Linearization

Mathematical Modeling · Axiom Academy

Unit 3: Continuous Models - Mathematical Modeling Nonlinear differential equations are notoriously difficult to solve analytically. Consider a general two-dimensional system: The functions f and g could be arbitrarily complex. However, near an equilibrium point where the system "wants to stay," the behavior simplifies dramatically. 2. Taylor Expansion for Functions of Two Variables The mathematical foundation of linearization is the multivariable Taylor series. For a function f(x, y) expanded around a point (x*, y*): The key terms are the partial derivatives evaluated at the equilibrium point. Higher-order terms (H.O.T.) become negligible when we're close to (x*, y*). 3. The Linearized System and the Jacobian For a system at equilibrium (x*, y*) where f(x*, y*) = 0 and g(x*, y*) = 0, the linearized dynamics are: where J is the Jacobian matrix , the matrix of all first partial derivatives: 4. Eigenvalues Determine Stability The eigenvalues of the Jacobian matrix tell us everything about local stability. For a 2x2 matrix, we find eigenvalues by solving: This gives us a quadratic equation. The eigenvalues determine the type of equilibrium: 5. The Hartman-Grobman Theorem When can we trust the linearization? The Hartman-Grobman theorem gives us a rigorous answer: If the equilibrium point x* is hyperbolic (all eigenvalues of J have nonzero real parts), then the nonlinear system is topologically equivalent to its linearization near x*.

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