7. Finite-Dimensional Dynamical Systems: Specialized Results

For autonomous and periodic continuous-time dynamical systems we show that stability and asymptotic stability imply uniform stability and uniform asymptotic stability, respectively. For such systems we also present specialized Converse Theorems. For continuous-time and discrete-time dynamical systems determined by semigroups, we present the LaSalle-Krasovskii invariance theory (involving monotonic Lyapunov functions). These results constitute sufficient conditions. For the special case of dynamical systems determined by linear autonomous homogeneous systems of differential equations and difference equations, we present invariance results which constitute necessary and sufficient conditions (involving monotonic Lyapunov functions). For general continuous-time and discrete-time dynamical systems we present invariance stability and boundedness results involving non-monotonic Lyapunov functions. We present results which make it possible to estimate the domain of attraction of an asymptotically stable equilibrium for dynamical systems determined by differential equations. We present stability results for dynamical systems determined by linear homogeneous differential equations and difference equations. Some of these results require knowledge of the state transition matrix while other results involve Lyapunov matrix equations. We present stability results for dynamical systems determined by linear periodic differential equations (the Floquet Theory). Also, we study in detail the stability properties of dynamical systems determined by second-order differential equations. We investigate various aspects of the qualitative properties of perturbed linear systems, including Lyapunov’s First Method (also called Lyapunov’s Indirect Method) for continuous-time and discrete-time dynamical systems; existence of stable and unstable manifolds in continuous-time linear perturbed systems; and stability properties of periodic solutions in continuous-time perturbed systems. We present a stability and boundedness comparison theory for finite-dimensional continuous-time and discrete-time dynamical systems.