Calculating Autocorrelation and Cross-Correlation Functions for Discrete Hyperchaotic Systems

Calculation of autocorrelation and cross-correlation functions for discrete hyperchaotic systems

When studying discrete hyperchaotic systems (such as the Kawakami system), analyzing their autocorrelation and cross-correlation functions helps understand the system's dynamic characteristics. These two functions can reveal the internal correlations within time series and the interaction patterns between different state variables.

Autocorrelation function: Reflects the linear correlation of the same state variable at different time points. For discrete sequences x(n) generated by the Kawakami system, autocorrelation can be obtained by calculating the covariance between the sequence and its time-shifted version. Specific steps include: zero-meaning the sequence, computing the inner product after time shifting, and normalization processing. The results can demonstrate the decay characteristics of the system's memory length, where hyperchaotic systems typically exhibit rapid decay with aperiodic oscillations.

Cross-correlation function: Used to analyze the time-delayed correlation between different variables (such as x(n) and y(n)). During calculation, both variables need to be zero-meaned separately, then the covariance strength under different time shifts is evaluated. In the Kawakami system, the cross-correlation function may reveal the nonlinear coupling strength between x-y subsystems, with peak positions corresponding to key interaction time delays.

Implementation key points: Preprocessing: Remove DC components from time series to highlight dynamic features Boundary handling: Use circular shift or zero-padding strategies for time-shift calculations with finite sequences Normalization: Constrain function values within the [-1,1] interval for cross-system comparisons Visualization: Identify system periodic windows or synchronization phenomena through function curves

Application significance: Correlation function analysis can verify the unpredictability of hyperchaotic systems and provide basis for parameter optimization in scenarios like chaotic secure communication. For high-dimensional mappings like the Kawakami system, cross-correlation analysis can also assist in determining whether the system is in a generalized synchronization state.