Classification of Critical Points

Differential Equations · Axiom Academy

LESSON Classification of Critical Points How the eigenvalues of A decide the shape of every trajectory near the origin: nodes, saddles, spirals, and centers. 1. The Big Picture: Read A Without Solving Consider where A is a matrix. Its eigenvalues come from — but you don't even need to solve for them individually. Two numbers you can read straight off A , the trace and determinant , place the system on a map that names its type instantly. → real eigenvalues of opposite sign → saddle and → real eigenvalues, same sign → node and → complex conjugate eigenvalues → spiral and → purely imaginary eigenvalues → center Eigenvalues from trace and determinant 2. Nodes: Real Eigenvalues, Same Sign When are real and share a sign , every trajectory approaches or leaves the origin along the straight eigenvector directions — a node . As , trajectories become tangent to the eigenvector of the eigenvalue with smaller magnitude (the "slow" direction dominates the picture near the origin). 3. Saddle Points: Real Eigenvalues, Opposite Sign When eigenvalues are real with opposite signs ( ), the origin is a saddle point — and a saddle is always unstable , regardless of how negative is. One negative eigenvalue can't outvote one positive eigenvalue: almost every trajectory eventually gets pulled out along the unstable direction. 4. Spirals: Complex Conjugate Eigenvalues

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