Logistic Chaotic System: Implementation and Analysis

The Logistic chaotic system represents a classic nonlinear dynamical model widely employed in chaos theory and complex system research. This system demonstrates complex chaotic behavior through a simple iterative formula, making it suitable for applications such as random number generation and encryption algorithms.

In MATLAB implementation, the Logistic chaotic system primarily utilizes its recurrence formula for numerical simulation. By adjusting parameters and initial values, this formula can generate diverse chaotic sequences. The simulation process typically involves three key steps: parameter configuration, iterative computation, and result visualization.

The implementation begins with defining the system's control parameter and initial conditions. The selection of control parameters directly influences the system's behavioral patterns—when parameters fall within specific ranges, the system enters a chaotic state. The iterative process is implemented through loop structures, where each cycle computes the current value and updates the next state according to the Logistic formula: x_n+1 = μx_n(1-x_n), where μ denotes the control parameter and x_n represents the current state value.

Finally, chaotic characteristics can be observed through time-series plots or phase space diagrams. Time-series plots display state evolution over iterations, while phase space diagrams reveal long-term behavioral patterns of the system. These visualization techniques facilitate intuitive understanding of the dynamic properties inherent to chaotic systems. Code implementation typically involves using MATLAB's plot() function for time-series visualization and scatter() for phase space representation, with proper axis labeling to enhance interpretability.