June 20 - July 1, 2011
Normally Hyperbolic Invariant Manifolds: Existence, Persistence, Approximation, and Their Applications
Quasi-periodic solutions in dynamical systems and their role in global dynamics
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 $mbbRtimes gO$.
Automatic differentiation methods in computational dynamical systems: invariant manifolds and normal forms
Invariant Objects in Dynamical Systems and their applications
Computation of limit cycles and their isochrons: Applications to biology
Differential equations with multiple lags
Generalized cyclic feedback system for the biomedical interaction network
Lyapunov exponents, periodic orbits, and horseshoes for semiflows on Hilbert space
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.
Space Mission Design with Dynamical Systems Theory
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
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 limit$epsilonrightarrow0$.
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].
Numerical study of regularity of functions related to critical objects
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.
Loss of normal hyperbolicity
Exchange lemmas
Ensemble Dynamics and Bred Vectors
Conditional Stability Theorems for Special Solutions of Nonlinear PDEs
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.