Connecting period-doubling cascades to chaos

The appearance of infinitely-many period-doubling cascades is one of the most prominent features observed in the study of maps depending on a parameter. They are associated with chaotic behavior, since bifurcation diagrams of a map with a parameter often reveal a complicated intermingling of period-doubling cascades and chaos. Period doubling can be studied at three levels of complexity. The first is an individual period-doubling bifurcation. The second is an infinite collection of period doublings that are connected together by periodic orbits in a pattern called a cascade. It was first described by Myrberg and later in more detail by Feigenbaum. The third involves infinitely many cascades and a parameter value $\mu_2$ of the map at which there is chaos. We show that often virtually all (i.e., all but finitely many) ``regular'' periodic orbits at $\mu_2$ are each connected to exactly one cascade by a path of regular periodic orbits; and virtually all cascades are either paired -- connected to exactly one other cascade, or solitary -- connected to exactly one regular periodic orbit at $\mu_2$. The solitary cascades are robust to large perturbations. Hence the investigation of infinitely many cascades is essentially reduced to studying the regular periodic orbits of $F(\mu_2, \cdot)$. Examples discussed include the forced-damped pendulum and the double-well Duffing equation.Comment: 29 pages, 13 figure

Sufficient Conditions for Polynomial Asymptotic Behaviour of the Stochastic Pantograph Equation

This paper studies the asymptotic growth and decay properties of solutions of the stochastic pantograph equation with multiplicative noise. We give sufficient conditions on the parameters for solutions to grow at a polynomial rate in $p$-th mean and in the almost sure sense. Under stronger conditions the solutions decay to zero with a polynomial rate in $p$-th mean and in the almost sure sense. When polynomial bounds cannot be achieved, we show for a different set of parameters that exponential growth bounds of solutions in $p$-th mean and an almost sure sense can be obtained. Analogous results are established for pantograph equations with several delays, and for general finite dimensional equations.Comment: 29 pages, to appear Electronic Journal of Qualitative Theory of Differential Equations, Proc. 10th Coll. Qualitative Theory of Diff. Equ. (July 1--4, 2015, Szeged, Hungary

Low-energy behavior of spin-liquid electron spectral functions

We calculate the electron spectral function for a spin-liquid with a spinon Fermi surface and a Dirac spin-liquid. Calculations are based upon the slave-rotor mean-field theory. We consider the effect of gauge fluctuations using a simple model and find the behavior is not strongly modified. The results, distinct from conventional Mott insulator or band theory predictions, suggest that measuring the spectral function e.g. via ARPES could help in the experimental verification and characterization of spin liquids.Comment: 7 pages, 7 figure