Preface to the Special Issue: Nonlinear systems theory and applications in engineering,control and life sciences

Nonlinear Dynamics -     

An ODE Model of the Motion of Pelagic Fish

A system of ordinary differential equations (ODEs) is derived from a discrete system of Vicsek, Czirók et al. [Phys. Rev. Lett. 75(6):1226–1229, 1995], describing the motion of a school of fish. Classes of linear and stationary solutions of the ODEs are found and their stability explored using equivariant bifurcation theory. The existence of periodic and toroidal solutions is also proven under deterministic perturbations and structurally stable heteroclinic connections are found. Applications of the model to the migration of the capelin, a pelagic fish that undertakes an extensive migration in the North Atlantic, are discussed and simulation of the ODEs presented.     

Qualitative analysis of radiating breathers

Perturbed sine-Gordon equations are investigated numerically to see if they have breather solutions. It is shown that breathers radiate, blow up, and split into kink-antikink pairs under most perturbations. The two perturbations proven by Birnir, McKean, and Weinstein not to produce radiation, of the first order in the perturbation parameter, a sin(u) + b ucos(u) and 1+3cos(u) - 4cos(u/2) + 4cos(u)log(cos(u/4)), stop radiating first-order radiation after adjusting the initial breather by the emission of such radiation. The first perturbation is a scaling of the breather, the second is shown to give a quasi-periodic orbit, which is a two-breather, on a torus. © 1994 John Wiley & Sons, Inc.     

The Stochastic Theory of Fluvial Landsurfaces

A stochastic theory of fluvial landsurfaces is developed for transport-limited erosion, using well-established models for the water and sediment fluxes. The mathematical models and analysis are developed showing that some aspects of landsurface evolution can be described by Markovian stochastic processes. The landsurfaces are described by nondeterministic stochastic processes, characterized by a statistical quantity, the variogram, that exhibits characteristic scalings. Thus the landsurfaces are shown to be self-organized critical (SOC) systems, possessing both an initial transient state and a stationary state, characterized by respectively temporal and spatial scalings. The mathematical theory of SOC systems is developed and used to identify three stochastic processes that shape the surface. The SOC theory of landsurfaces reproduces established numerical results and measurements from digital elevation models (DEMs).     

The Kolmogorov–Obukhov Statistical Theory of Turbulence

In 1941 Kolmogorov and Obukhov postulated the existence of a statistical theory of turbulence, which allows the computation of statistical quantities that can be simulated and measured in a turbulent system. These are quantities such as the moments, the structure functions and the probability density functions (PDFs) of the turbulent velocity field. In this paper we will outline how to construct this statistical theory from the stochastic Navier–Stokes equation. The additive noise in the stochastic Navier–Stokes equation is generic noise given by the central limit theorem and the large deviation principle. The multiplicative noise consists of jumps multiplying the velocity, modeling jumps in the velocity gradient. We first estimate the structure functions of turbulence and establish the Kolmogorov–Obukhov 1962 scaling hypothesis with the She–Leveque intermittency corrections. Then we compute the invariant measure of turbulence, writing the stochastic Navier–Stokes equation as an infinite-dimensional Ito process, and solving the linear Kolmogorov–Hopf functional differential equation for the invariant measure. Finally we project the invariant measure onto the PDF. The PDFs turn out to be the normalized inverse Gaussian (NIG) distributions of Barndorff-Nilsen, and compare well with PDFs from simulations and experiments.