Collected Bifurcation Diagrams for the Lorenz System

The bifurcation diagram of the Lorenz system serves as a crucial tool for studying the behavior of this classical chaotic system. It visually demonstrates transitions between steady states, periodic motions, and chaotic regimes as system parameters vary. Common analytical methods include: Parameter scanning method: By fixing other parameters and incrementally adjusting a key parameter (e.g., the Rayleigh number r), one observes the distribution of extreme values in system variables to construct bifurcation diagrams. In code implementation, this typically involves nested loops where the outer loop iterates through parameter values while the inner loop integrates the Lorenz equations using methods like Runge-Kutta (e.g., ode45 in MATLAB). The resulting diagram clearly reveals routes to chaos through period-doubling bifurcations. Poincaré section method: This technique records intersections between system trajectories and a specific plane in phase space. As parameters change, the number of intersection points transitions from finite (periodic solutions) to infinite (chaotic behavior). Code implementation requires defining a cross-section plane (e.g., z=constant) and detecting trajectory crossings using event detection algorithms, with bifurcation diagrams illustrating this evolutionary pattern. Lyapunov exponent method: By incorporating calculations of the largest Lyapunov exponent, bifurcation diagrams can annotate chaotic regions (positive exponents) and stable regions (negative exponents), enhancing quantitative analysis. Implementation involves linearizing the system equations and monitoring the divergence of nearby trajectories through QR decomposition or Wolf's algorithm to compute exponents efficiently. For the Lorenz system, bifurcation diagrams typically reveal characteristic "butterfly effect" structures, including stable fixed points at small r values, periodic oscillations at intermediate parameters, and chaotic attractors at large r values. Comparative analysis of multiple bifurcation diagrams helps identify parameter sensitivity intervals, providing foundations for chaos control strategies. Code implementations often utilize visualization techniques like scatter plots with color-coding to represent different dynamic regimes.