Analyzing a discrete dynamical system for stable orbits

Question

I have a discrete dynamical system $x_{n+1} = f(x_n)$ where $f(x,a) = \frac{2}{x} + \frac{x}{4} + a$ where $a = -1.5, -1.75, \cdots$ is a varying parameter. I wish to determine if a stable periodic orbit exists per each parameter $a$ and if so, find its period.

Let $a = -1.5$. So we have $f(x) = \frac{2}{x} + \frac{x}{4} -1.5$. This trajectory has equilibrium points at $x =-2.91485, \; 0.914854$ seen by this plot.

From the following Mathematica code,

orb   = NestList[f, .714854, 30 ]
orb2  = NestList[f, -2.61485, 30]

{0.714854, 1.47649, 0.223688, 7.49695, 0.641012, 1.78032, 0.0684742, \
27.7252, 5.50343, 0.239268, 6.91864, 0.518734, 2.48522, -0.0739374, \
-28.5684, -8.7121, -3.90759, -2.98872, -2.91636, -2.91488, -2.91485, \
-2.91485, -2.91485, -2.91485, -2.91485, -2.91485, -2.91485, -2.91485, \
-2.91485, -2.91485, -2.91485}

{-2.61485, -2.91857, -2.91491, -2.91486, -2.91485, -2.91485, \
-2.91485, -2.91485, -2.91485, -2.91485, -2.91485, -2.91485, -2.91485, \
-2.91485, -2.91485, -2.91485, -2.91485, -2.91485, -2.91485, -2.91485, \
-2.91485, -2.91485, -2.91485, -2.91485, -2.91485, -2.91485, -2.91485, \
-2.91485, -2.91485, -2.91485, -2.91485}

ListPlot[orb, PlotMarkers -> {Automatic, 10}, PlotRange -> All, Joined -> True  ]

we can see that despite the starting point $x_0$, the trajectory always converges to $x =-2.91485$. Why? Should I have expected this? I initially thought that starting near 0.9... would cause the trajectory to converge to the $x = 0.914854$ equilibrium point. Is this wrong intuition? Can anyone please explain?