We investigate the phase transition of a dynamical system generating a possibly infinite orbit of points. The points of the orbit are generated according to the following basic operation. Given a positive real number $a$, called the expansion factor, and two points $p, q$ at Euclidean distance $|pq|$ we determine the unique point $p'$ on the straight line passing through $p$ and $q$ which is antipodal to the point $p$ with respect to $q$ and at a Euclidean distance $a |pq|$ from $q$. The operation on points previously defined is denoted by $p \Rightarrow_{a,q} p'$. Let ${\bf a} := a_0,a_1, \ldots, a_{n-1} $ be arbitrary but fixed positive real numbers and ${\bf q} := q_0, q_1, \ldots, q_{n-1}$, be $n$ (anchor) points. An orbit consisting of an infinite sequence $p_0, p_1, \ldots , p_m, \ldots$ of points in the plane is generated by using the anchor points as follows. The orbit is initiated with an arbitrary point $p_0:=p$ and for all integers $m \geq 0$, satisfies $p_m \Rightarrow_{a_{m \bmod n},q_{m \bmod n}} p_{m+1}$ so that $p_{m+1} := (p_m)'$. The resulting sequence of points is called the $({\bf a}, {\bf q})$-orbit of $p$. For any starting point $p$ and any pair $({\bf a}, {\bf q})$ we characterize the boundedness of $({\bf a}, {\bf q})$-orbits. Namely, we show that there is a phase transition concerning the boundedness of the resulting $({\bf a}, {\bf q})$-orbit which depends on whether the product $a_0 a_1 \cdots a_{n-1}$ of the expansion factors is less or larger than one. We also characterize the behaviour of the orbits when $a_0 a_1 \cdots a_{n-1} = 1$. The “boundedness” phase transition phenomenon described above is shown to be valid for any dimension $d=1,2,3$ in Euclidean space. In addition, we propose variants of this approach for generating orbits on convex polygons, and propose several open problems corresponding to phase transition phenomena.