April 09
Bifurcation Pendulum
This code takes the pendulum equation ive been working with, and uses the solver ive created to obtain values for various parameters input into the equation. Random values are used for both initial conditions as well as the variable parameter, which in this case i use the driving frequency. This outputs a listplot of the driving amplitude vs the output value obtained by the runge solver. This gives a nice plot of where we can expect chaotic solutions. There are obvioius areas of singular valued answers, and obvious areas of period doubling cascade leading to chaos. This not only is a nice set of pictures but is helpful in determining which parameter values to use in other calculations.
Logistic
This code creates a solver for the logistic equation difference equation. Web diagrams are drawn to show periodic/chaotic behavior. Also included are solutions as a function of time, also showing the difference in periodic/chaotic behavior, as well as a period doubling sequence. Finally bifurcation diagrams are plotted in a similar way to the above calculations, with the only difference being the equation being solved. Initial values and the equation parameter are selected randomly.
Lorenz
This code uses essentially the same code as the initial runge solver, with the exception of the input derivative functions. This shows the utility of my code solver as well as detailing another important chaotic system. Only one solution is shown, but many others could easily be calculated.
Poincare
In this code I revolutionize the way my solver works. Many of the changes make the code run more accuratly and faster. Essentially i define the period of the driving oscillation, and define the stepsize in terms of that. There are then an integer number of steps in a driving period, giving slices at the right times. Also i changed the way that data is stored, namely only one point per driving cycle is saved, while the rest is still used to calculate up to the driving cycle. This leaves me with far less unused data clogging memory. I also added a part where the equations are solved for an integer number of drivecycles. In fact one of the inputs is now how many drive cycles. This gives me a chance to calculate out for 30000 drivecycles, saving only 30000 points, instead of the 100 or 1000 times as many points which are used in the calculation. Many poincare plots are shown for various values of parameter inputs. Also I began making animations of poincare plots as the parameters are varied. This gives nice smooth images. One of the best so far is changing the phase of the driver cycle. This gives some real nice animation cycles which show the successive folding of the fractal in the phase space. Unfortunatly, this was done most recently out of everything, and ill have to go back and perform similar calculations as those ive already done, but using the corrected algorithm.
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