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.

What is a system? What is a dynamical system? Systems are characterized by a few central notions: their state and their behavior foremost, and then some derived notions such as reachability and observability. These notions pop up in many fields, so it is important to understand them in nontechnical terms. This chapter therefore introduces what people call a narrative that aims at describing the central ideas. In the remainder of the book, the ideas presented here are made mathematically precise in concrete numerical situations. It turns out that a sharp understanding of just the notion of state suffices to develop most if not the whole mathematical machinery needed to solve the main engineering problems related to systems and their dynamics.

Lorenz discovered that the atmosphere, like any dynamical system with instabilities, has a finite limit of predictability. In this chapter, we briefly review the fundamental concepts of chaotic systems, from which we introduce the concepts of global and local Lyapunov vectors. We then discuss singular and bred vectors, two concepts closely related to Lyapunov vectors. To represent the uncertainties associated with each deterministic forecasting, ensemble forecasting methods are developed. We review early studies of ensemble forecasting and operational ensemble forecasting methods. The growth rate of errors and predictability of the atmosphere in different regions are discussed. We then discuss the role of different earth components in predictability for different time-scale phenomena, highlighting the dominant role of humans in the coupled Earth-Human system. We also give an outlook on the potential application of data assimilation to the coupled Earth-Human systems. While the chaotic nature of the atmosphere reveals the intrinsic difficulty of making accurate long-range predictions, it also indicates the possibility of deploying an initial small control that can grow large enough to alternate the future flow trajectory. Under this context, we introduce the Control Simulation Experiment (CSE), where we insert small perturbations into a chaotic system to let it evolve as we expect, essentially "controlling" the weather.

This book is about the geometry and topology of symplectic manifolds equipped with pairs of complementary Lagrangian foliations. The resulting structure is very rich, intertwining symplectic geometry, the theory of foliations, dynamical systems, and pseudo-Riemannian geometry in interesting ways.

Before describing the contents of the book in detail, we want to discuss a few motivating vignettes. The first two of these are to be kept in mind as motivational background, whereas the third and fourth ones will be taken up again and again later in the book.

In this chapter we present dynamical systems and their probabilistic description. We distinguish between system descriptions with discrete and continuous state-spaces as well as discrete and continuous time. We formulate examples of statistical models including Markov models, Markov jump processes, and stochastic differential equations. In doing so, we describe fundamental equations governing the evolution of the probability of dynamical systems. These equations include the master equation, Langevin equation, and Fokker–Plank equation. We also present sampling methods to simulate realizations of a stochastic dynamical process such as the Gillespie algorithm. We end with case studies relevant to chemistry and physics.

Appearance of rectifiability in miscellaneous topics of mathematics

Previous chapters have all developed in different ways the core idea that cognition is information processing. This chapter looks at a very different approach, using dynamical systems theory's mathematical and conceptual tools to model cognitive skills and abilities. The first section explains how how dynamical systems theory can describe cognitive skills and abilities without using the framework of representation and information processing. The second section examines how dynamical systems theory explains two examples of child development, with particular attention to the time-sensitive nature of the dynamic system theory in these examples.

Outside the immediate statistical applications, two notable implications of the Christoffel-Darboux kernel are sketched: on the effective semialgebraic approximation of nonsmooth functions and on the spectral analysis of Koopman's operator attached to some intricate dynamical systems.

Ergodic theory is concerned with dynamical systems -- collections of points together with a rule governing how the system changes over time. Much of the theory is concerned with the long term behavior of typical points-- how points behave over time, ignoring anomalous behavior from a small number of exceptional points. Computability theory has a family of precise notions of randomness: a point is "algorithmically random'' if no computable test can demonstrate that it is not random. These notions capture something essential about the informal notion of randomness: algorithmically random points are precisely the ones that have typical orbits in computable dynamical systems. For computable dynamical systems with or without assumptions of ergodicity, the measure 0 set of exceptional points for various theorems (such as Poincaré's Recurrence Theorem or the pointwise ergodic theorem) are precisely the Schnorr or Martin-Löf random points identified in algorithmic randomness.

After a brief discussion of invariant measures and entropy, we introduce semidynamical and dynamical systems. We use Koopmanism to show how to obtain semigroups/groups of linear operators on function spaces, when we have a quasi-invariant measure. Applications are given to solution flows of differential equations. In the last part we discuss “dilations” as a mathematical approach to the origins of irreversibility.