8.2 Lyapunov exponents and sensitivity to initial conditions

Chaos theory explores systems that are highly sensitive to initial conditions. Lyapunov exponents measure how quickly nearby trajectories diverge, quantifying this sensitivity. Positive exponents indicate chaos, while negative ones show stability.

Phase space represents all possible states of a system, with trajectories showing its evolution over time. This visual tool helps analyze dynamical systems, revealing fixed points, limit cycles, and strange attractors that characterize chaotic behavior.

Lyapunov Exponents and Exponential Divergence

Measuring Sensitivity to Initial Conditions

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• Lyapunov exponents quantify the average rate of divergence or convergence of nearby trajectories in phase space
  • Positive Lyapunov exponents indicate exponential divergence and chaos (Lorenz system)
  • Negative Lyapunov exponents indicate exponential convergence and stability (damped pendulum)
  • Zero Lyapunov exponents indicate neutral stability (undamped pendulum)
• Exponential divergence occurs when initially close trajectories separate exponentially fast over time
  • Small perturbations in initial conditions lead to vastly different outcomes (double pendulum)
  • Sensitivity to initial conditions is a hallmark of chaotic systems
• The butterfly effect illustrates the concept of sensitive dependence on initial conditions
  • A small change, like a butterfly flapping its wings, can lead to large-scale effects (weather patterns) over time
  • Demonstrates the unpredictability and complexity of chaotic systems

Calculating Lyapunov Exponents

• Lyapunov exponents are calculated by averaging the logarithm of the divergence rate over time
  • $\lambda = \lim_{t \to \infty} \frac{1}{t} \ln \frac{||\delta \mathbf{x}(t)||}{||\delta \mathbf{x}(0)||}$, where $\lambda$ is the Lyapunov exponent and $\delta \mathbf{x}(t)$ is the separation vector between nearby trajectories
• Positive Lyapunov exponents indicate chaos, while negative exponents indicate stability
  • Chaotic systems have at least one positive Lyapunov exponent (Rössler system)
  • Stable systems have all negative Lyapunov exponents (stable fixed point)
• The largest Lyapunov exponent determines the overall behavior of the system
  • A positive largest Lyapunov exponent implies chaos, even if other exponents are negative (Hénon map)
  • The sum of Lyapunov exponents is related to the divergence of the flow in phase space

Phase Space and Trajectories

Representing Dynamical Systems in Phase Space

• Phase space is an abstract space in which all possible states of a system are represented
  • Each point in phase space corresponds to a unique state of the system (position and velocity)
  • The dimension of the phase space is determined by the number of variables needed to describe the system (2D for a simple pendulum)
• Trajectories are the paths that the system follows in phase space over time
  • Trajectories represent the evolution of the system from an initial state (path of a pendulum in phase space)
  • The shape and behavior of trajectories provide insights into the system's dynamics (closed orbits for periodic motion)

Analyzing Dynamical Systems using Phase Space

• Phase space allows for the visualization and analysis of the qualitative behavior of dynamical systems
  • Fixed points, limit cycles, and strange attractors can be identified in phase space (Van der Pol oscillator)
  • Bifurcations and transitions between different behaviors can be studied using phase space (logistic map)
• Poincaré sections are used to reduce the dimensionality of phase space and simplify the analysis
  • A Poincaré section is a lower-dimensional slice of the phase space (2D section of a 3D phase space)
  • Poincaré maps capture the essential dynamics of the system and reveal important features (periodic points and chaos in the Hénon map)

Stability Analysis

Local Stability of Fixed Points

• Local stability refers to the behavior of a system near a fixed point or equilibrium
  • A fixed point is locally stable if nearby trajectories converge to it over time (stable node)
  • A fixed point is locally unstable if nearby trajectories diverge from it over time (unstable node)
• The stability of a fixed point is determined by the eigenvalues of the Jacobian matrix at that point
  • Negative real parts of eigenvalues indicate local stability (stable spiral)
  • Positive real parts of eigenvalues indicate local instability (unstable spiral)
  • Imaginary parts of eigenvalues indicate oscillatory behavior near the fixed point (center)

Global Stability and Basin of Attraction

• Global stability considers the behavior of a system over the entire phase space
  • A system is globally stable if all trajectories converge to a single fixed point or limit cycle (damped pendulum)
  • A system with multiple attractors is not globally stable, even if each attractor is locally stable (bistable system)
• The basin of attraction is the set of initial conditions that lead to a particular attractor
  • Each attractor has its own basin of attraction, which may be bounded or unbounded (Duffing oscillator)
  • Basins of attraction can have complex shapes and fractal boundaries in chaotic systems (Hénon map)
• Lyapunov functions are used to prove global stability by showing that a positive definite function decreases along trajectories
  • If a Lyapunov function exists, the system is globally stable (stable linear system)
  • The absence of a Lyapunov function does not imply instability, as the system may still be stable (nonlinear systems)