1a)
Lets look at the phase diagram of the system in the limit where Which has root solutions of the form We can plot the fixed points (in the limit) as a function of r and check for when r = 2.75.
1b)
From the BFD and trajectories below we can approximate the behaviour over our BF parameter.
stable
2 cycle
4 cycle
8 cycle
Chaotic
Graphs below are shifted.
1c)
r = 3.0, t approx 3 iterations
r = 3.2, t approx 5 iterations
r = 3.5, t approx 5 iterations
1d)
We want to model the fixed point as a function of the BFP. In order to capture cycles, we need to scatter a few iterations after the system has tended to the fixed point. See function above.
1e)
BFD
1f)
The pitchfork BF are a good example of self sym. These occur at the cycle doubling BFPs listed above.
1g)
We see that the two trajectories are symmetric for the first few iterations before diverging. This indicated sensitivity to initial conditions.
2a)
From the BFD and trajectories below we can approximate the behaviour over our BF parameter.
stable
2 cycle
4 cycle
8 cycle
Chaotic