Antiferromagnetic phase transition in a nonequilibrium lattice of Rydberg atoms

We study a driven-dissipative system of atoms in the presence of laser excitation to a Rydberg state and spontaneous emission. The atoms interact via the blockade effect, whereby an atom in the Rydberg state shifts the Rydberg level of neighboring atoms. We use mean-field theory to study how the Rydberg population varies in space. As the laser frequency changes, there is a continuous transition between the uniform and antiferromagnetic phases. The nonequilibrium nature also leads to a novel oscillatory phase and bistability between the uniform and antiferromagnetic phases.Comment: 4 pages + appendi

Bifurcations and dynamics emergent from lattice and continuum models of bioactive porous media

We study dynamics emergent from a two-dimensional reaction--diffusion process modelled via a finite lattice dynamical system, as well as an analogous PDE system, involving spatially nonlocal interactions. These models govern the evolution of cells in a bioactive porous medium, with evolution of the local cell density depending on a coupled quasi--static fluid flow problem. We demonstrate differences emergent from the choice of a discrete lattice or a continuum for the spatial domain of such a process. We find long--time oscillations and steady states in cell density in both lattice and continuum models, but that the continuum model only exhibits solutions with vertical symmetry, independent of initial data, whereas the finite lattice admits asymmetric oscillations and steady states arising from symmetry-breaking bifurcations. We conjecture that it is the structure of the finite lattice which allows for more complicated asymmetric dynamics. Our analysis suggests that the origin of both types of oscillations is a nonlocal reaction-diffusion mechanism mediated by quasi-static fluid flow.Comment: 30 pages, 21 figure

Complex oscillations in the delayed Fitzhugh-Nagumo equation

Motivated by the dynamics of neuronal responses, we analyze the dynamics of the Fitzhugh-Nagumo slow-fast system with delayed self-coupling. This system provides a canonical example of a canard explosion for sufficiently small delays. Beyond this regime, delays significantly enrich the dynamics, leading to mixed-mode oscillations, bursting and chaos. These behaviors emerge from a delay-induced subcritical Bogdanov-Takens instability arising at the fold points of the S-shaped critical manifold. Underlying the transition from canard-induced to delay-induced dynamics is an abrupt switch in the nature of the Hopf bifurcation

A phenomenological approach to normal form modeling: a case study in laser induced nematodynamics

An experimental setting for the polarimetric study of optically induced dynamical behavior in nematic liquid crystal films has allowed to identify most notably some behavior which was recognized as gluing bifurcations leading to chaos. This analysis of the data used a comparison with a model for the transition to chaos via gluing bifurcations in optically excited nematic liquid crystals previously proposed by G. Demeter and L. Kramer. The model of these last authors, proposed about twenty years before, does not have the central symmetry which one would expect for minimal dimensional models for chaos in nematics in view of the time series. What we show here is that the simplest truncated normal forms for gluing, with the appropriate symmetry and minimal dimension, do exhibit time signals that are embarrassingly similar to the ones found using the above mentioned experimental settings. The gluing bifurcation scenario itself is only visible in limited parameter ranges and substantial aspect of the chaos that can be observed is due to other factors. First, out of the immediate neighborhood of the homoclinic curve, nonlinearity can produce expansion leading to chaos when combined with the recurrence induced by the homoclinic behavior. Also, pairs of symmetric homoclinic orbits create extreme sensitivity to noise, so that when the noiseless approach contains a rich behavior, minute noise can transform the complex damping into sustained chaos. Leonid Shil'nikov taught us that combining global considerations and local spectral analysis near critical points is crucial to understand the phenomenology associated to homoclinic bifurcations. Here this helps us construct a phenomenological approach to modeling experiments in nonlinear dissipative contexts.Comment: 25 pages, 9 figure

Abelian Integral Method and its Application

Oscillation is a common natural phenomenon in real world problems. The most efficient mathematical models to describe these cyclic phenomena are based on dynamical systems. Exploring the periodic solutions is an important task in theoretical and practical studies of dynamical systems. Abelian integral is an integral of a polynomial differential 1-form over the real ovals of a polynomial Hamiltonian, which is a basic tool in complex algebraic geometry. In dynamical system theory, it is generalized to be a continuous function as a tool to study the periodic solutions in planar dynamical systems. The zeros of Abelian integral and their distributions provide the number of limit cycles and their locations. In this thesis, we apply the Abelian integral method to study the limit cycles bifurcating from the periodic annuli for some hyperelliptic Hamiltonian systems. For two kinds of quartic hyperelliptic Hamiltonian systems, the periodic annulus is bounded by either a homoclinic loop connecting a nilpotent saddle, or a heteroclinic loop connecting a nilpotent cusp to a hyperbolic saddle. For a quintic hyperelliptic Hamiltonian system, the periodic annulus is bounded by a more degenerate heteroclinic loop, which connects a nilpotent saddle to a hyperbolic saddle. We bound the number of zeros of the three associated Abelian integrals constructed on the periodic structure by employing the combination technique developed in this thesis and Chebyshev criteria. The exact bound for each system is obtained, which is three. Our results give answers to the open questions whether the sharp bound is three or four. We also study a quintic hyperelliptic Hamiltonian system with two periodic annuli bounded by a double homoclinic loop to a hyperbolic saddle, one of the periodic annuli surrounds a nilpotent center. On this type periodic annulus, the exact number of limit cycles via Poincar{\\u27e} bifurcation, which is one, is obtained by analyzing the monotonicity of the related Abelian integral ratios with the help of techniques in polynomial boundary theory. Our results give positive answers to the conjecture in a previous work. We also extend the methods of Abelian integrals to study the traveling waves in two weakly dissipative partial differential equations, which are a perturbed, generalized BBM equation and a cubic-quintic nonlinear, dissipative Schr\ {o}dinger equation. The dissipative PDEs are reduced to singularly perturbed ODE systems. On the associated critical manifold, the Abelian integrals are constructed globally on the periodic structure of the related Hamiltonians. The existence of solitary, kink and periodic waves and their coexistence are established by tracking the vanishment of the Abelian integrals along the homoclinic loop, heteroclinic loop and periodic orbits. Our method is novel and easily applied to solve real problems compared to the variational analysis