Physics 513 / Computer Science 524
Dynamics of Complex Systems
Fall 2022
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Syllabus
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Welcome to Physics 513 / Computer Science 524. This course is
broadly interested in temporal patterns that arise in macroscopic
physical systems, and how to quantify, classify, model, understand,
predict, simulate, and control such patterns. The subject is
interdisciplinary and has proved to be interesting and useful for
students studying physics, biophysics, engineering, biology,
neuroscience, medicine, chemistry, mathematics, computer science,
geology, meteorology, fluid dynamics, and also studying some subjects
in the humanities like economics and political science.
A main goal of the course will be to study the successes, limitations,
and implications of a late 20th-century discovery, that deterministic
nonlinear evolution equations with just a few independent variables
can generate complex temporal behaviors that can
quantitatively agree with experimental observations. It will
take a while to give you the background to appreciate the scope and
significance of this discovery and so the course will introduce and
discuss many of the following topics:
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Concepts related to a geometric and global way of thinking about the
solutions of nonlinear evolution equations. These concepts include:
phase space, dissipative versus conservative systems, attractors,
basins of attraction, bifurcation theory, linear stability theory,
Poincare sections and maps, strange attractors, Lyapunov exponents,
transition scenarios (Feigenbaum, Landau, Ruelle-Takens, and
intermittency), universality, fractals, fractal dimensions, and the
analysis of time series by embedding.
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Comparisons of theory with high-precision laboratory experiments and
with computer simulations, e.g., the sequences of transitions that
lead to chaos in a convecting fluid of moderate lateral extent.
- Applications of the ideas to different disciplines such as
physics, engineering, biology, chemistry, meteorology, and
neurobiology.
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Strategies and algorithms for simulating, analyzing, and controlling
complex systems, including the numerical integration of differential
equations, and the numerical calculation of power spectra, Lyapunov
exponents, and fractal dimensions. We will also discuss fundamental
limitations imposed on numerical simulation by nonlinear dynamics,
e.g., the impossibility of accurate long-term forecasting for chaotic
dynamics.
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