In this chapter, we present the microscopic (Langevin equation) and macroscopic (Fokker–Planck equation) descriptions of Brownian motion and confirm their consistency. Furthermore, we provide a detailed introduction to the Poisson process, which forms the foundation of chemical reactions. Subsequently, we introduce the chemical Langevin equation and its corresponding Fokker–Planck equation, which are utilized in modeling molecular number fluctuations in chemical reactions. We also explain stochastic differential equations with both the Ito and Stratonovich types of integrals. Exploring mechanisms arising from the presence of noise, we discuss noise-induced transitions and attractor selection and adaptation in dynamical systems, elucidating their functional significance in cells. Finally, as an advanced topic, we introduce adiabatic elimination in stochastic systems.

This chapter serves as an intuitive introduction to dynamical systems within the realm of biological systems, through visual representations of state space dynamics. Biological examples and experimental realizations are described to demonstrate how dynamical systems concepts are applicable in solving fundamental problems in cell biology. Differential equations are taken as typical of dynamical systems, and we explain topics such as nullcline and fixed points, linear stability analysis, and attractors, elucidating their significance using systems such as gene toggle switches. The introduction of limit cycles and the Poincaré–Bendixson theorem in two-dimensional systems is followed by examples such as the Brusselator and the repressilator system. Furthermore, we explore the basin structure in multi-attractor systems and provide detailed explanations using toggle switch systems to illustrate time-scale separation between variables and adiabatic elimination of variables. Several instances of co-dimension 1 bifurcations commonly observed in biological systems are presented, with a discussion of their biological significance in processes like cell differentiation. Finally, chaos theory is introduced.