The logistic equation is one of the most familiar nonlinear differential equations in the biological and social sciences. Herein we provide an exact solution to an extension of this equation to incorporate memory through the use of fractional derivatives in time. The solution to the fractional...

Exact first and second order statistics of a damped mechanical oscillator with a fluctuating frequency and driven by a fluctuating force are obtained. All the cumulants of the fluctuating frequency are assumed to be delta correlated, while those of the driving force are of arbitrary form. The...

Scale invariance, the property relating time series across multiple scales, has provided a new perspective of physiological phenomena and their underlying control systems. The traditional “signal plus noise” paradigm of the engineer was first replaced with a model in which biological time...

The allometric relationship between the basal metabolic rate B and total body mass M is B = aMb, where the allometry coefficient a and the allometry exponent b have been fit to various data sets for over 150 years. The best fit of the allometry exponent to 391 mammalian species is given by...

The dynamics of a two-state decision-making model (DMM) with a tunable control parameter K is described. On all-to-all (ATA) networks, the system undergoes a phase transition at a critical value of Kc=1. Scale-free networks were also found to undergo phase transitions, but the value of Kc...

We examine the stability properties of equilibrium moments of all orders for the damped mechanical oscillator with a delta correlated fluctuating frequency. A Markovian master equation is derived startimg from a frequency fluctuation process with finite correlation time τc and the limit τc→0...

We address the issue of the dynamical origin of scale-free link distributions. We study a two-dimensional lattice of cooperatively interacting units. Although the units interact only with the four nearest neighbors, a sufficiently large cooperation strength generates dynamically a scale-free...

The brain is probably the most interesting example of a complex network having 1/f variability as determined through the analysis of EEG time series and magnetoencephalogram recordings. Herein we develop a theory of 1/f noise of human cognition to explain the recent experimental observations...

A formally exact expression for a lattice model of the perturbed strain field in a grainy polycrystalline material is constructed using a discrete T-matrix. The matrix elements of T are expressed in terms of a self-consistent lattice Green's function and the “effective”, i.e., renormalized,...

We determine an approximate renormalized equation of evolution for an arbitrary nonlinear single-degree-of-freedom system externally driven by Gaussian parametric fluctuations of finite correlation time. The renormalization scheme used here gives a second order equation with a...