Fractal dimension and control of Julia set generated by the discrete competitive model

The aim of the article is to study the fractal behavior of a general non-linear discrete competitive model. The model represents the growth of two species that have allelopathic effects on each other. The discrete competitive allelopathic model has not yet been examined from a fractal perspective. Furthermore, Julia sets have not yet been controlled using the new control mechanism presented in this article, and this control scheme may be applicable to other ecological models of various orders. We first convert the model into the corresponding discrete model and then, using the escape-time approach, we generate the Julia set of this discrete version. The control of the Julia set is then obtained by altering the control parameters using the gradient control method and the state control method. By controlling the fixed-point stability, we can control the Julia set. Our main focus is on controlling the fixed-point stability. In the gradient control method, we observed that the Julia set of the controlled system shrinks as the value of the control parameter decreases. It should be noted that this phenomenon is due to the instability of fixed points. In the second control scheme, we observed that the Julia set shrinks vertically as the value of control parameters is taken in a triangular stable region and the value of parameters increases. This is again due to the instability of fixed points. Here, the complexity and irregularity of the Julia sets are characterized by the box-counting dimension, which is empirically estimated and mathematically easy to calculate. The synchronization of two different Julia sets of the models with different parameters is discussed by designing two nonlinear coupling items which make one Julia set change to another, and synchronization of Julia sets is achieved by its trajectories synchronization. In addition, the efficiency of two different synchronous controllers is discussed. Various main theoretical results and the effectiveness and correctness of methodologies used are validated using numerical simulations. The dependence of the structure of Julia set and results on the model parameter values are also discussed in this article. Moreover, the analysis is applicable to other similar mathematical models.