Piecewise Contracting Maps

Assume that a piecewise affine map is defined on a piecewise affine space equipped with some distance $d(\cdot,\cdot)$, perhaps given by some Euclidean structure.

A piecewise affine map $T$ is called piecewise contracting if its restriction to each piece $A$ satisfies:

\begin{align} \exists a<1 \forall x,y\in A\quad d(T_Ax,T_Ay)\leq a d(x,y) \end{align}

Of course, a globally contracting map is not very interesting from a dynamical point of view. When two pieces are involved, Gambaudo & Tresser showed that the symbolic dynamics in the $w$-limit set is identical to one of a circle rotation (see Bol. Soc. Brasil. Mat. 19 (1988), 61-114). All cases of rotation numbers can appear.

When there are more than two pieces, the dynamics may present intriguing features. This is particularly the case of the following map of the unit square $[0,1]^2$ with 4 pieces (and inspired by the modelling of regulatory processes in Biology, see J. Math. Bio. 52 (2006), 524-570):

\begin{equation} T(x_1, x_2) = a(x_1,x_2) + (1-a)(H(c_2-x_2),H(x_1-c_1)) \end{equation}

Here $H$ denotes the Heaviside function and the linear and threshold parameters $a$ and $,c_1,c_2$ are chosen between 0 and 1. The basic features of this map are easy to understand. The lines $x_1=c_1$ and $x_2=c_2$ define a partition of the square into 4 rectangles. The image of every rectangle exclusively intersects the counter-clockwise subsequent rectangle and, possibly, the initial rectangle itself depending on parameters. In any case, all iterates must end up in the subsequent rectangle after a while. Since this property holds for every rectangle, all orbits must circumvent forever around the singularity $(c_1,c_2)$.

A substantial amount of the asymptotic behavior is known for this system. It has been proved that, just as for rigid rotations on the circle, full families of uniformly winding motions exist across the parameter range (for exceptional set of parameter values, these sequences consist of ghost orbits). The corresponding rotation number has continuous and monotonic variations with threshold parameters. Unlike for circle rotations however, several orbits with distinct rotation numbers may coexist (actually, as many as desired when the parameters are suitably chosen). In addition, depending on parameters, the map is known to also possess orbits with non-uniform rotation properties. However, knowledge is limited beyond uniform orbits and a systematic analysis of all possible rotating motions remains to be done.

Since every periodic orbit (lying out of singularities) must be asymptotically stable, numerical plots of the basin of attractions when various rotations coexist are expected to produce fairly intricate pictures. They actually do !

Basins 1

Basin of attractions of periodic orbits for parameters $a = 0.98, c_1=c_2= 0.5$. There are 4 colors that correspond to 4 stable orbits (barely visible red domains are located in the center of the picture).

Basins 2

Basin of attractions (zoom in the neighborhood of the singularity $(c_1,c_2)$) of periodic orbits for parameters $a = 0.98, c_1=c_2= 0.5$. There are 5 stable orbits in this case.

Basins 3

A simple case where only 3 stable orbits exist: $a = 0.98, c_1=c_2= 0.12$.

Basins 4

Zoom in some region of the square for $a = 0.999, c_1=0.5006$ and $c_2= 0.5003$.

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