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Abstracts and Talk Materials
Invariant Objects in Dynamical Systems and their Applications
June 20-July 1, 2011


Peter W. Bates
http://www.math.msu.edu/~bates/

Normally Hyperbolic Invariant Manifolds: Existence, Persistence, Approximation, and Their Applications
June 20, 2011

Normally Hyperbolic Invariant Manifolds: Existence, Persistence, Approximation, and Their Applications

Rafael de la Llave
http://www.ma.utexas.edu/text/webpages/llave.html

Quasi-periodic solutions in dynamical systems and their role in global dynamics
June 20, 2011

Quasi-periodic solutions in dynamical systems and their role in global dynamics

Mohamed Sami ElBialy
http://math.utoledo.edu/~melbialy/

Stable and unstable manifolds for bisemigroups
June 30, 2011

We show the existence of local Lipschitzian stable and unstable manifolds for the ill posed problem of perturbations of hyperbolic bisemigroups. We do not assume backward nor forward uniqueness of solutions. We do not use cut-off functions because we do not assume global smallness conditions on the nonlinearities. We introduce what we call dichotomous flows which recovers the symmetry between the past and the future. Thus, we need to prove only a stable manifold theorem. We modify the tit{Conley-McGehee} approach to avoid appealing to Wazewski principle and Brouwer degree theory. Hence we allow both the stable and unstable directions to be infinite dimensional. We illustrate  our theorem by a simple example, namely  the elliptic system  $uxixid  +  gD u  = g(u,  uxid)$ in an infinite cylinder  $mbbRtimes  gO$.

Alex Haro
www.maia.ub.es/~alex

Automatic differentiation methods in computational dynamical systems: invariant manifolds and normal forms
June 20, 2011

Automatic differentiation methods in computational dynamical systems: invariant manifolds and normal forms

Invariant Objects in Dynamical Systems and their applications

Gemma Huguet
http://www.pagines.ma1.upc.edu/~huguet/

Computation of limit cycles and their isochrons: Applications to biology
June 22, 2011

Computation of limit cycles and their isochrons: Applications to biology

Gabor Kiss

Differential equations with multiple lags
July 1, 2011

Differential equations with multiple lags

Guang-Tsai Lei

Generalized cyclic feedback system for the biomedical interaction network
June 30, 2011

Generalized cyclic feedback system for the biomedical interaction network

Zeng Lian
http://www.lboro.ac.uk/departments/ma/people/Lian.html

Lyapunov exponents, periodic orbits, and horseshoes for semiflows on Hilbert space
June 30, 2011

Lyapunov exponents, periodic orbits, and horseshoes for semiflows on Hilbert space

Zhiwu Lin
http://people.math.gatech.edu/~zlin/

Nonlinear Landau damping and inviscid damping
June 30, 2011

Consider electrostatic plasmas described by 1D Vlasov-Poisson with a fixed ion background. In 1946, Landau discovered the linear decay of electric field near a stable homogeneous state. The nonlinear Landau damping was recently proved for analytic perturbations by Villani and Mouhot, but for general perturbations the problem is still largely open. With Chongchun Zeng at Georgia Tech, we construct nontrivial traveling waves (BGK waves) with any spatial period which are arbitrarily near any homogeneous state in H^s (s3/2) spaces might be much simpler and the nonlinear damping might be hopeful. We also obtained similar results for the problem of nonlinear inviscid damping of Couette flow, for which the linear decay was first observed by Orr in 1907.

Martin Wen-Yu Lo
www.gg.caltech.edu/~mwl

Space Mission Design with Dynamical Systems Theory
June 28, 2011

Space Mission Design with Dynamical Systems Theory

Jose-Maria Mondelo
http://mat.uab.cat/~jmm/

Numerical Fourier analysis of quasi-periodic functions
June 30, 2011

A procedure for the numerical computation of frequencies and amplitudes of quasi-periodic functions from equally spaced samples will be presented. It is based on a collocation-like strategy in frequency domain, using the Discrete Fourier Transform (DFT). Comments will be made on the practical choice of parameters in order to obtain high precision, avoiding DFT-related phenomena (leakage, aliasing). An application will be given to the study of the dynamics in the (practical) stability zone around the triangular libration points of the planar, circular RTBP for the Sun-Jupiter mass ratio.

