Part H: Quantifying Chaos

Exercises

These exercises build on the 1D map, 2D map, and 3D ODE simulation tools you made last week. Those tools let you look at trajectories and bifurcation diagrams for these classes of dynamical system.

The goal here is to augment these simulation and visualization tools with a set of analysis methods that measure the degrees of stability and instability in these same dynamical systems. The resulting programs illustrate the basic skeleton of any dynamical system exploration tool: they will consist of a simulation engine, a visualization subsystem, and a set of quantitative analysis tools. More sophisticated tools merely elaborate on this architecture, adding ease-of-use, flexibility, and individual features.

  1. LCE for Logistic map: Modify the LCE program for the Tent map to calculate the LCE, as a function of parameter, for the Logistic map.
  2. Integration: In the 1D map tool you created last week, that simulated a range of 1D maps, add a module that will plot the LCE as a function of parameter. That is, generalize the 1D map LCE program here to work on any map and then integrate this into the 1D Map tool.
  3. LCEs, fractal dimension, and contraction rate for the Lozi map: Modify the LCE program for the Henon map to calculate the LCEs, fractal dimension, and contraction rate for the Lozi map.
  4. Integrate the 2D map LCEs program into your 2D map tool from last week.
  5. The LCEs are averages, how do they converge? Add to the LCE program for the Lorenz ODEs a plot of the time series of the local stretching and shrinking factors. That is, plot log(d) for each of the three Lyapunov exponents. The graphical output of the program will have both a sample of the trajectory, which is already part of the program, and then another plot with three time series. How fast do the LCEs converge? Does the convergence to a stable average appear to be the same for the different LCEs?
  6. LCEs and dissipation for the Rossler ODEs: Modify the LCE program for the Lorenz ODEs to calculate the LCEs for the Rossler system.
  7. Integrate the 3D ODE LCE program into your 3D ODE tool from last week.
Table of Contents