Cobweb Diagram

What is a Cobweb Diagram?

A cobweb diagram (also called a Verhulst diagram or staircase diagram) is a graphical method for visualizing the iterates of a one-dimensional map \(f : \mathbb{R} \to \mathbb{R}\). Given an initial condition \(x_0\), the sequence \(x_1=f(x_0),\; x_2=f(x_1),\ldots\) is generated by repeated function application. The diagram draws two curves — the graph of \(y=f(x)\) and the diagonal \(y=x\) — and traces a zigzag path that encodes every iterate:

• Start at \((x_0, x_0)\) on the diagonal.
• Move vertically to \((x_0, f(x_0))\) on the curve — this gives \(x_1\).
• Move horizontally to \((x_1, x_1)\) on the diagonal — this "transfers" \(x_1\) to the \(x\)-axis.
• Repeat. The path's long-term behavior reveals stability, periodicity, or chaos.

Fixed Points & Their Location

A fixed point \(x^*\) satisfies \(f(x^*)=x^*\), i.e., it lies at the intersection of the curve \(y=f(x)\) and the diagonal \(y=x\). Every fixed point appears as a corner where the cobweb path could get "stuck" permanently.

Stability Classification via the Derivative

The local stability of \(x^*\) is governed entirely by the magnitude of \(f'(x^*)\):

Reading the Cobweb Shape

• Inward spiral: \(f'(x^*) \in (-1, 0)\) — oscillatory stable convergence.
• Staircase to the right: \(f'(x^*) \in (0, 1)\) — monotone stable convergence.
• Outward spiral: \(f'(x^*) \in (-\infty,-1)\) — oscillatory divergence.
• Rectangle / 2-cycle: orbit bounces between two values — period-2 orbit.
• Dense zig-zag filling a region: chaotic orbit, sensitive to initial conditions.

Periodic Orbits & Period-Doubling Cascade

A period-\(n\) orbit satisfies \(f^n(x^*)=x^*\) but \(f^k(x^*)\ne x^*\) for \(1 \le k < n\). For the logistic map \(f(x)=rx(1-x)\), the parameter \(r\) controls behavior:

• \(r \in (1,3)\): unique stable fixed point \(x^*=(r-1)/r\).
• \(r \in (3, 1+\sqrt{6})\approx 3.449\): stable period-2 orbit.
• Successive period-doublings at \(r\approx 3.449,\,3.544,\,3.5644,\ldots\)
• \(r\approx 3.5699\ldots\) (Feigenbaum constant): onset of chaos.
• \(r=4\): fully chaotic; Lyapunov exponent \(\lambda=\ln 2\).

Lyapunov Exponent & Chaos

The Lyapunov exponent measures the average rate of separation of nearby orbits:

• \(\lambda < 0\): attracting — orbits converge (stable fixed point or periodic orbit).
• \(\lambda = 0\): neutral — bifurcation boundary or neutral fixed point.
• \(\lambda > 0\): chaos — exponential divergence of nearby orbits.

The Lyapunov exponent is displayed in real time below the animation controls.

How to Interact

• Scroll or pinch on the cobweb canvas to zoom in/out.
• Drag the canvas to pan the view.
• Double-click the canvas to reset the view.
• Use Play/Step/All to animate, advance one step, or jump to all iterations.
• Drag parameter sliders to see the cobweb change in real time.
• Click anywhere on the cobweb canvas to set a new initial condition \(x_0\).