Linear Differential Systems Overview

Overview

This lecture introduces solving systems of first-order linear differential equations using matrix algebra, explores solution strategies for homogeneous and nonhomogeneous systems, and discusses stability analysis via phase plane methods and linearization.

Matrix Formulation of Systems

• A system of linear first-order ODEs can be written as x' = Ax, where x is a vector and A is a matrix of coefficients.
• The matrix formulation enables use of linear algebra techniques, like eigenvalues and eigenvectors.

Matrix Exponential Solutions

• The general solution for x' = Ax involves the matrix exponential: x(t) = exp(At)x(0).
• The matrix exponential exp(At) can be computed using the Taylor series and diagonalization, if A is diagonalizable.
• For diagonalizable A, exp(At) = U exp(Dt) U⁻¹, with U of eigenvectors and D diagonal of eigenvalues.

Eigenvalues and Solution Cases

• Solutions are of the form e^(λt)u, with λ eigenvalues and u eigenvectors of A.
• Three cases occur: real distinct eigenvalues, complex eigenvalues (use Euler’s formula), and real repeated defective eigenvalues.
• Defective eigenvalues require finding an extra vector v and using a t-multiplied ansatz.

Nonhomogeneous Systems

• Nonhomogeneous systems take the form x' = Ax + f(t).
• The general solution combines the complementary function (from the homogeneous part) and a particular integral.
• The method of undetermined coefficients adapts trial solutions to vector coefficients.
• Variation of parameters uses the formula: particular = X ∫(X⁻¹ f(t)) dt, where X collects independent solutions.

Stability and the Phase Plane

• The phase plane visualizes trajectories of systems with two variables, showing system evolution over time.
• Nullclines are curves where dx/dt or dy/dt = 0; equilibrium (fixed) points occur where nullclines intersect.
• In homogeneous linear systems, the only equilibrium is at the origin.
• Stability is determined by eigenvalues: negative values = stable, positive = unstable.
• The Poincaré diagram uses matrix trace and determinant to classify system behavior.

Linearization of Nonlinear Systems

• Nonlinear systems can be approximated near a point by linearization, using the Jacobian matrix.
• The Jacobian matrix contains partial derivatives of the system and evaluates system stability locally via eigenvalues.

Key Terms & Definitions

• Matrix Exponential (exp(At)) — Generalization of the exponential function to matrices, used to solve linear systems.
• Eigenvalue (λ) — Scalar for which Av = λv for some nonzero vector v.
• Eigenvector (u or v) — Nonzero vector unchanged in direction by matrix A, scaled by eigenvalue λ.
• Defective Eigenvalue — Eigenvalue without enough independent eigenvectors for diagonalization.
• Phase Plane — Graphical representation of trajectories of a dynamical system in state-space.
• Nullcline — Curve where one component of the system's derivative is zero.
• Equilibrium Point — State where all derivatives are zero; system remains at rest.
• Jacobian Matrix — Matrix of first partial derivatives of a vector function; used for linearization.

Action Items / Next Steps

• Practice solving a system of first-order ODEs using the matrix exponential method.
• Try finding eigenvalues and eigenvectors for 2x2 matrices.
• Attempt linearization of a given nonlinear system using the Jacobian matrix.
• Review phase plane plots and sketch nullclines for simple systems.