In the first chapter the authors consider the homogeneous linear systems (1.1) \(\dot x=A(t)x\) with a continuous matrix function \(A(t)\) bounded on the whole line \(R\), and the corresponding nonhomogeneous system (1.2) \(\dot x=A(t)x+f(t)\) with a bounded vector function \(f(t)\). The chapter is devoted to the connections between the following properties: (a) System (1.1) is exponentially dichotomic, (b) for every bounded \(f\) there exists at least one bounded solution of (1.2), (c) there exists a quadratic form \(V(t,x)=\langle S(t)x,x\rangle\) with a continuously differentiable and bounded matrix function \(S(t)\) such that the derivative of \(V\) with respect to (1.1) is negative definite. These properties are considered over \(R_ +\), \(R_ -\) and \(R\). The main result of the chapter says that if (c) is satisfied over \(R\) and det \(S(t)\neq 0\) for all \(t\in R\), then (a) holds over \(R\).
In Chapter 2 the so-called linear extension of a dynamical system on a torus is studied. This is defined in the following way. Let \(\dot\varphi=a(\varphi)\) define a dynamical system on the \(m\)-dimensional torus. Its linear extension is called the system of differential equations \(\dot\varphi=a(\varphi)\), \(\dot x=A(\varphi)x+f(\varphi)\), where \(A(\varphi)\) and \(f(\varphi)\) are given and continuous matrix and vector functions, respectively, on the \(m\)-dimensional torus. A surface \(x=u(\varphi)\) is said to be an invariant torus of the linear extension if \((d/dt)u(\varphi(t))=A(\varphi (t))u(\varphi(t))+f(\varphi(t))\) holds for every solution \(\varphi(t)\) of the equation \(\dot\varphi=a(\varphi)\). There are given conditions of the existence of an invariant torus, of its exponential dichotomy and stability.
The final chapter is devoted to the decomposition of the linear extension of a dynamical system on a torus.