10345 Introduction to Complex Systems and Chaos
|Introduktion til komplekse systemer og kaos|
|Taught under open university|
Scope and form:
Lectures, practicals, computer exercises.
Duration of Course:
Date of examination:
Type of assessment:
Not applicable together with:
General course objectives:
To give the participants a fundamental insight into nonlinear phenomena in complex systems; to introduce analytical and computational tools for the analysis of dynamical phenomena in complex systems; practicals and computer exercises will enable the participants to perform nonlinear analyses of mathematical models.
|A student who has met the objectives of the course will be able to:|
- Define fixed points and stability for one- and two-dimensional dynamical systems and compute the fixed points location and stabilty.
- Perform scaling of models and determine their dimensionless parameters.
- Perform a detailed qualitative phase plane analysis applying linearisation and null clines.
- Explain the properties of stable and unstable manifolds for saddle points.
- Explain the most typical local bifurcations in one- and two-dimensional dynamical systems and determine these in simple systems.
- Demonstrate the absence of periodic solutions in gradient systems using Lyapunov exponents and Dulac's criterion.
- Demonstrate the existence of periodic solutions using Poincaré-Bendicson's theorem.
- Compute fixed points and their stability for one-dimensional iterated maps and explain their bifurcation diagrams.
- Apply numerical methods to determine fixed points and their stability, bifurcation points, approximation of stable and unstable manifolds of saddle points, and to help perform a phase plane analysis.
- Compute the fractal dimension of self-similar objects and explain the most important properties of chaotic solutions.
Nonlinear phenomena: phase plane analysis, limit cycles, chaotic dynamics. Scaling. Index theory. Lyapunov functions. Stability of equilibria and periodic solutions. Computer simulation. Local bifurcations: saddle-node, transcritical, pitchfork, Hopf and period-doubling. Poincare sections and Poincare maps. Stable and unstable manifolds. Global bifurcations. Chaos and predictability; computation of Lyapunov exponents. Fractal geometry; computation of fractal dimension.
Steven H. Strogatz, Nonlinear Dynamics and Chaos, ISBN 0-7382-0453-6.
|, 309, 256, (+45) 4525 3310,
, 303B, 157, (+45) 4525 3067,
|10 Department of Physics|
|01 Department of Mathematics|
Registration Sign up:
|Mathematical models, dynamical systems, nonlinear phenomena, phase plane analysis, bifurcations, symmetry breaking, local bifurcations, global bifurcations, chaos, fractals, Lyapunov exponents, computer simulation.|
August 14, 2012|
See course in DTU Course base