Learning objectives
By the end of this lesson, you should be able to:
- Identify a fixed point.
- Distinguish stable and unstable equilibria.
- Relate local slope or linearization to stability.
Prerequisites: What Is a Dynamical System?
EXPERIMENT Interactive dynamics laboratory
Perturb → Evolve → CompareChange one parameter at a time. Predict the qualitative behavior before running or moving the control.
Equilibrium is not enough
Knowing that a fixed point exists does not tell us whether nearby states return to it or move away. Stability answers that second question.
Linearization
Near a one-dimensional fixed point, a map can be approximated by \(\delta_{n+1}\approx f'(x^*)\delta_n\). Therefore perturbations shrink when \(|f'(x^*)|<1\) and grow when \(|f'(x^*)|>1\).
Why local analysis matters
Many nonlinear systems are hard to solve globally, but local linearization often explains how equilibria lose stability and how bifurcations begin.
Key takeaways
What should stay with you
- A fixed point satisfies f(x*)=x*.
- Stability asks what happens after a small perturbation.
- Linearization provides a local approximation to nonlinear dynamics.
Knowledge check
For a one-dimensional map xₙ₊₁=f(xₙ), which local condition indicates a stable fixed point x*?
Small perturbations shrink when the local multiplier has magnitude below one.