Dynamical Systems and Iteration

Key idea

Fixed points, cycles and chaos.

In depth

A dynamical system repeatedly applies a map xₙ₊₁ = f(xₙ). Fixed points satisfy f(x) = x and are stable if |f′(x)| < 1. Orbits can converge, cycle, or behave chaotically.

Worked example

f(x) = x²: fixed points 0 (stable) and 1 (unstable).

Real-world uses

• Modelling population cycles, weather and repeated calculations in computing

Common mistake

Assuming simple rules lead to simple behaviour.

How to learn this

1. Read the key idea and say it in your own words.
2. Work through the worked example, then try a similar problem.
3. Use the interactive lesson’s visual and mini-quiz to check the pattern.

Frequently asked questions

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Related expert topics in Analysis & Topology

• Axioms of the Real Numbers – ℝ is a complete ordered field.
• Completeness and Supremum – Every nonempty bounded-above set has a least upper bound.
• Archimedean Property – For any real x there is an integer n > x.
• Density of Rationals – Between any two reals there is a rational.
• Sequences and Convergence (ε–N) – aₙ → L if for every ε>0 eventually |aₙ − L| < ε.
• Limit Theorems for Sequences – Algebra of limits and squeeze theorem.