Joint work with G. Gómez and C. Simó

Benson Muite
http://www.math.lsa.umich.edu/~muite

Some observations from computations of the Kohn-M\"{u}ller model
July 1, 2011

We describe some computational results for the partial differential equation

rho u_{tt} - beta Delta u = 15(u_x^3 -u_x)_x +gamma u_{yy} - epsilon^2 u_{xxxx}

which arises as a simplified model for phase transformations. We describe some features of the equation in the limitepsilonrightarrow0.

Bruce B. Peckham
http://www.d.umn.edu/~bpeckham/

Breakup of an invariant circle in a noninvertible map of the plane
July 1, 2011

In a 1989 paper, E. Lorenz studied the application of Euler's method to a certain 2-dimensional system of ODEs. As the time step was increased, the corresponding map progressed from exhibiting an attracting fixed point to an invariant circle to "full chaos." Of special interest is the parameter range including the breakup of the invariant circle and the first appearance of sensitivity to initial conditions. The "invariant circle" develops bumps, then cusps, then loops. A follow-up numerical study [Frouzakis, Kevrekidis, P 2003] revealed some details of this transition, including the interaction of the invariant circle with stable and unstable manifolds of saddles for periodic points. Additional investigation is still being performed. This talk will discuss computation of Arnold tongues and (if this is a good week) the paths corresponding to invariant circles with fixed irrational rotation numbers that lie "in between" the tongues. Techniques follow [Schilder and P, 2007].

Nikola Petrov
http://www.math.ou.edu/~npetrov/

Numerical study of regularity of functions related to critical objects
July 1, 2011

Numerical study of regularity of functions related to critical objects

Tuoc Van Phan
www.math.utk.edu/~phan

Small solutions of nonlinear Schrodinger equations near first excited states
June 30, 2011

Consider a nonlinear Schrodinger equation in R3 whose linear part has three or more eigenvalues satisfying some resonance conditions. Solutions which are initially small in H1 L1(R3) and inside a neighborhood of the first excited state family are shown to converge to either a first excited state or a ground state at time infinity. An essential part of our analysis is on the linear and nonlinear estimates near nonlinear excited states, around which the linearized operators have eigenvalues with nonzero real parts and their corresponding eigenfunctions are not uniformly localized in space.

Stephen Schecter

Exchange lemmas
June 29, 2011

Exchange lemmas

Stephen Schecter

Loss of normal hyperbolicity
June 30, 2011

Loss of normal hyperbolicity

George R Sell
http://www.math.umn.edu/~sell/

Ensemble Dynamics and Bred Vectors
June 21, 2011

Ensemble Dynamics and Bred Vectors

Milena Stanislavova

Conditional Stability Theorems for Special Solutions of Nonlinear PDEs
June 30, 2011

Conditional Stability Theorems for Special Solutions of Nonlinear PDEs

Zhifu Xie
http://sest.vsu.edu/~zxie

Central Configurations of the N-body problem.
July 1, 2011

I recently discovered a new phenomenon on central configurations in the collinear three-body problem and in the collinear four-body problem. There exists a configuration that is a central configuration for at least two different arrangements of a given mass vector. Such central configuration is called the super central configuration in the $n$-body problem and it may lead some surprising dynamical behaviors. Super central configurations do not exist in the planar four-body problem. The existence and classifications of super central configuration in the collinear cases are also very important for counting the number of central configurations under different equivalent classes. When I extend to study the super central configurations in general homogeneous potential, the existence of super central configurations has a relation to the Mathematical Beatuy- Golden Ratio.

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