Dynamical Systems

The field of dynamics concerns the study of systems whose internal parameters, called states, obey a set of temporal rules, essentially encompassing all observable phenomena. Over its long history, the field developed into three distinct subdisciplines: applied dynamics, which concerns the modeling process that transforms actual system observations into an idealized mathematical dynamical system; mathematical dynamics, which focuses on the qualitative analysis of the model; and experimental dynamics, which ranges from controlled laboratory experiments to numerical simulation of state equations on computers.

The study of dynamics dates back to at least Galileo (1564–1642), who essentially founded it as a branch of natural philosophy now called physics. Galileo established the close interplay between theory and experiment and was one of the first to fully study the concept of acceleration. Galileo and others, including Johannes Kepler (1571–1630), addressed the concepts of change, rate of change, and rate of rate of change as they were ubiquitously observed in natural phenomena. These researchers found that nature is fraught with basic laws relating to changes in physical states that could be observed and measured.

Research on dynamics further developed with Isaac Newton (1642–1727) and Gottfried Leibniz (1646–1716), who independently formalized the notion of derivatives that served to couch these dynamical laws or systems in a mathematical form, after which flourished the development of a whole spectrum of formal qualitative analysis tools. These early contributions were of a purely analytical form that dealt with classical differential equations and eventually gave rise to more geometrical and topological methods that have dominated the field ever since. Next, with the rapid advancement of measurement and computational technologies, the area of experiments (at first physical and then numerical) became an important component of the overall study of dynamics.

Dynamics was exclusively studied in the domain of classical physics until the 1920s, when applications to biological and social sciences began to appear. This expansion has continued to today, where dynamics is now studied in virtually all areas of science. As an example, such a perspective can be fruitfully applied to the electrical, mechanical, orbital, and propulsion subsystems found in spacecraft vehicles. It may come as a surprise that the complex behavior called chaos did not formally arrive on the scene until around 1950.

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