# Poincaré plane and Logistic Map

How can we draw Poincaré plane and phase portrait for the Logistic Map for different parameter values?

For dynamical systems in general, you typically simulate the dynamics in a computer - for instance, numerically integrating differential equations, or iterating a map - and then you can analyze your data. The tools mentioned in the question are primarily used for visualizing the solutions:

• a phase portrait is usually the plot of one or more trajectories in the system's phase space (which is its state space, i.e., a space where each point denotes a different system state);
• a Poincaré plot is (for 1-D systems) a plot of, e.g., $$x_{n+1}$$ against $$x_n$$, i.e., for a trajectory $$x_0, x_1, x_2, x_3, ...$$ the Poincaré plot will be 2-D with points $$(x_0,x_1), (x_1,x_2), (x_2,x_3), ...$$;
• don't mix up the Poincaré plot with the similarly named Poincaré map, which displays trajectories at a lower dimensionality than that of the system, by means of a section of its phase space (for details see answers 1 and 2, Wikipedia, ...). As for logistic map, it's 1-D, so it has a 0-D Poincaré section/map - not that useful for visualization purposes.

How can we draw Poincare plane and phase portrait to Logistic Map for different parameter values?

1. choose the value of the parameter $$r$$;
2. choose an initial condition for the state variable $$x$$: that's $$x_{n=0}$$;
3. obtain $$x_0$$'s trajectory by iterating $$x_{n+1}=rx_n(1-x_n)$$.

Now, with this trajectory in hands, you can obtain plots (by hand or software) such as the Poincaré plot for $$r=4$$ displayed in the Logistic Map's Wikipedia entry:

For a different parameter value, you'd obtain a different plot. A 3-D plot with axes $$r$$, $$x_{n+1}$$, and $$x_n$$ would also be a possibility.

The logistical phase portrait is simply a segment of line, $$x_n \in [0,1]$$. It can show which parts of it are more densely populated by trajectories, and a 2-D plot of $$\{x_n\}$$ as a function of the parameter $$r$$ is known as a bifurcation diagram. Again from Wikipedia:

Also worth of mention, for 1-D maps at least, is the cobweb plot, which is essentially a graphical representation of the iteration process that is very useful for gaining an intuition about the process. Wikipedia examples use precisely the logistic map: