Dynamical Systems - Chaos: Stability condition for a 2-cycle system

In summary: Could you explain both?In summary, the conversation is about a question from a self-teaching course that involves finding the 2-cycle system and fixed points of a given system. The person is having trouble understanding the condition used in the solution and the theory behind it. They ask for help in understanding the concept of fixed points and the application of the chain rule in determining the stability of the system.
  • #1
Master1022
611
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Homework Statement
Find the values of the parameter ##\lambda## such that the 2-cycle is stable
Relevant Equations
None
Hi,

(This question is part of the same example as a previous post of mine, but I have a question about a different part of it)

I was looking at a question from an exam for a course I am self-teaching. There is a sub-question which asks us to find the values of a parameter for which the 2-cycle system is stable. However, when looking at the solution I don't understand the condition they have used (and I can't see any reference to it in the materials I am using).

Question & Attempt:
Let us imagine we have a system: ## x_{n + 1} = F(x_n) = \lambda ( 1 - \alpha x_n ^2) ##. Then there are some sub-questions about finding the 2-cycle system and the fixed points of it, before the parts I specifically I don't understand.

(i) Find the 2-cycle system
We can do this by doing ## F(F(x_n)) = \lambda - \alpha \lambda ^3 - 2 \alpha ^2 \lambda ^3 x^2 - \alpha ^3 \lambda ^3 x ^4 ##

(ii) Then we find the fixed points of the 2-cycle system
Then we look for the fixed points by setting ## F(F(x)) = x ## which results in the following:
[tex] (\lambda - \alpha \lambda x^2 - x) (1 - \alpha x \lambda - \alpha \lambda ^2 + \alpha ^2 x^2 \lambda ^2) = 0 [/tex]

Then the second term is the condition for ##F(F(x)) = x ##, which I thought was also a fixed point.

This leads to two solutions:
[tex] \hat x_{1, 2} = \frac{\alpha \lambda \pm \alpha \lambda \sqrt{ 4 \alpha \lambda ^2 - 3}}{2 \alpha ^2 \lambda ^2} [/tex]

We are told to use ## \alpha = \frac{1}{2} ## from here onwards.
Thus the solutions reduce to: ## \hat x_{1, 2} = \frac{1}{\lambda} \left( 1 \pm \sqrt{ 2 \lambda ^2 - 3} \right) ##

Below are the parts I don't understand the theory for:
(iii) Find the values of ##\lambda## for which the 2-cycle exists (subject of previous question)

The solution exists when ## 2 \lambda ^2 - 3 > 0 \rightarrow \lambda > \sqrt{\frac{3}{2}} ##

(iv) Find the values of ## \lambda ## for which the 2-cycle is stable

The solution scheme says that we do:
[tex] \left| \frac{dF(F(x))}{dx} |_{\hat x_{1, 2}} \right| = \left| F'(\hat x_1) F'(\hat x_2) \right| < 1 [/tex]


and then it arrives at the answer: ## -\sqrt{\frac{3}{2}} < \lambda < \sqrt{\frac{5}{2}} ##

I am not sure where this condition comes from... Any help would be greatly appreciated.
 
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  • #2
(iii): The fixed points of [itex]F \circ F [/itex] must be real for a 2-cycle to exist.

(iv): Which do you not understand: why [itex]|f'(x)| < 1[/itex] is the condition for [itex]x_{n+1} = f(x_n)[/itex] to be stable, or why the chain rule gives [itex](F \circ F)'(x_1) = (F \circ F)'(x_2) = F'(x_1)F'(x_2)[/itex]?
 
  • #3
Thanks @pasmith !

pasmith said:
(iii): The fixed points of [itex]F \circ F [/itex] must be real for a 2-cycle to exist.
Sure thing!

pasmith said:
(iv): Which do you not understand: why [itex]|f'(x)| < 1[/itex] is the condition for [itex]x_{n+1} = f(x_n)[/itex] to be stable, or why the chain rule gives [itex](F \circ F)'(x_1) = (F \circ F)'(x_2) = F'(x_1)F'(x_2)[/itex]?
Neither unfortunately.
 

1. What is a dynamical system?

A dynamical system is a mathematical model that describes the behavior of a system over time. It is a set of equations that represent the evolution of the system's state variables as time progresses.

2. What is chaos in a dynamical system?

Chaos in a dynamical system refers to a state of unpredictable and irregular behavior, even though the system's equations are deterministic and have no random elements. This means that small changes in the initial conditions of the system can lead to drastically different outcomes.

3. What is a 2-cycle system?

A 2-cycle system is a type of dynamical system that has two equilibrium points or cycles. These points or cycles are reached when the system's state variables remain constant over time, and they represent stable or unstable states of the system.

4. What is the stability condition for a 2-cycle system?

The stability condition for a 2-cycle system is a mathematical criterion that determines whether the system's equilibrium points or cycles are stable or unstable. It involves analyzing the eigenvalues of the system's Jacobian matrix, and if the eigenvalues have a magnitude less than 1, the equilibrium points or cycles are stable.

5. How does the stability condition affect the behavior of a 2-cycle system?

The stability condition plays a crucial role in determining the behavior of a 2-cycle system. If the equilibrium points or cycles are stable, the system will tend to stay near those points or cycles, and small perturbations will not significantly affect its behavior. On the other hand, if the equilibrium points or cycles are unstable, the system will exhibit chaotic behavior and be highly sensitive to initial conditions.

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