Stability of Dynamical Systems: Linear Approach

Abstract

Our understanding of the stability of a particular operating mode of a dynamical system is formed intuitively as we build up our experience and understanding of everyday life and nature. The first steps of a small child give him or her very real representations of the stability of walking, although these representations may not yet enter consciousness. Looking at the famous painting entitled Young Acrobat on a Ball by P. Picasso, we have a distinct feeling that the girl’s equilibrium is not quite stable. As adults, we can already discuss the stability of a ship on a stormy sea, the stability of economic trends in relation to the actions of managers and politicians, the stability of our nervous system with regard to stressful perturbation, etc. In each case, we talk about different properties that are specific to the considered systems. However, if we think about it carefully, we can find something in common, inherent in any system. The common feature is that, when we talk about stability, we understand the way the dynamical system reacts to a small perturbation of its state. If arbitrarily small changes in the system state begin to grow in time, the system is unstable. Otherwise, small perturbations decay with time and the system is stable.