Chaotic Systems Analysis - Advanced Mathematical Toolkit for Dynamical Systems

MATHEMATICAL FRAMEWORK

RÖSSLER SYSTEM EQUATIONS

dx/dt = -y - z dy/dt = x + ay dz/dt = b + z(x - c) Default Parameters: a = 0.2, b = 0.2, c = 5.7

TAKENS' EMBEDDING THEOREM

X(t) = [x(t), x(t+τ), x(t+2τ), ..., x(t+(m-1)τ)] Where: - m = embedding dimension - τ = time delay - X(t) = reconstructed phase space vector

CORRELATION DIMENSION

C(r) = lim(N→∞) (1/N²) × Σ H(r - ||xi - xj||) D₂ = lim(r→0) log C(r) / log r Where H is the Heaviside step function

CORE CAPABILITIES

CHAOTIC ATTRACTOR ANALYSIS

Complete implementation of the Rössler system with numerical integration using Runge-Kutta methods. Generate time series data and analyze chaotic behavior through Lyapunov exponents and phase space reconstruction.

TIME-DELAY EMBEDDING

Implementation of Takens' embedding theorem for phase space reconstruction from scalar time series. Automatic parameter selection using mutual information and false nearest neighbors algorithms.

FRACTAL DIMENSION ESTIMATION

Multiple algorithms for estimating correlation dimension including Grassberger-Procaccia method with automated scaling region detection and statistical validation of results.

SCIENTIFIC VISUALIZATION

Publication-quality 3D visualizations of chaotic attractors, phase portraits, embedding analysis plots, and correlation dimension scaling with customizable styling and export options.

AUTOMATED PARAMETER OPTIMIZATION

Intelligent algorithms for optimal embedding parameter selection including time delay estimation via mutual information minimization and embedding dimension via false nearest neighbors analysis.

COMPREHENSIVE TESTING

Full test suite with 100% coverage ensuring mathematical correctness, numerical stability, and reproducible results across different computational environments and parameter ranges.

ANALYSIS VISUALIZATIONS

Three-dimensional visualization of the Rössler chaotic attractor showing the characteristic spiral structure and strange attractor geometry.

Complete analysis dashboard showing time series, phase portraits, embedding analysis, and correlation dimension estimation with statistical validation.

TECHNICAL SPECIFICATIONS

SYSTEM FOCUS

Rössler Chaotic System

INTEGRATION METHOD

4th Order Runge-Kutta

EMBEDDING ALGORITHM

Takens' Theorem Implementation

DIMENSION ESTIMATION

Grassberger-Procaccia Method

PARAMETER OPTIMIZATION

Mutual Information & FNN

VISUALIZATION

3D Phase Space & 2D Projections

FRAMEWORK

NumPy, SciPy, Matplotlib

TEST COVERAGE

100% Code Coverage