Chapter 1 Autonomous Linear Differential and Difference Equations An autonomous linear differential equation or difference equation is deter- mined by a fixed matrix. Here linear algebra can directly be used to derive properties of the corresponding solutions. This is due to the fact that these equations can be solved explicitly if one knows the eigenvalues and a basis of eigenvectors (and generalized eigenvectors, if necessary). The key idea is to use the (real) Jordan form of a matrix. The real parts of the eigenvec- tors determine the exponential growth behavior of the solutions, described by the Lyapunov exponents and the corresponding Lyapunov subspaces. In this chapter we recall the necessary concepts and results from linear alge- bra and linear differential and difference equations. Section 1.1 establishes existence and uniqueness of solutions for initial value problems for linear differential equations and shows continuous dependence of the solution on the initial value and the matrix. Section 1.2 recalls the Jordan normal form over the complex numbers and derives the Jordan normal form over the reals. This is used in Section 1.3 to write down explicit formulas for the solutions. Section 1.4 introduces Lyapunov exponents and relates them to eigenvalues. Finally, Section 1.5 presents analogous results for linear difference equations. 1.1. Existence of Solutions This section presents basic results on existence of solutions for linear au- tonomous differential equations and their continuous dependence on the ini- tial value and the matrix. 3
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