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We study resonance curves of nonlinear dynamical systems with chaotic forcing functions. We use the calculus of variations to determine the forcing function that induces the largest response. We compute the resonant forcing for a set of model systems and determine the response of the dynamical system to each forcing function. We show that the response is largest if the model system matches the dynamical system. We find that the signal to noise ratio is particularly large if one of the Lyapunov exponents is large.

We review localization with non-Hermitian time evolution as applied to simple models of population biology with spatially varying growth profiles and convection. Convection leads to a constant imaginary vector potential in the Schrodinger-like operator which appears in linearized growth models. We illustrate the basic ideas by reviewing how convection affects the evolution of a population influenced by a simple square well growth profile. Results from discrete lattice growth models in both one and two dimensions are presented. A set of similarity transformations which lead to exact results for the spectrum and winding numbers of eigenfunctions for random growth rates in one dimension is described in detail. We discuss the influence of boundary conditions, and argue that periodic boundary conditions lead to results which are in fact typical of a broad class of growth problems with convection.

We’ve been working on the crackling noise in hysteresis loops [1, 2, 3, 4, 5, 6]. Hysteresis occurs when you push and pull on a system with an external force, and the response lags behind the force. The hysteresis loop is the graph of force (say, an external magnetic field H) versus the response (say, the magnetization M of the material). In many materials, the hysteresis loop is not actually microscopically smooth: it is composed of small bursts, or avalanches (figure 1). In many first-order phase transitions, these bursts cause acoustic emission (crackling noise); in magnets, they are called Barkhausen noise.