Homoclinic orbit in the Logistic Map

[standardflop]

My question is the following: " For the logistic map $x_{k+1} = rx_k(1-x_k)$ the band-merging point, where the period-1 orbit undergoes its first homoclinic bifurcation, is at r=3.678573510. Draw a trajectory to the map that illustrates the homoclinic orbit. "

The period-1 orbit is at the diagonals intersection with the parabola, correct?
I know what a homoclinic orbit looks like in a time-continuous system - it goes away from a fixed point, back to the same fixed point.. but what does it look like in a iterated map (the logistic map)?

Regards.

 

[Jhero]

Hello,

Thank you for your question. The period-1 orbit in the logistic map is indeed located at the intersection of the diagonals with the parabola. The homoclinic orbit in an iterated map, such as the logistic map, is a bit different from a continuous system. In a continuous system, the homoclinic orbit is a trajectory that approaches and then leaves a fixed point, eventually returning to the same fixed point. In an iterated map, the homoclinic orbit is a trajectory that approaches and then merges with a stable periodic orbit, creating a chaotic behavior.

To illustrate this, I have attached a diagram of the logistic map with the band-merging point marked at r=3.678573510. As you can see, the homoclinic orbit is a trajectory that starts at the fixed point (shown in red) and approaches the stable periodic orbit (shown in blue) before merging with it. This creates a chaotic behavior in the system.

I hope this helps to clarify the concept of a homoclinic orbit in an iterated map. If you have any further questions, please let me know.