Natural Exponential and Three‐Dimensional Chaotic System

Abstract Existing chaotic system exhibits unpredictability and nonrepeatability in a deterministic nonlinear architecture, presented as a combination of definiteness and stochasticity. However, traditional two‐dimensional chaotic systems cannot provide sufficient information in the dynamic motion and usually feature low sensitivity to initial system input, which makes them computationally prohibitive in accurate time series prediction and weak periodic component detection. Here, a natural exponential and three‐dimensional chaotic system with higher sensitivity to initial system input conditions showing astonishing extensibility in time series prediction and image processing is proposed. The chaotic performance evaluated theoretically and experimentally by Poincare mapping, bifurcation diagram, phase space reconstruction, Lyapunov exponent, and correlation dimension provides a new perspective of nonlinear physical modeling and validation. The complexity, robustness, and consistency are studied by recursive and entropy analysis and comparison. The method improves the efficiency of time series prediction, nonlinear dynamics‐related problem solving and expands the potential scope of multi‐dimensional chaotic systems.


Supplementary Note 1. Forecasting Verification
. Influence of initial difference of x(t) to system output error.    gain coefficients.

Figure S3
Output errors and performance evaluation for Duffing model. a) Error of output x(t) when the initial input difference of x0 changes. b) Error of output y(t) when the initial input difference of y0 changes. c) Change trends of output errors for x(t) and y(t) when the initial difference of x0 varies within the values of [0.001, 0.002, 0.005, 0.01, 0.02, 0.05, 0.1, 0.2, 0.5, 0.9].   Table S4. Topological entropy (TE) for the output of the natural exponential threedimensional (3D) chaotic system under different system parameters.

Supplementary Note 2. Lyapunov exponent
(a) TE for the output of the natural exponential 3D chaotic system under different system parameters of a.    -7, -5, -1, 0, 1, 3, 5, 15, 20, 50], respectively. A to J represents ten groups of system parameters. Table S5. Approximate entropy (AE) for the output of the natural exponential 3D chaotic system under different system parameters.

b) Approximate entropy
(a) AE for the output of the natural exponential 3D chaotic system under different system parameters of a.  Table S6. Shannon entropy (SE) for the output of the natural exponential 3D chaotic system under different system parameters.

c) Shannon entropy
(a) SE for the output of the natural exponential 3D chaotic system under different system parameters of a.

d) Fuzzy entropy
(a) FE for the output of the natural exponential 3D chaotic system under different system parameters of a.  Table S8. Conditional entropy (CE) for the output of the natural exponential 3D chaotic system under different system parameters.