I asked the [following](https://math.stackexchange.com/questions/4136248/bounding-the-nth-term-of-a-sequence-given-a-recursive-non-linear-bound) question in MSE:



> 
> Let $a,b\in\mathbb{R}^+$.
> Suppose that $\{x\_n\}\_{n=0}^\infty$ is a sequence satisfying
> $$|x\_n|\leq a|x\_{n-1}|+b|x\_{n-1}|^2, $$
> for all $n\in\mathbb{N}$. How can we bound $|x\_n|$ with a number $M\_n$ depending on $n$, $a$, $b$, and $x\_0$?
> 
> 
> 


However, I did not get an answer.
It seems that there should be some obvious observation to give a naive upper bound? Although it would be nice to have an asymptotically tight bound in closed form, if possible of course!



Bounding the $n$-th term of a sequence, given a non-linear recursive bound
Let us define the sequence $y\_n$ by $y\_0=|x\_0|$, $y\_{n+1}=ay\_n+by\_n^2$. Then we have
$|x\_n|\leq y\_n$ for all $n$, since $ay+by^2$ is increasing for positive $a,b$,
and it is enough to estimate the positive sequence $y\_n$. Setting $z\_n=by\_n$, we obtain a simpler recurrent relation
$z\_{n+1}=az\_n+z\_n^2.$ For this last relation, the behavior depends on $a$. If $a\geq 1$, all orbits tend to $+\infty$ and $z\_n\sim \exp(2^nu(a,z\_0))$ where $u$ is a positive
function. If $a<1$, then there is a positive fixed point $z^\*=1-a$. When $z\_0>z^\*$
the behavior is as above, when $z\_0<z^\*$, the orbit tends to $0$ as the geometric
progression: $z\_n\sim a^n$, and when $z\_0=z^\*$ then $z\_n=z^\*$.


For the function $u(a,z)$, there is no explicit expression, it is variously known as
the Green function of the complement of the Julia set, or the equilibrium potential of the Julia set. This function is actually easy to compute by the formula
$$u(a,z)=\lim\_{n\to\infty}2^{-n}\log|f^n(z)|,$$
where $f^n$ is the $n$-th iteration of the function $f^1(z)=az+z^2.$ This expression converges very fast.


Ref:
Douady, A.; Hubbard, J. H.
Étude dynamique des polynômes complexes. Partie I.
Publications Mathématiques d'Orsay, 84-2. Université de Paris-Sud, Département de Mathématiques, Orsay, 1984. 75 pp.
[available online](http://sites.mathdoc.fr/PMO/feuilleter.php?id=PMO_1984)


