A robust numerical method to study oscillatory instability of gap solitary waves

The spectral problem associated with the linearization about solitary waves of spinor systems or optical coupled mode equations supporting gap solitons is formulated in terms of the Evans function, a complex analytic function whose zeros correspond to eigenvalues. These problems may exhibit oscillatory instabilities where eigenvalues detach from the edges of the continuous spectrum, so called edge bifurcations. A numerical framework, based on a fast robust shooting algorithm using exterior algebra is described. The complete algorithm is robust in the sense that it does not produce spurious unstable eigenvalues. The algorithm allows to locate exactly where the unstable discrete eigenvalues detach from the continuous spectrum. Moreover, the algorithm allows for stable shooting along multi-dimensional stable and unstable manifolds. The method is illustrated by computing the stability and instability of gap solitary waves of a coupled mode model.Comment: key words: gap solitary wave, numerical Evans function, edge bifurcation, exterior algebra, oscillatory instability, massive Thirring model. accepted for publication in SIAD

A normal form for excitable media

We present a normal form for travelling waves in one-dimensional excitable media in form of a differential delay equation. The normal form is built around the well-known saddle-node bifurcation generically present in excitable media. Finite wavelength effects are captured by a delay. The normal form describes the behaviour of single pulses in a periodic domain and also the richer behaviour of wave trains. The normal form exhibits a symmetry preserving Hopf bifurcation which may coalesce with the saddle-node in a Bogdanov-Takens point, and a symmetry breaking spatially inhomogeneous pitchfork bifurcation. We verify the existence of these bifurcations in numerical simulations. The parameters of the normal form are determined and its predictions are tested against numerical simulations of partial differential equation models of excitable media with good agreement.Comment: 22 pages, accepted for publication in Chao

Does individual behavior converge to policy recommendations in times of pandemic? Evidence from COVID-19 in US states

The COVID-19 pandemic is an exceptional shock on human habitual behavior and provides a rare opportunity to analyze resilience in preferences. We use Google\u27s mobility and policy stringency indices to investigate if policy maker and resident „preferences" align over the period. Differences in utility across the ten largest states in the United States should lead to idiosyncratic response on perceived cost of restrictions and associated risk attitudes in policy respond. We conduct structural break and rolling unit root tests on estimated residual. Our results suggest that individual behavior converges to the policy prescriptions within the time span up to 18 months

The Compact UV Nucleus of M33

The most luminous X-ray source in the Local Group is associated with the nucleus of M33. This source, M33 X-8, appears modulated by ~20% over a ~106 day period, making it unlikely that the combined emission from unresolved sources could explain the otherwise persistent ~1e39 erg/s X-ray flux (Dubus et al. 1997, Hernquist et al. 1991). We present here high resolution UV imaging of the nucleus with the Planetary Camera of the HST undertaken in order to search for the counterpart to X-8. The nucleus is bluer and more compact than at longer wavelength images but it is still extended with half of its 3e38 erg/s UV luminosity coming from the inner 0.14". We cannot distinguish between a concentrated blue population and emission from a single object.Comment: 3 figures, accepted for publication in ApJ Letter

A test for a conjecture on the nature of attractors for smooth dynamical systems

Dynamics arising persistently in smooth dynamical systems ranges from regular dynamics (periodic, quasiperiodic) to strongly chaotic dynamics (Anosov, uniformly hyperbolic, nonuniformly hyperbolic modelled by Young towers). The latter include many classical examples such as Lorenz and H\'enon-like attractors and enjoy strong statistical properties. It is natural to conjecture (or at least hope) that most dynamical systems fall into these two extreme situations. We describe a numerical test for such a conjecture/hope and apply this to the logistic map where the conjecture holds by a theorem of Lyubich, and to the Lorenz-96 system in 40 dimensions where there is no rigorous theory. The numerical outcome is almost identical for both (except for the amount of data required) and provides evidence for the validity of the conjecture.Comment: Accepted version. Minor modifications from previous versio