Mathematical Preliminaries

This chapter establishes the mathematical framework necessary for the analysis and characterization of bioprocess dynamics. It explains the fundamental concepts and tools from the theory of the stability of dynamical systems and provides an overview of the non‐smooth dynamical systems. Nonlinear dynamical system theory which includes specific techniques and concepts such as chaos, dissipative structures, bifurcation, catastrophe theory, and fractals shows that extreme complexity can also arise due to the nonlinear dynamics and couplings of relatively simple systems represented by relatively small equation systems. The main purpose of bifurcation analysis is to characterize changes in the qualitative behavior of a mathematical model defined by ordinary differential equation's and/or partial differential equations by varying key parameters. The chapter deals with the mathematical approach to describe the evolution in time of the bioprocess under consideration. It investigates the concept of Lyapunov stability and defines the notions: stable, asymptotically stable, equilibrium point.