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

Top images from around the web for Technique of cobweb plots

• 

  Chaos theory - Wikipedia View original

  Is this image relevant?

• 

  Cobweb plot - Wikipedia View original

  Is this image relevant?

• 

  An Elementary Study of Chaotic Behaviors in 1-D Maps View original

  Is this image relevant?

• 

  Chaos theory - Wikipedia View original

  Is this image relevant?

• 

  Cobweb plot - Wikipedia View original

  Is this image relevant?

1 of 3

Top images from around the web for Technique of cobweb plots

• 

  Chaos theory - Wikipedia View original

  Is this image relevant?

• 

  Cobweb plot - Wikipedia View original

  Is this image relevant?

• 

  An Elementary Study of Chaotic Behaviors in 1-D Maps View original

  Is this image relevant?

• 

  Chaos theory - Wikipedia View original

  Is this image relevant?

• 

  Cobweb plot - Wikipedia View original

  Is this image relevant?

1 of 3

• 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)