Differential Equations, Dynamical Systems, and an Introduction to Chaos
by Morris W. Hirsch, Stephen Smale, Robert L. Devaney · Academic Press (Elsevier)
Upper-undergraduate textbook (2nd edition, 2004) on linear and nonlinear systems of ODEs: canonical forms, matrix exponentials, phase portraits, equilibria and stability, Lyapunov functions, limit cycles, bifurcations, and the Lorenz attractor. Readers learn to analyze the qualitative behavior of dynamical systems rigorously.
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More resources on Systems of ODEs
Wolfram MathWorld
MathWorld is an online mathematics encyclopedia from Wolfram Research offering detailed, browsable articles on topics across the math spectrum, including algebra, geometry, calculus, and number theory. Each entry includes definitions, theorems, formulas, diagrams, worked examples, and links to further reading.
Introduction to Differential Equations
Learn differential equations: explore methods for solving, modeling, and analyzing them. Taught by the Khan Academy Team.
Differential Equations and Linear Algebra
Textbook by Gilbert Strang (Wellesley-Cambridge Press, 2014) that teaches ODEs and linear algebra together. Its eigenvalue chapter covers du/dt = Au, the matrix exponential, and stability. Paired with free MIT video lectures for each section. The intermediate-level counterpart to the more advanced Hirsch/Smale/Devaney.
How (and why) to raise e to the power of a matrix | DE6
A 27-minute animated lecture, chapter 6 of 3Blue1Brown's differential equations series. It defines the matrix exponential through its Taylor series and shows why exp(Mt) solves linear systems x' = Mx, visualized as flow in state space. Best geometric intuition for the topic's central object, the matrix exponential, and its link to state-space flow.
Gilbert Strang: Eigenvalues and Eigenvectors (Differential Equations and Linear Algebra video lectures)
Eight short MIT lectures by Gilbert Strang that apply eigenvalues to differential equations. Topics are diagonalization, matrix powers and Markov matrices, solving linear systems du/dt = Au, the matrix exponential, similar and symmetric matrices, and second-order systems.
Notes on Diffy Qs, Chapter 3: Systems of ODEs
Free open textbook chapter by Jiří Lebl. Covers matrices and linear systems, the eigenvalue method, 2D vector fields, second-order systems and mass-spring applications, repeated eigenvalues, matrix exponentials, and nonhomogeneous systems. Includes exercises with answers. More rigorous than Paul's Notes on the eigenvalue method and matrix exponential while staying readable.