- #1

Master1022

- 611

- 117

- Homework Statement
- Find the values of the parameter ##\lambda## such that the 2-cycle is stable

- Relevant Equations
- None

Hi,

(

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).

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) ##

(iii) Find the values of ##\lambda## for which the 2-cycle exists (subject of previous question)

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

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.

(

*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][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.