4.3 Cobweb Plots and Fixed Points

Cobweb plots are a visual tool for understanding one-dimensional maps in chaos theory. They show how values change over time by iterating a function, helping us see patterns and predict long-term behavior.

Fixed points are key in cobweb plots. These are where the function intersects the y=x line. By looking at how the plot behaves around these points, we can determine if they're stable, unstable, or neutral, giving insights into the system's dynamics.

Cobweb Plots

Technique of cobweb plots

• Graphical tool for analyzing dynamics of one-dimensional maps which are functions that map a domain value to a range value
• Function typically written as $x_{n+1} = f(x_n)$ where $x_n$ is current value and $x_{n+1}$ is next value
• To create a cobweb plot:
  1. Plot function $f(x)$ and line $y = x$ on same graph
  2. Start with initial value $x_0$ on x-axis
  3. Draw vertical line from $x_0$ to function $f(x)$
  4. From that point, draw horizontal line to line $y = x$
  5. Repeat process, drawing vertical lines to $f(x)$ and horizontal lines to $y = x$, creating "cobweb" pattern
• Visualizes iteration of function showing how values evolve over time (logistic map, population growth models)

Fixed points in cobweb plots

• Values $x^*$ where $f(x^*) = x^*$ and function intersects line $y = x$
• Stability determined using cobweb plots:
  • Convergence to fixed point indicates stability
  • Divergence from fixed point indicates instability
  • Oscillation around fixed point without convergence or divergence indicates neutral stability
• Slope of function at fixed point determines stability:
  • $|f'(x^*)| < 1$ indicates stable fixed point
  • $|f'(x^*)| > 1$ indicates unstable fixed point
  • $|f'(x^*)| = 1$ indicates neutral fixed point (transcritical bifurcation, pitchfork bifurcation)

Fixed Points

Types of fixed point stability

• Stable fixed points:
  • Nearby points attracted to fixed point
  • Small perturbations eventually return to fixed point
  • Cobweb plot iterations converge to fixed point (attractors, sinks)
• Unstable fixed points:
  • Nearby points repelled from fixed point
  • Small perturbations grow, moving away from fixed point
  • Cobweb plot iterations diverge from fixed point (repellers, sources)
• Neutral fixed points:
  • Nearby points neither converge to nor diverge from fixed point
  • Small perturbations may result in oscillations or other non-convergent behavior around fixed point
  • Cobweb plot iterations may form closed loops or exhibit other patterns around fixed point (centers, saddles)

Cobweb plots and iterative processes

• Provide insights into long-term behavior of iterative processes governed by one-dimensional maps
• Behavior of cobweb plot as iterations increase reflects asymptotic behavior of system:
  • Convergence to fixed point indicates system will eventually reach stable equilibrium (damped oscillations, steady states)
  • Divergence from fixed point suggests system will grow without bound or exhibit chaotic behavior (exponential growth, sensitivity to initial conditions)
  • Oscillations or other patterns may indicate presence of periodic orbits or other complex dynamics (limit cycles, strange attractors)
• Help identify basins of attraction for different fixed points or attractors:
  • Basin of attraction is set of initial conditions that eventually lead to particular attractor
  • Different initial conditions may result in different long-term behaviors depending on basins of attraction they belong to (bifurcation diagrams, phase portraits)