Stability of Equilibria

Mathematical Modeling · Axiom Academy

LESSON Stability of Equilibria Analyzing the long-term behavior of continuous dynamical systems through linearization and eigenvalue analysis An equilibrium point is Lyapunov stable (or simply "stable") if solutions starting nearby remain nearby for all future time. Formally: Think of it as a "containment" property: nearby trajectories stay nearby, but don't necessarily approach the equilibrium. Asymptotic stability is a stronger condition: not only do nearby solutions stay nearby, but they actually converge to the equilibrium as time goes to infinity. This is the behavior we typically want in applications: the system naturally returns to its equilibrium state after perturbations. For a system of differential equations, the Jacobian matrix captures how the system changes near an equilibrium point. It is the key to linearization. The Jacobian matrix at a point (x*, y*) is: The eigenvalues of the Jacobian matrix determine the stability of the equilibrium. For a 2x2 matrix, we find them by solving: The trace and determinant of J provide useful shortcuts: 5. Classification: Nodes, Saddles, Spirals, Centers The eigenvalues determine the geometric character of trajectories near the equilibrium: 6. The Trace-Determinant Plane We can visualize all possible equilibrium types in a single diagram using the trace (T) and determinant (D) of the Jacobian: D Saddle point (eigenvalues have opposite signs) D > 0, T Stable (both eigenvalues have negative real parts)

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