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.

In this chapter, we discuss dynamical system approaches for cellular differentiation. We explain how intracellular reaction dynamics can give rise to various attractors using a simple discrete-time and discrete-state reaction model known as a Boolean network. Subsequently, we outline the behavior of a simple stochastic differentiation model of stem cells, where the scaling law discovered therein aligns well with that observed in the distribution of clonal cell populations generated by epidermal stem cells. To integate both approaches, we introduce a theory wherein cell–cell interactions induce transitions between attractors, and stability at the cell-population level emerges through the regulation of these dynamic transitions. Such a circular relationship satisfies the consistency between the cell and the cell population. We expound on three types of differentiation processes, that by Turing instability, transition from an oscillatory state (limit-cycle) to a fixed point, and retaining oscillatory expression dynamics. Additionally, we analyze stability at the cell population level through the regulation of differentiation ratios and the differentiation dynamics of stem cells. Finally, we engage in a discussion of unresolved issues in the field.