Dynamical Systems in One and Two Dimensions: A Geometrical Approach

This chapter is intended as an introduction or tutorial to nonlinear dynamical systems in one and two dimensions with an emphasis on keeping the mathematics as elementary as possible. By its nature such an approach does not have the mathematical rigor that can be found in most textbooks dealing with this topic. On the other hand it may allow readers with a less extensive background in math to develop an intuitive understanding of the rich variety of phenomena that can be described and modeled by nonlinear dynamical systems. Even though this chapter does not deal explicitly with applications – except for the modeling of human limb movements with nonlinear oscillators in the last section – it nevertheless provides the basic concepts and modeling strategies all applications are build upon. The chapter is divided into two major parts that deal with oneand two-dimensional systems, respectively. Main emphasis is put on the dynamical features that can be obtained from graphs in phase space and plots of the potential landscape, rather than equations and their solutions. After discussing linear systems in both sections, we apply the knowledge gained to their nonlinear counterparts and introduce the concepts of stability and multistability, bifurcation types and hysteresis, heteroand homoclinic orbits as well as limit cycles, and elaborate on the role of nonlinear terms in oscillators. 1 One-Dimensional Dynamical Systems The one-dimensional dynamical systems we are dealing with here are systems that can be written in the form dx(t) dt = ẋ(t) = f [x(t),{λ}] (1) In (1) x(t) is a function, which, as indicated by its argument, depends on the variable t representing time. The left and middle part of (1) are two ways of expressing Armin Fuchs Center for Complex Systems & Brain Sciences, Department of Physics, Florida Atlantic University e-mail: [email protected] R. Huys and V.K. Jirsa (Eds.): Nonlinear Dynamics in Human Behavior, SCI 328, pp. 1–33. springerlink.com c © Springer-Verlag Berlin Heidelberg 2010