Cobweb Diagrams

Mathematical Modeling · Axiom Academy

Unit 2: Discrete Models - A graphical technique for analyzing discrete dynamical systems and understanding iteration behavior A cobweb diagram is a graphical method for visualizing iterations of a discrete dynamical system. Given a function f(x), we want to understand what happens when we repeatedly apply it: The diagram consists of two curves: The function curve: y = f(x) - shows how the system transforms each value The diagonal line: y = x - serves as a "mirror" to reflect outputs back to inputs Fixed points occur where these curves intersect, i.e., where f(x) = x. 2. Constructing Cobweb Diagrams Step by Step Building a cobweb diagram follows a simple, repetitive process: Step 1: Start at the initial value x_0 on the x-axis Step 2: Draw a vertical line up (or down) to the curve y = f(x) Step 3: Draw a horizontal line to the diagonal y = x Step 4: Repeat from the new x-value (which equals the previous f(x)) The key insight: the horizontal line to the diagonal "converts" the y-value (output) back into an x-value (input) for the next iteration. 3. The Iteration Process Graphically Each "stair step" in the cobweb represents one iteration of the map. Watch how the sequence of values unfolds: The cobweb pattern reveals the iteration sequence without computing any numbers. The shape tells us immediately whether the sequence is: Spiraling inward: Converging to a fixed point (stable) Spiraling outward: Diverging from a fixed point (unstable)

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