Large normally hyperbolic cylinders in a priori stable Hamiltonian systems

We prove the existence of normally hyperbolic invariant cylinders in nearly integrable hamiltonian systems

Quasiperiodic perturbations of heteroclinic attractor networks

We consider heteroclinic attractor networks motivated by models of competition between neural populations during binocular rivalry. We show that gamma distributions of dominance times observed experimentally in binocular rivalry and other forms of bistable perception, commonly explained by means of noise in the models, can be achieved with quasiperiodic perturbations. For this purpose, we present a methodology based on the separatrix map to model the dynamics close to heteroclinic networks with quasiperiodic perturbations. Our methodology unifies two different approaches, one based on Melnikov integrals and the other one based on variational equations. We apply it to two models: first, to the Duffing equation, which comes from the perturbation of a Hamiltonian system and, second, to a heteroclinic attractor network for binocular rivalry, for which we develop a suitable method based on Melnikov integrals for non-Hamiltonian systems. In both models, the perturbed system shows chaotic behavior, while dominance times achieve good agreement with gamma distributions. Moreover, the separatrix map provides a new (discrete) model for bistable perception which, in addition, replaces the numerical integration of time-continuous models and, consequently, reduces the computational cost and avoids numerical instabilitiesPeer ReviewedPostprint (author's final draft

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

Arnold diffusion for a complete family of perturbations

In this work we illustrate the Arnold diffusion in a concrete example — the a priori unstable Hamiltonian system of 2 + 1/2 degrees of freedom H(p, q, I, f, s) = p2/2+ cos q - 1 + I2/2 + h(q, f, s; e) — proving that for any small periodic perturbation of the form h(q, f, s; e) = e cos q (a00 + a10 cosf + a01 cos s) (a10a01 ¿ 0) there is global instability for the action. For the proof we apply a geometrical mechanism based on the so-called scattering map. This work has the following structure: In the first stage, for a more restricted case (I* ~ p/2µ, µ = a10/a01), we use only one scattering map, with a special property: the existence of simple paths of diffusion called highways. Later, in the general case we combine a scattering map with the inner map (inner dynamics) to prove the more general result (the existence of instability for any µ). The bifurcations of the scattering map are also studied as a function of µ. Finally, we give an estimate for the time of diffusion, and we show that this time is primarily the time spent under the scattering map.Peer ReviewedPostprint (published version

Separatrix splitting at a Hamiltonian 02iω0^2 i\omega bifurcation

We discuss the splitting of a separatrix in a generic unfolding of a degenerate equilibrium in a Hamiltonian system with two degrees of freedom. We assume that the unperturbed fixed point has two purely imaginary eigenvalues and a double zero one. It is well known that an one-parametric unfolding of the corresponding Hamiltonian can be described by an integrable normal form. The normal form has a normally elliptic invariant manifold of dimension two. On this manifold, the truncated normal form has a separatrix loop. This loop shrinks to a point when the unfolding parameter vanishes. Unlike the normal form, in the original system the stable and unstable trajectories of the equilibrium do not coincide in general. The splitting of this loop is exponentially small compared to the small parameter. This phenomenon implies non-existence of single-round homoclinic orbits and divergence of series in the normal form theory. We derive an asymptotic expression for the separatrix splitting. We also discuss relations with behaviour of analytic continuation of the system in a complex neighbourhood of the equilibrium

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