Homoclinic orbit in the Logistic Map

• standardflop
Expert SummarizerIn summary, the logistic map has a band-merging point at r=3.678573510, where the period-1 orbit undergoes its first homoclinic bifurcation. In an iterated map, the homoclinic orbit is a trajectory that approaches and merges with a stable periodic orbit, creating a chaotic behavior. This can be seen in the attached diagram of 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.

Last edited:

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.

1. What is a homoclinic orbit in the Logistic Map?

A homoclinic orbit in the Logistic Map is a type of chaotic behavior that occurs in a logistic equation, where the population of a species grows and declines repeatedly in a seemingly random pattern.

2. How is a homoclinic orbit different from other types of chaotic behavior?

A homoclinic orbit is different from other types of chaotic behavior because it involves a specific type of orbit that connects a fixed point to itself. This results in a more complex and unpredictable pattern compared to other types of chaotic behavior.

3. What factors influence the formation of a homoclinic orbit in the Logistic Map?

The formation of a homoclinic orbit in the Logistic Map is influenced by the growth rate and carrying capacity of the species being modeled. Additionally, the initial population size and the values of the parameters in the logistic equation can also impact the formation of a homoclinic orbit.

4. Can a homoclinic orbit be predicted or controlled?

No, a homoclinic orbit cannot be predicted or controlled. Due to the sensitive nature of chaotic systems, even small changes in initial conditions or parameters can lead to vastly different outcomes. This makes it impossible to accurately predict or control the formation of a homoclinic orbit in the Logistic Map.

5. What practical applications does the study of homoclinic orbits have?

The study of homoclinic orbits in the Logistic Map can have practical applications in various fields, such as ecology, economics, and physics. Understanding chaotic behavior and the formation of homoclinic orbits can help researchers better understand and predict complex systems in these fields.

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