I A nonlinear recurrence relation

AI Thread Summary
The discussion revolves around a nonlinear recurrence relation, specifically $$a_{n+1}a_n^2 = 1$$, and the quest for its fixed points. It is established that there is exactly one real fixed point at ##a_n=1##, which is not stable, while two complex solutions also exist. The sequence does not converge if ##a_n## is not equal to 1, indicating no other fixed points. Participants clarify that the fixed point must satisfy the condition of the recurrence relation, leading to the conclusion about the nature of the solutions. The original poster finds clarity through the discussion and references a book by Strogatz for further understanding.
Wuberdall
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Hi Physics Forums,

I am stuck on the following nonlinear recurrence relation
$$a_{n+1}a_n^2 = a_0,$$
for ##n\geq0##.

Any ideas on how to defeat this innocent looking monster?

I have re-edited the recurrence relation
 
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Can you provide some context here? Is this homework? In what book / course did you come across this? Is this for a particular fractal graph?

It seems that it would oscillate from very small to very large until you're dividing by zero or by infinity.

##a_{n+1} = 1 / { a_n^2 } ## where ##a_n \neq 0##
 
Wuberdall said:
Hi Physics Forums,

I am stuck on the following nonlinear recurrence relation
$$a_{n+1}a_n^2 = 1,$$
for ##n\geq0##.

Any ideas on how to defeat this innocent looking monster?
What do you want to know about it? Have you tried isolating ##a_{n+1}## on one side of the equation?
 
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Wuberdall said:
I am stuck
In what way ?
This homework ? Please use the template and provide a full problem description and an attempt at solution tohat shows where you are stuck ...
 
Hi, this is not homework or course related. I am trying to determine if a fixed point for a certain dynamical system is unique. In doing so I come across the above recurrence relation.

So what I am really looking for, is a solution and whether or not this solution is unique
 
Wuberdall said:
Hi, it is not homework or course related. I am trying to determine if a fixed point for a certain dynamical system is unique. In doing so I come across the above recurrence relation.

So what I am really looking for, is a solution and whether or nor this solution is unique
It has exactly one fixed point at ##a_n=1##, though it is not a stable fixed point.
 
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Where do the ##a_n## live and are there initial conditions?
 
tnich said:
It has exactly one fixed point at ##a_n=1##, though it is not a stable fixed point.
Thanks, this is exactly what I was looking for and also what my intuition told me.

How do you conclude that their is exactly one fixed point ?
 
Wuberdall said:
Thanks, this is exactly what I was looking for and also what my intuition told me.

How do you conclude that their is exactly one fixed point ?
If ##a_n## is not 1, then the sequence does not converge, so there can be no other fixed point.
 
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  • #10
tnich said:
If ##a_n## is not 1, then the sequence does not converge, so there can be no other fixed point.
A fixed point must satisfy ##a_{n+1}=a_n##. In this case that results in ##a_n a_n^2=1## which has three solutions (two of which are complex), but only ##a_n=1## results in a fixed point.
 
  • #11
tnich said:
If ##a_n## is not 1, then the sequence does not converge, so there can be no other fixed point.

Thanks, for your time.

I have figured it out now. It turned out that I was a bit rusty. So I found my old and dusty book by Strogatz on my bookshelf. All your comments make complete sense now and I see why they are true.

I wish you a happy and sunny weekend.
 
  • #12
tnich said:
A fixed point must satisfy ##a_{n+1}=a_n##. In this case that results in ##a_n a_n^2=1## which has three solutions (two of which are complex), but only ##a_n=1## results in a fixed point.
Oops, no I think the two complex cube roots of 1 also are fixed points.
 
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  • #13
Wuberdall said:
I wish you a happy and sunny weekend.

It's only Monday. The weekend is a long way off.
 

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