PHY254H1 Lecture Notes - Double Pendulum, Chaos Theory, Bifurcation Diagram
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Lecture 18: nonlinearity & chaos: relevant reading: these notes and python programs are your main re- source. For further reading, see bg: parts of chapters 1-4. Or for a brief chapter, i recommend chapter 12 of classical mechanics by. Finally, going through tutorial 9 is recommended if you want some practice with the concepts discussed. So this equation gives you a sequence of x values. Remember our for- mula for numerical integration in python: x[i + 1] = x[i] + v[i]dt. This has exactly the form of a di erence equation. Basically, any dif- ferential equation can be represented by a di erence equation which is why they are relevant for physics. 1: another di erence equation (or map) we"ve encountered (although not stated as such) is a poincare section. Remember poincare sections plot the phase space (e. g. and values) at speci c times (multiples of the driving frequency). This can be represented in a 2d map like:
