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.
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