Dynamical Systems And Chaos: The Lorenz Attractor Part 1

These are videos form the online course ‘Introduction to Dynamical Systems and Chaos’ hosted on Complexity Explorer. With these videos you'll gain an introduction to the modern study of dynamical systems, the interdisciplinary field of applied mathematics that studies systems that change over time.

Topics to be covered include: phase space, bifurcations, chaos, the butterfly effect, strange attractors, and pattern formation. The course will focus on some of the realizations from the study of dynamical systems that are of particular relevance to complex systems:

1. Dynamical systems undergo bifurcations, where a small change in a system parameter such as the temperature or the harvest rate in a fishery leads to a large and qualitative change in the system's behavior.

2. Deterministic dynamical systems can behave randomly. This property, known as sensitive dependence or the butterfly effect, places strong limits on our ability to predict some phenomena.

3. Disordered behavior can be stable. Non-periodic systems with the butterfly effect can have stable average properties. So the average or statistical properties of a system can be predictable, even if its details are not.

4. Complex behavior can arise from simple rules. Simple dynamical systems do not necessarily lead to simple results. In particular, we will see that simple rules can produce patterns and structures of surprising complexity.

[MLDawn]

Great video. Could you please tell me how one can use the Rayleigh parameter to generate the following dynamics: 1) Fixed point 2) Quasi-periodic and 3) Chaotic. If there are acceptable ranges for Rayleigh so that each dynamic may happen, would you please be very specific with those ranges? Many thanks.