# Complex Systems

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## General information

Course start: See schedule

Course book: Steven H Strogatz, Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering (Westview Press, 2014)

[there are two editions of that book, from 2000 (Perseus Books Group) and 2014 (Westview Press), with only minor differences; both editions will do as course literature]

Credits: 7.5p

Examination: Homework examination and oral examination (the latter only for PhD students)

Prerequisites: basic course in differential equations is recommended

Lecturer: Jack Lidmar, Department of physics, AlbaNova, email: jlidmar@kth.se

## Course description

Complex systems (dynamical systems) are mathematical models describing the time evolution of systems by means of equations of motion and initial values. Models are typically formulated in terms of coupled nonlinear differential equations (continuous time) or iterated maps (discrete time). Even for models with relative simple mathematical structure, the solutions may show surprisingly complex features, such as nonlinear oscillations, fractal structures and chaotic behavior.
The course gives an introduction to the analysis of such complex systems and their behavior. Key concepts are, amongst others, the phase space perspective, attractors, sensitive dependence on initial conditions, chaos, bifurcations, Poincare maps, and numerical solutions.

## Objectives

After the course you shall

• be familiar with analytical and numerical methods of coupled nonlinear differential equations
• be able to interpret and characterize different solution types
• be able to apply these methods to systems from physics, biology, chemistry and engineering

## Keywords

Coupled nonlinear differential equations, phase space, trajectories, attractors (fixed points, limit cycle, tori, strange attractors etc), invariant sets, stability analysis, bifurcations, chaos, Lyapunov exponents, Poincare-Bendixson theorem, Lorenz equations, Rössler system, Poincare map, iterative maps, logistic map, Sharkovskii’s theorem.

## Examination

Home assignments. For PhD students there will also be an oral examination.

## Literature

Steven H Strogatz, Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering (Westview Press, 2014)

Apart from the Strogatz book, there are a few others which present the course material from slightly different perspectives on a comparable level:

• Edward Ott, Chaos in Dynamical Systems (Cambridge University Press, 2009)
• John Argyris, Gunther Faust, Maria Haase, Rudolf Friedrich, An Exploration of Dynamical Systems and Chaos (Elsevier, 1994; new edition: Springer, 2015)
• Julien C Sprott, Chaos And Time-Series Analysis (Oxford University Press, 2001)
• Robert Hilborn, Chaos And Nonlinear Dynamics: An Introduction for Scientists and Engineers (Oxford University Press, 2001)

Further reading (popular science account on the development of chaos theory):
James Gleick, Chaos: Making a New Science (Penguin Books, 2008)