Uniqueness of fixed points for rational transformations

Question

Background

Let $a,b,c,d$ be nonnegative constants and consider the map $T\colon [0,1]\times[0,1] \rightarrow [0,1]\times[0,1]$ defined by

$$ T(x,y) := \left( \frac{1}{1 + ax + by}, \frac{1}{1 + cx + dy} \right), \quad (x,y) \in [0,1] \times [0,1]. $$

I'm interested in this map (and other similar ones) in the context of monotone dynamical systems. They come up, for instance, as the input to output characteristic of controlled dynamical systems with outputs modeling simple biochemical networks.

Question

By Brower Fixed Point Theorem, $T^2:= T \circ T$ has a fixed point. Now is this fixed point unique? If it is not unique, is there a counterexample? Are there reasonable hypotheses on $a,b,c,d$ which would guarantee that to be true?

Of course I'm also interested in the general finite-dimensional case

$$ T(x_1, \ldots, x_n) := \left( \frac{1}{1 + a_{11}x_1 + \cdots + a_{1n}x_n}, \ldots, \frac{1}{1 + a_{n1}x_1 + \cdots + a_{nn}x_n} \right), $$ as well as replacing the $1$'s in the numerator and denominator by general constants. But I'm hoping to find something conceptual in the simplest case which could then be applied in the general scenario.

Progress

(1) Computer simulations with randomly generated coefficients $a,b,c,d$ seem to indicate that this is true in arbitrary finite dimensions, with any nonnegative $a,b,c,d$.

(2) If $a + c < 1$ and $b + d < 1$, then $T$ is a contraction (with respect to the sum-norm $|(x,y)| := |x| + |y|$. In particular, it has no period-2 points, and so $T^2$ has a unique equilibrium. But these hypotheses seem too restrictive. In fact, taking into consideration where $a,b,c,d$ come from, this is not always true.

(3) This is also true if $b = c$ and $a = d = 0$. This makes me wonder whether there could be some sort of diagonalisation argument. Though I have no idea what to do with the nonlinearity.

(4) I'm also attempting to approach this as a global optimization problem. More precisely, I'm looking at the map $$ (x,y) \longmapsto \|T^2(x,y) - (x,y)\|^2 $$ and tried to compute the Hessian using Maple but haven't gotten anywhere so far.

Answer 1

I deleted the other answer in favor of a more elementary approach outlined below. Some details are missing, but they can be filled in easily.

$\newcommand{\R}{\mathbb{R}} \newcommand{\nlsum}{\sum\nolimits}$

We'll show that under fairly mild assumptions, your map has a unique fixed point towards which it converges.

We start by introducing the hyperbolic metric \begin{equation*} \delta : \R_{++}^n \times \R_{++}^n \to \R_+\quad (x,y) \mapsto \|\log x - \log y\|_{\infty}, \end{equation*} where $\log x$ denotes the vector $(\log x_1,\ldots,\log x_n)$.

A few key lemmas about this metric are listed below.

Symmetry under inversion.

(Lemma) Let $x, y > 0$, then $\delta(x^{-1},y^{-1}) = \delta(x,y)$, (the inverses are taken elementwise).

Scale invariance.

(Lemma) Let $x, y \in \R_{++}^n$, and $\alpha > 0$ be a scalar. Then, $\delta(\alpha x, \alpha y) = \delta(x,y)$.

Negative curvature properties.

(Lemma) Let $x, y, w \in \R_{++}^n$. Then, \begin{equation*} \delta\left(\nlsum_i w_i x_i, \nlsum_i w_iy_i \right) \le \max_{1 \le i \le n} \delta(x_i,y_i). \end{equation*}

Contraction.

(Lemma) Let $x, y > 0$ and $z \ge 0$. Then, \begin{equation*} \delta(z+x,z+y) \le \gamma(x,y,z)\delta(x,y), \end{equation*} where $\gamma(x,y,z) \le 1$ (with strict inequality if $z > 0$).

Now let $T : \R_{++}^{n} \to \R_{++}^{n}$ be the map of interest, so that in particular for a fixed $n \times n$ coefficient matrix $[a_{ij}]$, we have \begin{equation*} Tx : x \mapsto \left(\frac{1}{1+a_{i1}x_{1}+a_{i2}x_{2}+\cdots + a_{in}x_n}\right)_{i=1}^n. \end{equation*}

(Theorem) The map $T$ is a strict contraction in the metric $\delta$.

Proof: The lemmas above are used in the various steps below.

\begin{eqnarray} \delta(Tu,Tv) &=& \delta((Tu)^{-1},(Tv)^{-1})\\\\ &=& \left\|\log \left(1+\nlsum_j a_{ij}u_j\right)-\log\left(1+\nlsum_ja_{ij}v_{j}\right)\right\|_{\infty}\\\\ &\le& \max_{1\le i \le n}\gamma_i\left|\log (\nlsum_j a_{ij}u_j)-\log(\nlsum_j a_{ij}v_{j})\right|\\\\ &\le& \max_{1 \le i \le n}\gamma_i\max_{1\le j\le n}|\log u_j-\log v_j|\\\\ &=& \max_{1\le i \le n}\gamma_i \delta(u,v)\\\\ &\le& \gamma\delta(u,v), \end{eqnarray}

where $\gamma = \max_{1\le i \le n}\gamma_i < 1$. Since our domain is compact, this suffices to ensure a unique fixed point via the Banach fixed-point theorem.

Answer 2

This may be an overkill, but the theory is available if you are willing to consider complex variables and view your map as a rational map of the complex projective space. There are several papers by Bedford and Kim devoted to problems similar to yours. For example,

MR2742672 (2012a:37093) Bedford, Eric; Kim, Kyounghee The number of periodic orbits of a rational difference equation. Complex analysis and digital geometry, 47–56, Acta Univ. Upsaliensis Skr. Uppsala Univ. C Organ. Hist., 86, Uppsala Universitet, Uppsala, 2009.

MR2858166 (2012h:37092) Bedford, Eric; Kim, Kyounghee Linear fractional recurrences: periodicities and integrability. (English, French summary) Ann. Fac. Sci. Toulouse Math. (6) 20 (2011), Fascicule Spécial, 33–56.

These should be accessible to someone with little background or interest in complex variables. If you want a full-blown theory (pun intended, blow-ups are used), and read French, there is also this paper:

MR1656005 (99i:32031) Favre, Charles Points périodiques d'applications birationnelles de P2. (French. English, French summary) [Periodic points of birational mappings of P2] Ann. Inst. Fourier (Grenoble) 48 (1998), no. 4, 999–1023.