Question

I have the following iterative process $$a_n=a_{n-1}(1-\phi(a_{n-1})),\quad 0< a_0<1,$$ where $\phi(x)$ is a continuous increasing function, $\phi(0)=0$, and if $x\in(0,1)$ then $0< \phi(x)<1$. For instance, if $\phi(x)=x$, then this is just the logistic map.

Clearly, $a_n$ converges to $0$ as $n\to\infty$ for any function $\phi$, but the convergence rate depends on $\phi$.

My question is how we can estimate the convergence rate. Are there any standard methods which can be used here?

Thanks.

Answer 1

The standard techniques is as follows. At first, we guess the growth rate by replacing the discrete dynamical system to ODE: $a_n-a_{n-1}=-a_{n-1}\phi(a_{n-1})$ is replaced to $A'(n)=-A(n)\phi(A(n))$, where function $A(n)$ imitates $a_n$. Nex, we solve this equation, rewriting it as $g(A(n))'=1$, where $g'(t)=-1/(t\phi(t))$. Our guess is that $g(A(n))$ behaves like $n$ (since it derivative behaves as 1), and the same for $g(a_n)$. For proving this guess rigorously we introduce a new sequence $b_n=g(a_n)$ and write down the formula $b_n-b_{n-1}=g(a_n)-g(a_{n-1})=(a_n-a_{n-1})g'(s)$ for some $s$ between $a_{n-1}$ and $a_n$. Next, $(a_n-a_{n-1})g'(s)=\frac{a_{n-1}\phi(a_{n-1})}{s\phi(s)}$. Assume that $\phi(x):\phi(y)$ tends to 1 when $x/y$ tends to 1 and $x,y$ tend to $+0$. Then we may conclude that our last fraction tends to $1$, and by usual Stolz theorem conclude that $b_n$ behaves like $n$.

Example: $\phi(x)=x$, $g(t)=1/t$, so we consider the sequence $b_n=1/a_n$. We get $b_{n}-b_{n-1}=\frac{a_{n-1}-a_{n}}{a_na_{n-1}}=\frac{a_{n-1}}{a_{n}}$, what clearly tends to 1, hence $b_n\sim n$.

Answer 2

If $\phi(x)=x^m$ your function is analytic at $0$, and convergence rate can be found with very high precision. Same happens if it is analytic in sufficiently large sector with vertex at $0$. See about this P. Fatou, Sur les equations fonctionnelles (1919-20). If $\phi(x)=x$, convergence is logarithmic, that is very slow. If $\phi(x)=x^m$ it is even slower, as you can see interating $\sin x$ on your calculator.