Taishan Yi.