# 12.1 Introduction

"As far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality."

-Albert Einstein

The interplay between the abstract world of mathematics and the real world is not as straightforward as it might seem at first. While it is true that mathematics can be used to make sense of and make testable predictions about certain real-life situations, such as solar eclipses, there are many types of natural phenomena, such as the turbulence of fluids in motion, for which current mathematical models are inadequate. Situations such as turbulence represent the frontier of how mathematics can be used to help us understand reality. Coming to an understanding of turbulence is challenging because of the complexity and dynamism of moving fluids. Turbulence involves the combined behavior of trillions of particles of fluid, each of which is subject to many types of forces and interactions. While mathematics can be used to describe the behavior of a single particle relatively comprehensively, the behavior of a group of associated particles is well understood only under certain, sometimes contrived, conditions.

How can we make progress in understanding large, complex, dynamic systems? It helps to start with certain special cases that lend themselves more readily than others to analysis and quantification with the currently available mathematical tools. An understanding of the behavior of a system in these special cases can then provide hints regarding the behavior of the system in more general situations. This is a common strategy in applied mathematics: First, find intriguing special cases that lend themselves readily to study and explanation, then explore how the results can be generalized. Spontaneous synchronization is one such special case of complicated dynamic phenomena. Understanding the mathematics of how, and under what circumstances, entities can come into synchronization with one another provides a starting point for exploring the vast world of nonlinear dynamics.

Our world is filled with all sorts of phenomena that amaze us with their regularity and baffle us with their complexity. For example, how is it that a school of fish can, seemingly simultaneously, all turn on a dime at a mere hint of a nearby predator? How is it that very large groups of East Asian fireflies, and some other varieties as well, when left to their own devices, spontaneously synchronize their flashes? How do the individual cells that make up your heart contract in a coordinated rhythmic fashion to keep your blood flowing? Even a system as simple and seemingly unrelated as an inanimate pair of grandfather clocks can exhibit a kind of synchronous behavior. It is clear that synchronization is a phenomenon that can be found in many different contexts.

The art of mathematical modeling involves identifying a few simple and quantifiable assumptions about a given system (or systems) of study that actually give rise to a good approximation of the phenomenon of interest. Mathematically capturing the complex, dynamic phenomena of the real world is a gargantuan task and is an area in which there is much opportunity for the advancement of our understanding. The study of synchronization represents one of the outposts on the frontier of this vast, unexplored territory.

In this chapter, we will begin by looking at some examples of natural phenomena that exhibit fascinating coordinated and synchronous behavior. Then we will learn a bit about the available mathematical tools that are useful in our quest to understand these phenomena, namely differential equations and calculus, the mathematics of change. From there we will investigate how one particular mathematical model of a system of coupled oscillators can be used to help us understand complex coordinated behavior. We will then be prepared to take a more in-depth look at a couple of examples from the realms of biology and physics to see how the study of synchronization is an example of using mathematics to describe the real world.