Non linear recursive relation....

[chisigma]

From mathhelpforum.com...

Hi. This is my first post here so I hope I've posted in the right place. My question concerns finding closed forms of nonlinear recurrence relations such as the following...

$\displaystyle a_{n+1}= a^{2}_{n}-1\ ;\ a_{0}=a$ (1)

This one is both nonlinear and nonhomogeneous. The even terms do form a homogeneous recurrence relation, which is nonetheless still nonlinear. Are there general methods for solving particular types of nonlinear recurrence relations? I've tried googling but the results aren't very helpful...

Sylvia A. Anderson

How to aid Sylvia?... there is a closed form solution to (1)?... if not, there is the way to find some informations of the solution, like the convergence-divergence and the limit in case of convergence?...

Kind regards

$\chi$ $\sigma$

 

[Amer]

My ideas

if a=1 or 0 or -1
it will diverge
$$a_1 = 1 -1 = 0$$
$$a_2 = 0 -1 = -1$$
$$a_3 = 1 - 1= 0$$

if $$\mid a \mid > 2$$
diverge, i choose a=2
$$a_1 = 4-1 = 3$$
$$a_2 = 9 -1 = 8$$
I tried to see if it is increasing or decreasing

$$\frac{a_{n+1}}{a_n} = 1$$
$$\frac{a_n ^2 - 1 }{a_n } = 1$$
$$a_n^2 - a_n -1 = 0$$ two zeros

$$a_n = \frac{1\mp \sqrt{1 +4}}{2}$$

decreasing between the two zeros, and increasing outside

 

[chisigma]

Let's proceed as explained in...
http://www.mathhelpboards.com/showthread.php?426-Difference-equation-tutorial-draft-of-part-I

... so that the first step is to write the deifference equation in the alternative form...

$\displaystyle \Delta_{n}=a_{n+1}-a_{n}= a^{2}_{n}-a_{n}-1= f(a_{n})\ ;\ a_{0}=a$ (1)

The function f(*) is represented in the figure...

https://www.physicsforums.com/attachments/125._xfImport

There are one 'attractive fixed point' [ a point where is f(x)=0 and f(x) crosses the x-axis with negative slope...] at $\displaystyle x_{-}= \frac{1-\sqrt{5}}{2}$ and one 'repulsive fixed point' [a point where is f(x)=0 and f(x) crosses the x-axis with positive slope...] at $\displaystyle x_{+}= \frac{1+\sqrt{5}}{2}$. The fact that there is an interval around $\displaystyle x_{-}$ where is $\displaystyle |f(x)|< |2\ (x_{-}-x)|$ however means that in general doesn't exist a solution which tends to $x_{-}$ [see theorems 4.1 and 4.2 of the tutorial post...] and 'almost all' the solutions diverge. As explained in the tutorial post a closed form solution of the (1) 'probably' doesn't exist and what we can do is the search of periodical solution. The solution of periodicity one are of course $a_{n}=x_{-}$ and $a_{n}=x_{+}$. The solution of periodicity two are generated for the values of a satisfying the equation...

$\displaystyle a^{4}-2\ a^{2}-a= a\ (a+1)\ (a^{2}-a-1)=0$ (2)

... that are $a=x_{-}$, $a+x_{+}$, $a=0$ and $a=-1$. The conclusion is the following...

a) for $\displaystyle |a|>x_{+}$ the solution diverges to $+ \infty$...

b) for $\displaystyle |a|<x_{+}\ , a \ne x_{-}$ the solution diverges but tends to the solution with periodicity two '0 -1 0 -1...'

c) for $ a=x_{-}$ we have the solution with periodicity one $x_{-}$ and for $ a=x_{+}$ we have the solution with periodicity one $x_{+}$...

Solutions with periodicity greater than two, if they exist, have to be found...

Kind regards



$\chi$ $\sigma$

 

[Amer]

wow, did i get 50% of the answer ?

 

[blue_raver22]

Hello Sylvia and welcome to the forum! Nonlinear recurrence relations can be challenging to solve, especially if they are nonhomogeneous. In general, there are no specific methods for solving particular types of nonlinear recurrence relations, but there are some general techniques that can be used.

One approach is to try to find a pattern in the terms of the sequence and see if it can be expressed in a closed form. In your example, it seems that the even terms of the sequence are related to the odd terms, so you could try to express $a_{2n}$ in terms of $a_{2n-1}$.

Another approach is to use generating functions, which can be helpful in finding closed forms for nonlinear recurrence relations. This involves converting the recurrence relation into a power series and then using algebraic manipulations to solve for the coefficients.

If there is no closed form solution to a nonlinear recurrence relation, it is still possible to find some information about the solution. For example, you could analyze the behavior of the sequence and determine if it converges or diverges, and if it converges, what is its limit. This can be done by looking at the behavior of the terms as $n$ approaches infinity and using techniques such as the squeeze theorem or the ratio test.

I hope this helps and good luck with your research on nonlinear recurrence relations!