(1+s)/(2+s)=(1+s)(2-s)/(4-s2). When s=√3, this becomes (2+s-s2)/(4-s2)=√3-1.
3/(1+s-x)=
3/(1-x+s)=
3(1-x-s)/((x-1)2-s2)=
3(1-x-√3)/(x2-2x-2), and the whole equation becomes:
z=3(1-x-√3)/(x2-2x-2)+1-√3.
This expression has an asymptote at z=1-√3, which means that if the magnitude of the input x is sufficiently large (|x|→∞), z→1-√3. This value is about -0.73205.
An attractor attains stability (converges to a specific value) for a particular input x; or it attains a set of specific values, each related to a specific input.
However, although z does converge to a specific value, the input has to have a large magnitude, an indefinite range of values for which z approaches this value, but will never attain it.
It's not apparent what a circle attractor is.
The only way I can think of this is to imagine a mass of swirling particles which eventually migrate to a circular orbit around a fixed point, rather like a belt of asteroids might migrate to a circular orbit around a star. Those asteroids which are initially close to the orbit will quickly move to that orbit, whilst those nearer to the star will take longer to migrate to the orbit. The speed of the asteroids may be the determining factor for the radius of the circle. The attractor would be the circle itself. The given attractor is a hyperbola, not a circle, so I can't see the connection in this case, unless it represents a differential equation or its solution.
The theories concerning attractors seem to be quite complicated, and "attractor" has more than one meaning, including physical definitions involving dynamics.
The given equation could represent a 3-dimensional hyperbolic surface (actually two curved surfaces generated by the same equation) which is the attractor. The two surfaces appear to stay the same distance apart and move together horizontally if s changes (but s≠2). Perhaps the curves represent dynamic magnetic or electric fields, and the fields have the same magnetic polarity so there is a fixed repulsion between them. Magnetically/electrically charged particle within the fields migrate might eventually migrate to the boundaries (the surfaces). The attractor could be the solution to a differential field equation which describes such a system in which randomly distributed charged particles migrate to one or other of the two surfaces over time.
Hope this helps.