6.3 The Rössler System

The Rössler system is a set of three equations that create chaos in a simple model. It shows how complex behavior can arise from basic rules, making it a key example in chaos theory.

The system uses three variables that change over time, creating a strange attractor. By tweaking the system's parameters, we can see different patterns emerge, from steady states to wild chaos.

The Rössler System

Introduction to Rössler system

• Set of three coupled nonlinear ordinary differential equations developed by Otto Rössler in 1976 exhibits chaotic behavior for certain parameter values
• Simplified model of chemical kinetics demonstrates the possibility of chaos in a relatively simple system (Belousov-Zhabotinsky reaction)
• Continuous-time dynamical system has state variables that evolve continuously over time contrasts with discrete-time systems (logistic map) where variables change at discrete time steps

Equations of Rössler system

• Defined by three differential equations:
  • $\frac{dx}{dt} = -y - z$
  • $\frac{dy}{dt} = x + ay$
  • $\frac{dz}{dt} = b + z(x - c)$
• Three state variables x, y, and z represent the concentrations of chemical species in the original context (activator, inhibitor, catalyst)
• Three parameters a, b, and c control the system's behavior and the emergence of chaos
• Typical parameter values for chaotic behavior: a = 0.2, b = 0.2, and c = 5.7 other combinations can also lead to chaos (a = 0.1, b = 0.1, c = 14) or other types of behavior (periodic orbits, fixed points)

Visualization of Rössler dynamics

• Exhibits chaotic behavior for certain parameter values with sensitivity to initial conditions and aperiodic, bounded trajectories
• Strange attractor is a complex geometric structure in the system's phase space where trajectories are attracted to and confined within this structure exhibiting fractal properties (self-similarity, non-integer dimension)
• Visualizing the Rössler attractor:
  1. Plot the system's trajectories in 3D phase space (x, y, z) reveals the intricate, layered structure of the attractor
  2. Poincaré section is a 2D slice through the 3D phase space helps analyze the attractor's structure and properties (fractal dimension, symbolic dynamics)
• Animations and interactive visualizations (Python libraries: matplotlib, plotly) help explore the system's dynamics and the effects of changing parameters

Mechanisms in Rössler attractor

• Stretching and folding are key mechanisms behind the Rössler system's chaotic dynamics responsible for generating the strange attractor
• Stretching mechanism causes nearby trajectories to diverge exponentially over time leading to sensitivity to initial conditions quantified by positive Lyapunov exponents (λ > 0)
• Folding mechanism keeps trajectories confined within a bounded region of phase space preventing trajectories from escaping to infinity results in the layered, fractal structure of the attractor (horseshoe map)
• Interplay between stretching and folding:
  1. Continuous stretching and folding of the system's phase space
  2. Generates the complex, chaotic dynamics of the Rössler system
  3. Produces the intricate, self-similar structure of the strange attractor
• Understanding these mechanisms provides insights into the origins of chaos in the Rössler system and its connection to other chaotic systems (Lorenz system, double pendulum)