Solution of $x_{k+1} = x_{k} (a x_{k} + b)$

Question

Could anyone help me to solve the equation

$x_{k+1} = x_{k} (a x_{k} + b)$,

for find the explicit solution of $x_{k}$?

BTW. Do you know a GOOD book for the classification for non linear difference equation? I've looked for something but most part speaks about the linear case, but i know there's some kind of classification like differential equation.

PS. i've fixed the mistake... sorry

$x_k=0$ satisfies that equation, but without any information on how $i$ relates to $k+1$ I don't think it's possible to say anything interesting. — Henrik supports the community, Dec 05 '14 at 13:48
Sorry... i miss the index the equation isn't $x_{k+1} = x_{i}(ax_{k} + b)$ but $x_{k+1} = x_{k}(ax_{k} + b)$, there isn't index $i$... sorry again — user8469759, Dec 05 '14 at 13:59
There'are instances of this problem that provides explicit solutions, if you set $a = -1$ and $b=2$ the equation has well known closed form solution. — user8469759, Feb 06 '19 at 12:42
Sure, there are some instances that have closed form solutions, but I think in general they don't. — Gerry Myerson, Feb 07 '19 at 09:23

score 1 · Accepted Answer · answered Feb 07 '19 at 09:36

According the Mathworld,

"A quadratic map is a quadratic recurrence equation of the form $x_{n+1}=a_2x_n^2+a_1x_n+a_0$. While some quadratic maps are solvable in closed form (for example, the three solvable cases of the logistic map), most are not."

"The well-known recurrence $x_{n+1}=x_n^2+c$ that is often called "the" quadratic map is not in general solvable in closed form."

There is some discussion there of the few cases that do have closed form solutions.

Solution of $x_{k+1} = x_{k} (a x_{k} + b)$

1 Answers1