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In this paper, a novel escape-time algorithm is proposed to calculate the connectivity’s degree of Julia sets generated from polynomial maps. The proposed algorithm contains both quantitative analysis and visual display to measure the connectivity of Julia sets. For the quantitative part, a connectivity criterion method is designed by exploring the distribution rule of the connected regions, with an output value Co in the range of [0,1]. The smaller the Co value outputs, the better the connectivity is. For the visual part, we modify the classical escape-time algorithm by highlighting and separating the initial point of each connected area. Finally, the Julia set is drawn into different brightnesses according to different Co values. The darker the color, the better the connectivity of the Julia set. Numerical results are included to assess the efficiency of the algorithm.

The Lotka-Volterra model plays an important role in the research area of population biology. This work presents the analysis of dynamical behaviours of a kind of three-species Gause-Lotka-Volterra (GLV for short) system from the viewpoint of fractals. First, the definition of Julia set which describes the initial distribution rule of the three species' densities is introduced. Second, the gradient control method which contains both giant parameter and state feedback is applied to realize the boundary control of the initial fractals area of three coexisting species. Third, we consider the upper bound of the controlled Julia set from a kind of weakly-coupled GLV system, i.e. NPZsystem, by analysing the growth pattern of the initial species. Finally, the nonlinear coupling terms are designed to realize the synchronization of two Julia sets, with the result that the dynamic behaviors of the controlled system can be guided to an ideal one. Numerical examples are included to verify the conclusions of the theoretical investigations. [ABSTRACT FROM AUTHOR]