CIM Thematic Term on Mathmatics and the Environment

Title: Chaotic dynamics and applications

Summary:

Lecture 1. Introduction and basic concepts.

Chaotic behavior is possible for a wide variety of systems. This lecture will begin by an illustration of this fact using examples from several situations where chaos occurs. Following that, important background material for subsequent lectures will be introduced including the following: deterministic dynamical systems, phase space, the Poincaré- surface of section technique, maps and flows, and the concept of attractors. Before the advent of chaotic dynamics it was widely believed that the possible configurations of attracting motions in phase space were geometrically rather simple (for example, a single point or a closed loop). It is now apparent, however, that the attractors of commonly encountered chaotic systems can have extremely singular structure. These sets are fractals and have the property that their dimension can be non-integer. Hence these attractors have been called strange attractors. From the practical point of view, chaos has an even more drastic effect. Namely, it leads to extreme sensitivity of the motion to small changes in initial conditions. The important consequence of this is that chaos can make the forecasting of the system state beyond a certain time virtually impossible.

Lecture 2. Fractal basin boundaries.

It is extremely common that a system can evolve to different final states (attractors) depending on the initial conditions that the system starts from. The region in the phase space corresponding to initial conditions which eventually result in a given final state is the basin of attraction for that final state. As a consequence of chaotic dynamics, the boundaries separating different basins can be very complicated, and, in fact, they can be fractal. This can occur even when the attractors themselves are not chaotic. Examples illustrating this for simple systems will be given, and the important consequence of the fractal nature of these boundaries as an obstruction to predictability will be illustrated.

Lecture 3. Obstruction to modelling and shadowing.

Scientists attempt to understand physical phenomena by constructing models. A model serves as a link between scientists and nature, and one fundamental goal is to develop models whose solutions accurately reflect the nature of the physical process. A dynamical model uses simplifying assumptions and approximations in the hope of capturing the essential characteristics of how a physical system evolves with time. The question of whether a model accurately reflects nature is one constantly faced by scientists. Recently, we have discovered that there exists a new level of mathematical difficulty, brought from the theory of dynamical systems, which can limit our ability to represent nature using deterministic models. Specifically, we have discovered that certain chaotic systems found in nature cannot be modeled, particularly higher dimensional chaotic systems. No model of such a system produces solutions of reasonable length that are realized by nature. (Furthermore, for these processes, the numerical solutions of the models do not approximate any actual model solutions.)

Lecture 4. Chaos: control and communication.

It is common for systems to evolve with time in a chaotic way. In practice, however, it is often desired that chaos be avoided or modified for the system to be optimized with respect to some performance criterion. Given a system which behaves chaotically, one approach might be to make some large (and possibly costly) alteration in the system which completely changes its dynamics in such a way as to achieve the desired objectives. Here we assume that this avenue is not available. Thus we address the following question: Given a chaotic system, how can we obtain improved performance and achieve a desired behavior by making only small controlling temporal perturbations in an accessible system parameter. Controlled chaotic systems offer an advantage in flexibility in that any one of a number of different behaviors, chaotic or not, it can be stabilized by the small control, and the choice can be switched from one to another depending on the current desired system performance. I will give many relevant applications to the sciences and engineering including applications to biological systems. In particular, I will show that we can use the close connection between the theory of chaotic systems and information theory in a way that is more than purely formal. I will show that small perturbations can be utilized to cause the symbolic dynamics of a chaotic system to track a prescribed symbol sequence thus allowing us to encode any desired message in the signal from a chaotic oscillator. The natural complexity of chaos thus provides a vehicle for information transmission in the usual sense. Furthermore, I will argue that this method of communication will often have technological advantages. Finally, I will present results of an experiment which demonstrates that chaos can be used to transmit information. In it the symbolic dynamics of a chaotic electrical oscillator is controlled to carry some desired message by using small perturbing current pulses. I will show a movie in which the communication experiment is done in real time.

Lecture 5. The plankton paradox and other issues.

Nature is permeated by phenomena in which active processes, such as chemical reactions and biological interactions, take place in environmental flows. They include the dynamics of growing population of plankton in the oceans and the evolving distribution of ozone in the polar stratosphere. I will show that if the dynamics of active particles in environmental flows is chaotic, then necessarily the concentration of particles have the observed fractal filamentary structures. These structures, in turn, are the skeletons and the dynamic catalysts of the active processes, yielding an unusual singularly enhanced productivity. I will then suggest that this singular productivity could be the hydrodynamic explanation for the plankton paradox, in which an extremely large number of species are able to coexist, negating the competitive exclusion principle that asserts the survival of only the most perfectly adapted to each limiting resource.

Title: Dynamical systems, numerical experiments and super-computing

Summary:

Dynamical systems study the evolution models of natural phenomena and the simplified models which help to understand them. They can be given in deterministic form, either by means of ordinary differential equations, partial differential equations or discrete maps. They are useful in all domains of science and technology.

In their study tools from all areas of mathematics are used. But for systems with some degree of complexity it is impossible to produce a fairly complete description of the evolution in the space of states, and its dependence with respect to parameters, without using numerical techniques. They are essential for concrete applications and very useful even for theoretical studies. They can be seen as an experimental part of mathematics.

In the last years it has become possible to achieve a generalisation in the systematic use of numerical experiments, due to the availability of large arrays of processors working in parallel with a reduced cost. But the impact of new algorithms has been even larger. This makes feasible to face problems of larger and larger complexity.

In the lectures some aspects of the general methodology and several examples will be presented.