Calculation of Autocorrelation and Cross-Correlation Functions for Discrete Hyperchaotic Systems

This article presents the calculation of autocorrelation and cross-correlation functions for discrete hyperchaotic systems. We employ the Kawakami hyperchaotic system to accomplish this task. First, it's essential to understand what discrete hyperchaotic systems are. Simply put, a discrete hyperchaotic system is a nonlinear dynamical system that can generate highly complex behaviors and seemingly random outputs. These systems are typically governed by a set of differential or difference equations that describe the relationships between various variables in the system. The Kawakami hyperchaotic system is a commonly used discrete hyperchaotic system known for its excellent properties and wide range of applications. It consists of four variables described by the following difference equations: x_{n+1} = sin(y_n) + c·sin(z_n) y_{n+1} = sin(z_n) + c·sin(x_n) z_{n+1} = sin(x_n) + c·sin(y_n) w_{n+1} = w_n + a·sin(x_n) where c and a are control parameters. By adjusting these parameters, we can obtain different system behaviors and outputs. In code implementation, these equations would be iterated numerically to generate the time series data, typically using a for-loop structure with initial conditions set for all four variables. In our program implementation, we simulate the discrete hyperchaotic system using the Kawakami model and calculate both autocorrelation and cross-correlation functions. Specifically, our algorithm first generates a time series from the Kawakami hyperchaotic system by iterating the equations over a specified number of time steps. The implementation would involve initializing the system with random or predefined starting values and then computing subsequent states using the difference equations. We then utilize this generated time series to compute the autocorrelation function (which measures the similarity between observations at different time lags) and cross-correlation functions (which measure the correlation between different variables in the system). The correlation calculations typically employ discrete correlation formulas, possibly using FFT-based methods for computational efficiency when dealing with large datasets. The final output provides the autocorrelation and cross-correlation functions for the discrete hyperchaotic system, which helps us better understand and analyze the characteristics and behaviors of these complex systems.