Shilnikov Lemma for a nondegenerate critical manifold of a Hamiltonian system

We prove an analog of Shilnikov Lemma for a normally hyperbolic symplectic critical manifold $M\subset H^{-1}(0)$ of a Hamiltonian system. Using this result, trajectories with small energy $H=\mu>0$ shadowing chains of homoclinic orbits to $M$ are represented as extremals of a discrete variational problem, and their existence is proved. This paper is motivated by applications to the Poincar\'e second species solutions of the 3 body problem with 2 masses small of order $\mu$. As $\mu\to 0$, double collisions of small bodies correspond to a symplectic critical manifold of the regularized Hamiltonian system

Scaling law in the Standard Map critical function. Interpolating hamiltonian and frequency map analysis

We study the behaviour of the Standard map critical function in a neighbourhood of a fixed resonance, that is the scaling law at the fixed resonance. We prove that for the fundamental resonance the scaling law is linear. We show numerical evidence that for the other resonances $p/q$, $q \geq 2$, $p \neq 0$ and $p$ and $q$ relatively prime, the scaling law follows a power--law with exponent $1/q$.Comment: AMS-LaTeX2e, 29 pages with 8 figures, submitted to Nonlinearit

Transfer of energy to high frequencies in the cubic defocusing nonlinear Schrodinger equation

We consider the cubic defocusing nonlinear Schrödinger equation on the two dimensional torus. We exhibit smooth solutions for which the support of the conserved energy moves to higher Fourier modes. This behavior is quantified by the growth of higher Sobolev norms: given any δ[much less-than]1,K [much greater-than] 1, s > 1, we construct smooth initial data u 0 with ||u0||Hs , so that the corresponding time evolution u satisfies u(T)Hs[greater than]K at some time T. This growth occurs despite the Hamiltonian’s bound on ||u(t)||H1 and despite the conservation of the quantity ||u(t)||L2. The proof contains two arguments which may be of interest beyond the particular result described above. The first is a construction of the solution’s frequency support that simplifies the system of ODE’s describing each Fourier mode’s evolution. The second is a construction of solutions to these simpler systems of ODE’s which begin near one invariant manifold and ricochet from arbitrarily small neighborhoods of an arbitrarily large number of other invariant manifolds. The techniques used here are related to but are distinct from those traditionally used to prove Arnold Diffusion in perturbations of Hamiltonian systems

Visible Light Responsive Photocatalyst Induces Progressive and Apical-Terminus Preferential Damages on Escherichia coli Surfaces

BACKGROUND: Recent research shows that visible-light responsive photocatalysts have potential usage in antimicrobial applications. However, the dynamic changes in the damage to photocatalyzed bacteria remain unclear. METHODOLOGY/PRINCIPAL FINDINGS: Facilitated by atomic force microscopy, this study analyzes the visible-light driven photocatalyst-mediated damage of Escherichia coli. Results show that antibacterial properties are associated with the appearance of hole-like structures on the bacteria surfaces. Unexpectedly, these hole-like structures were preferentially induced at the apical terminus of rod shaped E. coli cells. Differentiating the damages into various levels and analyzing the percentage of damage to the cells showed that photocatalysis was likely to elicit sequential damages in E. coli cells. The process began with changing the surface properties on bacterial cells, as indicated in surface roughness measurements using atomic force microscopy, and holes then formed at the apical terminus of the cells. The holes were then subsequently enlarged until the cells were totally transformed into a flattened shape. Parallel experiments indicated that photocatalysis-induced bacterial protein leakage is associated with the progression of hole-like damages, further suggesting pore formation. Control experiments using ultraviolet light responsive titanium-dioxide substrates also obtained similar observations, suggesting that this is a general phenomenon of E. coli in response to photocatalysis. CONCLUSION/SIGNIFICANCE: The photocatalysis-mediated localization-preferential damage to E. coli cells reveals the weak points of the bacteria. This might facilitate the investigation of antibacterial mechanism of the photocatalysis