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Logistic Map

The logistic map is defined by \( x_{n+1} = f_r({x_n})\) with \( f_r(x) = r x (1-x) \), with \( 0 \leq x \leq 1 \) and \( 0 \leq r \leq 4 \).

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Bifurcation Diagrams

Bifurcation diagrams show steady-state orbits at different \(r\) values. These plot \(x\) vs. \(r\) (with parameter \(r\) on the horizontal axis). Several initial data are chosen randomly at each \(r\) value and propogated for thousands of steps. Only the final several hundred steps are plotted at each \(r\) value to isolate steady state behavior. Number of attractors doubles repeatedly as chaos is approached.

Cobweb Diagrams

Cobweb diagrams provide a simple way to visualize time evolution of the map. They can be produced easily by hand by repeatedly reflecting along the x=y diagonal (blue), then tracing back to the plot of \(f(x)\) (black). The horizontal axis is \(x\). You can observe different types of and numbers of attractors at various parameter values.

Chirikov Map

Images below are for the Chirikov map \[ p_{n+1} = p_{n} - K \sin(x_{n}) \] \[ x_{n+1} = x_{n} + p_{n+1} \] with \( -\pi \leq x < \pi \) and \( -\infty < p < \infty \).

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Poincare Plot

The approach to chaos is evident as certain orbits become space-filling.

Chaotic Art

Some fun images generated by adding randomness to bifurcation diagrams and Poincare plots.