Bifurcation calculation in population chaotic dynamics

Chaotic population dynamics present interesting behaviors that are the object of study for biologists, microbiologists, ecologists, epidemiologists and bioanalysts. With the proposal of the mathematician and biologist Pierre Verhulst on the logistic equation as a population discrete mathematical model, studies were initiated on chaotic dynamics regarding the growth factor immersed in the equation. When assigning different values to the factor, orbits are observed that are initially convergent, however, there are values for which there is no convergence due to the bifurcations suffered by the orbit. For values greater than or equal to 3 and less than 4, the forks bend, but not consecutively. Through the logistic equation and the identity function it is possible to find the values of the orbit corresponding to a specific value of the growth factor, however, these data are evidenced graphically, so it is important to know the explicit calculation of these bifurcations. The calculation of the fixed points of the quadratic iterator of the logistic equation is shown by equations.