## About

It is only in recent years that the scientific community has focused significant
resources in understanding the complexity of non-linear, distributed systems involving
many interacting components. While the toolbox of methods for dealing with such systems
is still relatively small, tools have been developed that provide useful qualitative
characterizations of the behavior of non-linear dynamical systems. Over the last thirty
years advances in the theory of nonlinear dynamics have made possible the qualitative
analysis of nonlinear models involving finitely many dimensions. Many of the concepts
that have emerged with the development of these tools are now well-known in nonlinear
science, and include *chaos*, *strange attractors*, *fractals*, and
*scaling*

More recently, an emerging focus has been on methods for characterizing the qualitative
behavior of high- and infinite-dimensional non-linear systems. These developments
promise to put within our reach the qualitative analysis of more realistic models of the
complex phenomena of interest to scientists. The power of these methods resides, in
part, in the realization that the qualitative behavior of many classes of complex
systems is insensitive to differences in the details of their underlying mechanisms and
processes. Many apparently "different" systems exhibit qualitively similar
(*universality classes*) of behaviours.

The purpose of the Center of Nonlinear Science (CNLS) is to make these techniques available and demonstrate to researcher in nonlinear science that, when used in conjunction with more traditional tools, these emerging methodologies are capable of providing relatively "complete" characterizations of important classes of nonlinear phenomena.

## Center for Complex and Nonlinear Science

## Recent Work (22)

### 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 is developed showing that landsurface evolution is described by Markovian stochastic processes. The landsurfaces are described by non-deterministic stochastic processes, characterized by a statistical quantity, the variogram, that exibits characteristic scalings. Thus the landsurfaces are shown to be SOC (Self-organized-critical) systems, or systems of color, possessing both an initial transient state and a stationary state. The theory reproduces established numerical results and measurements from DEMs (digital elevation models).

### Discrete and continuous models of the dynamics of pelagic fish: application to the capelin

In this paper, we study simulations of the schooling and swarming behavior of a mathematical model for the motion of pelagic fish. We use a derivative of a discrete model of interacting particles originated by Vicsek, Czir´ok et al. [6] [5] [23] [24]. Recently, a system of ODEs was derived from this model [2], and using these ODEs, we find transitory and long-term behavior of the discrete system. In particular, we numerically find stationary, migratory, and circling behavior in both the discrete and the ODE model and two types of swarming behavior in the discrete model. The migratory solutions are numerically stable and the circling solutions are metastable. We find a stable circulating ring solution of the discrete system where the fish travel in opposite directions within an annulus. We also find the origin of noise-driven swarming when repulsion and attraction are absent and the fish interact solely via orientation.

### Modeling and Simulations of the Spawning Migration of Pelagic Fish

We model the spawning migration of the Icelandic capelin stock using an interacting particle model with added environmental fields. Without artificial forcing terms or a homing instinct, we qualitatively reproduce several observed spawning migrations using available temperature data and approximated currents. The simulations include orders of magnitude more particles than many similar models, affecting the global behavior of the system. Without environmental fields, we analyze how various parameters scale with the number of particles. In particular we present scaling behavior between the size of the time step, radii of the sensory zones and the number of particles in the system. We then discuss incorporating environmental data into the model.