An Introduction to Dynamical Systems Theory

Abstract

As we discussed in Chap. 1, the onset of thermoacoustic instability is a nonlinear phenomenon resulting from the coupled interaction between the acoustic field in a combustor and the heat release rate field of the flame. A thermoacoustic system can undergo different dynamical transitions (called bifurcations), due to a change in any system (control) parameter. We observe that the characteristics of such dynamical transitions are governed primarily by the type of the underlying flow field, i.e., laminar or turbulent flows, present in a combustor. As a result, we notice the occurrence of different dynamical states during the onset of thermoacoustic instability in laminar and turbulent thermoacoustic systems. Furthermore, tools based on linear time series analysis are not sufficient to reveal many features exhibited by nonlinear thermoacoustic systems. In order to overcome the limitations of linear approaches in analyzing the data from experiments or numerical simulations, we need different methodologies based on dynamical systems theory to accurately characterize the nonlinear behaviors of thermoacoustic systems. In this chapter, we will introduce the basics of dynamical systems theory and cover different topics such as stability analysis, bifurcations, attractors, nonlinear measures, phase space reconstruction, Poincaré section, and recurrence plot. In the subsequent chapters, we will explore the application of these methodologies to perform time series analysis of the data obtained from experiments and numerical simulations performed on both laminar and turbulent thermoacoustic systems. Dynamical SystemThe properties of most natural and man-made systems vary with time. Some examples include the displacement of a swinging pendulum, the growth of a particular species in forest, the temperature of the earth's surface in a day, heartbeats, the local velocity of a turbulent flow, etc. A system whose behavior changes as