Heart rate variability (HRV) measures cycle-to-cycle correlations in the instantaneous oscillation period of the
heart. In this paper it is shown that a simple model process, consisting of a sum of uncoupled sinusoidal oscillators
with slightly different frequencies, has a HRV spectrum with a 1/f scaling over a range of frequencies. This implies
that the appearance of 1/f HRV spectra in experiments should not be considered evidence of oscillator coupling
or other more complex dynamics. The origin of the 1/f scaling in the model is examined analytically, and its
dependence upon the sampling of low-amplitude fluctuations of the process is highlighted.
The expected steady-state fraction of active nodes in Watts' model of threshold dynamics on random networks is
determined analytically. The analysis applies to random graphs with arbitrary degree distributions, and includes
the effect of finite seed fractions. The seed fraction is shown to have a strong impact upon the existence of global
cascades and Watts' cascade condition is extended to include these effects.
The motion of elastically coupled Brownian particles in ratchet-like potentials has attracted much recent interest due to its application to transport processes in many fields, including models of DNA polymers. We consider the influence of the type of interacting force on the transport of two particles in a one-dimensional flashing ratchet. Our aim is to examine whether the common assumption of elastic coupling captures the important features of ratchet transport when the inter-particle forces are more complex. We compare Lennard-Jones type interaction to the classical case of elastically coupled particles. Numerical simulations agree with analytical formulas for the limiting cases where the coupling is very weak or very strong. Parameter values where the Lennard-Jones force is not well approximated by a linearization of the force about the equilibrium distance are identified.
In this paper we discuss the effects of white noise on the spectral lineshape of a simple two dimensional oscillator and compare to the phase noise spectrum predicted by the methods of Demir and Kaertner. A numerical method based on the Fokker-Planck equation is employed to directly calculate the spectrum of the dynamical system. These results are then used as a benchmark to assess the efficiency of a novel small-noise perturbation method we have developed. Due to the simplicity of the oscillator it is possible to solve the approximated partial differential equation using both characteristics; these solutions are then compared to the exact numerical results. We elucidate the effect of the amplitude-phase coupling term which causes the spectral lineshape to become non-Lorentzian.
Motion in Gaussian random space-time fields ("dynamical disorder") is proposed as a model for certain dynamical systems where fluctuations play an important role. Analytical and numerical methods adapted from the study of passive scalar turbulence are applied to two examples: phase diffusion in noisy nonlinear oscillators and demonstrating the existence of 1/f phase noise in the mean field of Kuramoto's coupled oscillators model.
Models of stock price fluctuations based on simple random walks do not agree with empirical stock price data. We point out an analogy with motion in a one-dimensional random field which generalizes the stock dynamics to include random dependence on the current price in a natural way. Results of an analytically tractable limit are presented, demonstrating that some of the characteristics of real stock data may be reproduced by such models. Shortcomings of the model are noted, and a numerical simulation method for extension beyond the analytically tractable case is presented.