Stephen Smale

From topology to computation : proceedings of the Smalefest

This volume contains papers given at a symposium in the honour of Stephen Smale at the University of California, Berkeley, ranging over fields in applied mathematics that reflect Smales interests. Topics discussed include dynamical systems, differential topology and computational theory.

Stability and Genericity in Dynamical Systems

A general reference to this subject, with examples, written about the summer of 1967 is [7], (reported in a recent Bourbaki seminar by C. Godbillon). Here I will try to emphasize developments since. An important source of much of this more recent work should appear in the immediate future in [1].

First-Order Equations

This chapter describes the behavior of first-order differential equations. The first example is the most basic differential equation—namely, the unlimited population growth model. Later models include the logistic population growth model. Where it is assumed that there is a carrying capacity for the given population. Then harvesting is introduced into the model. This leads to the concept of bifurcation. To understand these bifurcations, numerous qualitative techniques are introduced, including slope field, solution graphs, phase line, and bifurcation diagram. Then the logistic model is modified to allow for periodic harvesting, which leads to a nonautonomous differential equation. Periodic solutions are introduced via this model. The Poincare map is then used to prove the existence of these types of solutions and to understand the nearby behavior.

Discrete Dynamical Systems

The Lorenz model from the previous chapter motivates the excursion in this chapter into discrete dynamical systems. In this chapter, iteration of functions on the real line is the central theme. Fixed and periodic points are characterized. New types of bifurcations arise. The discrete logistic population model is investigated in depth. This family of maps is shown to behave chaotically for certain parameters. The method for understanding this chaotic behavior called symbolic dynamics is described in detail. The Cantor middle-thirds set plays a prominent role in understanding the chaos. Finally, the orbit diagram provides a qualitative picture of the behavior of iterated maps.

The Lorenz System

This chapter investigates the famous Lorenz system from meteorology. This was one of the first examples of a differential equation that was shown to exhibit chaotic behavior. Linearization provides a mechanism to understand the behavior near the equilibria of this system. Global techniques then show that there is an attractor for this system that is neither an equilibrium point nor a limit cycle. A specific model for this attractor is then constructed. This reduces the three-dimensional system of differential equations to a two-dimensional iterated function, which, in turn, is reduced to a one-dimensional mapping.

Applications in Biology

Various examples of differential equations that arise in biology are discussed in this section. The first is a model for infectious diseases known as the SIRS model (susceptible, infected, and recovered). The second is the predator–prey system in which it is shown that all solutions lie on closed orbits. This changes when the model is modified to account for limited growth of the population. The third model is a competitive species model where it is shown that all solutions now tend to one of a finite number of equilibria. Previously introduced techniques such as linearization, nullclines, and the Poincare-Bendixson Theorem are used to investigate these models. Later explorations include a competition and harvesting model and a SIRS model using zombies.

Applications in Circuit Theory

Various differential equations that arise in circuit theory are investigated in this chapter. The first example is the RLC circuit equation. A more general type of system is the Lienard equation. The final example is the van der Pol equation. Here we show that all nonzero solutions tend to a periodic solution. Again the associated Poincare map is the tool that provides this result. A Hopf bifurcation often arises in circuit equations, and these new types of bifurcations are described in this chapter. A final exploration involves a system from neurodynamics, the Fitzhugh–Nagumo equations.

Global Nonlinear Techniques

Unlike the previous chapter, which focused on local techniques for nonlinear systems, this chapter introduces global ideas to understand the behavior of these systems. In particular, nullclines are used to partition the phase space into regions where the vector field points in certain directions. This allows us to understand heteroclinic bifurcations. A further notion of stability, Liapunov stability, is introduced. The phase plane for the ideal pendulum is described using the total energy function. Finally, two special types of systems are introduced: gradient systems and Hamiltonian systems.

Higher-Dimensional Linear Systems

This chapter tackles higher-dimensional linear systems of differential equations. Here the system involves an n × n matrix, so all of the linear algebra of the previous chapter comes into play.

Higher-Dimensional Linear Algebra

This chapter moves on to higher-dimensional linear algebra. The concepts of linear independence, subspace, and spanning set arise.