This paper aims to investigate the complex behavior of the logistic map, which experiences impulsive perturbations dependent on its state. The conditions of stability and border-collision bifurcation (BCB) of periodic orbits are presented. The main contributions of this paper are as follows: (1) The mathematical model of the logistic map involving impulsive forces is presented; (2) The BCB of periodic orbits is examined by creating the two-parameter bifurcation diagrams of the map; (3) Several bifurcation scenarios related to the existence of BCB are demonstrated.
The system model consider the logistic map written as the difference equation \[ x(k+1)=rx(k)(1-x(k)), \] where \(x(k)\in [0,1] (k=0,1,2,\dots)\) denotes the system state and \(r\) is a system parameter. Assuming that the logistic equation is perturbed by an impulsive force when its state \(x\) is less than the fixed value \(h\), one can express the resulting difference equation as \[ \begin{cases} x(k+1)= r x(k)(1-x(k)) \quad x(k)>h,\\ x(k+1) = x(k) +\alpha \quad x(k)\leq h, \end{cases} \] where \(0 < h < 1\) and \(0 < \alpha < 1- h\) determine the strength of the impulsive force.