5 Constant-Coefficient Systems

When the coefficient matrix is constant, the general theory of the previous section becomes completely explicit. The fundamental matrix is a matrix exponential, its entries are combinations of exponentials, polynomials and trigonometric functions determined by the eigenvalues and the Jordan structure of \(A\), and the long-run behaviour of every solution can be read off from the eigenvalues. In the plane, two numbers, the trace and the determinant, classify every system. This is the most important special case in the course, because near an equilibrium every smooth system looks like a constant-coefficient linear one.

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