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Fixed Points and Stability

Undergraduate

Definition

A fixed point is a state that doesn't change over time in a dynamical system. Fixed points can be stable (attractors), unstable, or saddle points.

f(x^*) = x^* (discrete), f(x^*) = 0 (continuous)

Fixed point condition

|f'(x^*)| < 1 : stable, |f'(x^*)| > 1 : unstable

Stability in discrete systems

Example 1

Analyze fixed points and stability of f(x) = x² - x - 1.

System stabilization

Equilibrium populations

Market equilibrium