Biomedical Engineering Reference
In-Depth Information
2
NONLINEAR DYNAMICAL SYSTEMS
Joshua E. S. Socolar
Physics Department, Duke University, Durham, North Carolina
The concepts and techniques developed by mathematicians, physicists, and engineers to
characterize and predict the behavior of nonlinear dynamical systems are now being ap-
plied to a wide variety of biomedical problems. This chapter serves as an introduction to
the central elements of the analysis of nonlinear dynamics systems. The fundamental dis-
tinctions between linear and nonlinear systems are described and the basic vocabulary
used in studies of nonlinear dynamics introduced. Key concepts are illustrated with clas-
sic examples ranging from simple bistability and hysteresis in a damped, driven oscillator
to spatiotemporal modes and chaos in large systems, and to multiple attractors in complex
Boolean networks. The goal is to give readers less familiar with nonlinear dynamics a
conceptual framework for understanding other chapters in this volume.
1.
INTRODUCTION
The latter half of the twentieth century saw remarkable advances in our un-
derstanding of physical systems governed by nonlinear equations of motion.
This development has changed the scientific worldview in profound ways, si-
multaneously supplying a dose of humility—the recognition that deterministic
equations do not guarantee quantitative predictability—and a great deal of in-
sight into the qualitative and statistical aspects of dynamical systems. One of the
byproducts has been the realization that the mathematical constructs developed
for modeling simple physical systems can be fruitfully applied to more complex
systems, some of which are of great interest to the biomedical community. Ex-
amples range from electrical signal propagation in cardiac tissue, where one
Address correspondence to: Joshua E.S. Socolar, Physics Department, Duke University, 107 Physics
Building, Durham, NC 27708; 919-660-2557 (socolar@phy.duke.edu).
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