Stability of Fixed Points

Mathematical Modeling · Axiom Academy

LESSON Stability of Fixed Points - Mathematical Modeling Understanding the Long-term Behavior of Discrete Dynamical Systems Consider a discrete dynamical system described by the iteration: A fixed point (also called an equilibrium point) is a value x^* that satisfies: At a fixed point, the system is in equilibrium: if you start at x^* , you stay at x^* forever! To find fixed points, solve the equation x = f(x) , or equivalently: Graphically, fixed points occur where the graph of y = f(x) intersects the line y = x . Find the fixed points of f(x) = 2x - 3 . Solving: -x = -3 , so x^* = 3 . Find the fixed points of f(x) = x^2 - 2 . Fixed points: x^*_1 = 2 and x^*_2 = -1 . Finding fixed points tells us where equilibria exist. But we also need to know: are they stable? Move away from x^* ? (Unstable) Consider a small perturbation from the fixed point: The question becomes: does grow or shrink as n increases? To analyze stability, we linearize f near the fixed point. Using Taylor expansion: Since f(x^*) = x^* , the perturbation evolves as: For a fixed point x^* of the map x_ n+1 = f(x_n) : Worked Examples: Stability Analysis Analyze the stability of fixed points for f(x) = rx(1-x) where r > 0 . x(1 - r + rx) = 0 , giving x^*_1 = 0 and Step 3: Evaluate at fixed points Summary: As r increases past 1, stability transfers from x^*_1 = 0 to . When r exceeds 3, x^*_2 also becomes unstable, leading to period-doubling bifurcations. Analyze stability for f(x) = ae^ -x where a > 0 .

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