I am trying to generate a bifurcation plot for the following system (a single oscillator given by the difference equation)
$x(n + 1) = f(x(n))$, with $f(x) = 1 - ax^2$, where $a$ is a constant positive real number such that $0 \leq a \leq 2$.
I don't know where I should start and would appreciate any advice on which type of computer software I should look into for the task of generating a bifurcation plot for this recursive difference equation. Also, the problem above has the following follow up questions:
- Determine the values of $a$ for which the system gives a stable two cycle.
- Determine a few values of $a$ where bifurcation occur for the above system.
- What can be said about the response of system for the specific values of $a = 0.5, a = 1,$ and $a = 1.9?$
I want to verify that the information I have deduced without generating the bifurcation plot and testing a few iterations is correct. It would seem the values $a = 0.5$ and $a = 1$ are both stable for values of $|x| \leq 1$, converging to the fixed point $x^* = 1$. On the other hand, testing a few iterations for $a = 1.9$ the system seems unstable for all values of $x$ and there are no fixed points. The bifurcation seems to occur at $a = 1$, and when $a$ is small enough it produces two cycles. Any further help with the followup problems and/or information on where I should look in regards to obtaining a bifurcation plot (ideally a computer software such as MATLAB or Wolfram Mathematica) would be appreciated, thanks.