Hopf bifurcations in time-delay systems with band-limited feedback

We investigate the steady-state solution and its bifurcations in time-delay systems with band-limited feedback. This is a first step in a rigorous study concerning the effects of AC-coupled components in nonlinear devices with time-delayed feedback. We show that the steady state is globally stable for small feedback gain and that local stability is lost, generically, through a Hopf bifurcation for larger feedback gain. We provide simple criteria that determine whether the Hopf bifurcation is supercritical or subcritical based on the knowledge of the first three terms in the Taylor-expansion of the nonlinearity. Furthermore, the presence of double-Hopf bifurcations of the steady state is shown, which indicates possible quasiperiodic and chaotic dynamics in these systems. As a result of this investigation, we find that AC-coupling introduces fundamental differences to systems of Ikeda-type [Ikeda et al., Physica D 29 (1987) 223-235] already at the level of steady-state bifurcations, e.g. bifurcations exist in which limit cycles are created with periods other than the fundamental ``period-2'' mode found in Ikeda-type systems.Comment: 32 pages, 5 figures, accepted for publication in Physica D: Nonlinear Phenomen

Engineering Gaussian states of light from a planar microcavity

Quantum fluids of light in a nonlinear planar microcavity can exhibit antibunched photon statistics at short distances due to repulsive polariton interactions. We show that, despite the weakness of the nonlinearity, the antibunching signal can be amplified orders of magnitude with an appropriate free-space optics scheme to select and interfere output modes. Our results are understood from the unconventional photon blockade perspective by analyzing the approximate Gaussian output state of the microcavity. In a second part, we illustrate how the temporal and spatial profile of the density-density correlation function of a fluid of light can be reconstructed with free-space optics. Also here the nontrivial (anti)bunching signal can be amplified significantly by shaping the light emitted by the microcavity

Prosperity and Stagnation in Capitalist Economies

The KMG growth dynamics in Chiarella and Flaschel (2000) assume that wages, prices and quantities adjust sluggishly to disequilibria in labor and goods markets. This paper modifies the KMG model by introducing Steindlian features of capital accumulation and income distribution. The resulting KMGS(teindl) model replaces the neoclassical medium- and long-run features of the originalKMG model by a Steindlian approach to capital accumulation, as developed in a paper by Flaschel and Skott (2005). The model is of dimension 4 or 5, depending on the specification of the labor supply. We prove stability assertions and show that loss of stability always occurs by way of Hopf-bifurcations. When global stability gets lost, a nonlinear form of the Steindlian reserve army mechanism can ensure bounded dynamics. These dynamics are studied numerically and shown to exhibit long phases of prosperity, but also long phases of stagnant growth. JEL Categories: E24, E31, E32.KMGS dynamics, accelerating growth, stagnant growth, normal / adverse income shares adjustment, reserve army mechanisms.

The Bramson delay in the non-local Fisher-KPP equation

We consider the non-local Fisher-KPP equation modeling a population with individuals competing with each other for resources with a strength related to their distance, and obtain the asymptotics for the position of the invasion front starting from a localized population. Depending on the behavior of the competition kernel at infinity, the location of the front is either $2t - ({3}/2)\log t + O(1)$, as in the local case, or $2t - O(t^\beta)$ for some explicit $\beta \in (0,1)$. Our main tools here are alocal-in-time Harnack inequality and an analysis of the linearized problem with a suitable moving Dirichlet boundary condition. Our analysis also yields, for any $\beta\in(0,1)$, examples of Fisher-KPP type non-linearities $f\_\beta$ such that the front for the localFisher-KPP equation with reaction term $f\_\beta$ is at $2t - O(t^\beta)$

The Bramson delay in the non-local Fisher-KPP equation

We consider the non-local Fisher-KPP equation modeling a population with individuals competing with each other for resources with a strength related to their distance, and obtain the asymptotics for the position of the invasion front starting from a localized population. Depending on the behavior of the competition kernel at infinity, the location of the front is either $2t - ({3}/2)\log t + O(1)$, as in the local case, or $2t - O(t^\beta)$ for some explicit $\beta \in (0,1)$. Our main tools here are a local-in-time Harnack inequality and an analysis of the linearized problem with a suitable moving Dirichlet boundary condition. Our analysis also yields, for any $\beta\in(0,1)$, examples of Fisher-KPP type non-linearities $f_\beta$ such that the front for the local Fisher-KPP equation with reaction term $f_\beta$ is at $2t - O(t^\beta)$