Introduction to Linear Systems of Differential Equations

The theory of linear systems of differential equations is one of the cornerstones of the whole theory of differential equations. At its root is the concept of the Lyapunov characteristic exponent.

In this book, Adrianova presents introductory material and further detailed discussions of Lyapunov exponents. She also discusses the structure of the space of solutions of linear systems. Classes of linear systems examined are from the narrowest to widest: 1) autonomous, 2) periodic, 3) reducible to autonomous, 4) nearly reducible to autonomous, 5) regular.

In addition, Adrianova considers the following:

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stability of linear systems and the influence of perturbations of the coefficients on the stability
the criteria of uniform stability and of uniform asymptotic stability in terms of properties of the solutions
several estimates of the growth rate of solutions of a linear system in terms of its coefficients

How perturbations of the coefficients change all the elements of the spectrum of the system is definitely the most complicated and involved problem in the whole theory of linear systems. Introduction to Linear Systems of Differential Equations presents the proof of the necessary and sufficient conditions for stability of the exponents for the simplest case of a two-dimensional diagonal system.