Introduction to Nonlinear Dynamics and Self-Organization

Nonlinear dynamics occur in a wide range of natural, technological, and social systems. They govern mechanical, hydrodynamic, and chemical processes as well as biological rhythms, neuronal activity, ecological systems, and collective social phenomena. Even simple models can give rise to surprising dynamics, ranging from extreme sensitivity to initial conditions and the coexistence of multiple stable states to sudden qualitative changes in dynamical behavior.

The study of nonlinear systems is one of the core areas of modern physics and at the same time provides an important foundation for understanding nonlinear and collective phenomena across many other scientific disciplines. Despite the wide range of specific applications, many of these systems exhibit similar behaviors and follow universal principles that can be described using a common mathematical framework.

This lecture provides an introduction to the fundamental concepts and methods of nonlinear dynamics. Topics include phase spaces, attractors, stability and multistability, bifurcations, oscillations, critical transitions, chaos, and stochastically driven systems. Building on these foundations, selected phenomena of self-organization in complex systems are discussed, including synchronization and spatial pattern formation. Examples from physics, biology, chemistry, and the neurosciences and social sciences illustrate the multidisciplinary relevance of nonlinear and complex systems.

The accompanying exercises are designed to deepen the lecture material through selected examples that are worked on independently by the students and subsequently presented and discussed during the in-person sessions.