Novel method for identifying the linear region in calculating the correlation dimension and the largest Lyapunov exponent

Identifying the linear region interval is crucial for calculating correlation dimension (D2) and the largest Lyapunov exponent (LLE). In this paper, we propose a dynamic range identification method based on the single-objective programming theory. First, calculating the correlation integral and the average divergence exponent using the algorithm of Grassberger and Procaccia and the algorithm of Rosenstein et al.. Then, utilizing single-objective programming theory and taking the maximization of interval length as the objective function, while imposing constraints on the model's fitting accuracy, correlation between variables, etc., accurate identification of the linear region under the condition of optimal linearity is achieved. Finally, we used the least square method to fit the linear region to obtain D2 and the LLE, respectively. Some examples are used for numerical simulation to verify the effectiveness of our method. Compared to some mainstream methods, our method does not require a complex calculation process, and the results obtained are similar to theirs. In addition, this method provides a new idea for calculating the linear region.