A nonlinear recurrence relation

[Wuberdall]

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

 

[jedishrfu]

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 $a_n \neq 0$

 

[tnich]

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?

 

[BvU]

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 ...

 

[Wuberdall]

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

 

[tnich]

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.

 

[fresh_42]

Where do the $a_n$ live and are there initial conditions?

 

[Wuberdall]

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 ?

 

[tnich]

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.

 

[tnich]

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.

 

[Wuberdall]

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.

 

[tnich]

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.

 

[PeroK]

I wish you a happy and sunny weekend.

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