## INTRODUCTION

A dynamical system is simply one that changes over time (see Box 1). It is apparent that many of the processes cognitive scientists study—development, learning, the spread of activation in a semantic network, changing patterns of cortical activity, motor behavior, and so on—are intrinsically dynamical. Dynamical systems theory supplies new tools and concepts for understanding these and other cognitive phenomena. The theory offers a full complement of analytical methods (e.g., nonlinear time series analyses) and modeling strategies (e.g., differential equations). Most importantly, it motivates novel theoretical and empirical questions: One may explore the relative stability of a cognitive activity, test for meta- or multi-stability, or test for empirical patterns consistent with scaling behavior, emergence, and self-organization.

BOX 1

**DYNAMICAL SYSTEMS**

Dynamical systems are systems whose *state* evolves over time according to a rule. These rules are often written as differential or difference equations. The evolution of the state is a *trajectory* in *state space*, whose coordinates are the variables that fully characterize the system. Trajectories are drawn toward *attractors*, subsets of phase space that are stable solutions to the system equation(s). *Repellers* are unstable solutions, and trajectories are driven away from them. A *bifurcation* is a change in the number or type of solutions, for example the appearance (or disappearance) of an attractor, as some parameter is varied.

Oscillators are one class of dynamical system. Dynamical models of oscillators vary in complexity, from simple, idealized, linear, harmonic motion to more realistic, nonlinear, *limit-cycles*— stable oscillations that result from a balance of energy lost (e.g, to friction) and energy injected into the system. Two variables comprise the state space for a limit cycle—the position and velocity of the oscillating body. A limit-cycle attractor is an elliptical orbit in this two-dimensional space (see Figure 1); if the initial position and velocity are not on the attractor, the system will evolve toward it, or if a perturbation temporarily bumps the system from the attractor, it will quickly ‘relax’ back onto it. These properties reflect the stability of limit cycles, and contrast with the behavior of *chaotic* dynamical systems, which exhibit different long-term behaviors when initial conditions change and diverge exponentially over time from the previous trajectory when perturbed.

These kinds of questions complement those posed by mainstream cognitive science—which traditionally has emphasized static properties of mind such as symbolic representations or structural mechanisms of information processing—by focusing explicitly on change1–9 (see, e.g., Elman10 for a dynamical interpretation of connectionist models). Much work in cognitive dynamics was inspired by parallel efforts to understand the dynamics of perception-action.11–15 Early and significant progress emerged in the domain of motor coordination, where the behavioral phenomena were more obviously dynamical and high-resolution measurement technologies were more readily available.16 The dynamical account of interlimb rhythmic coordination12,14 is an example of an early success.

Research in cognitive dynamics is often allied with some version of embodiment.17–21 Dynamical systems were sometimes framed as replacements for computational-representational accounts,8,19 as distinct but complementary,22 or even as consistent with information processing given the implicit computations performed by dynamical systems.23–25 Whether and how dynamical systems accounts might be reconciled with traditional accounts, they provide a fresh perspective on many foundational problems in cognitive science,1 including perception-action,12,14,26 memory,27,28 word recognition,29–32 decision making,33–35 learning,36–38 problem solving,39 and language.10,40

Progress in applying models, analyses, and the theoretical concepts of dynamical systems to cognition is accelerating and these approaches are now broadly represented in cognitive science. Naturally, the discipline entertains debates about the merits of the perspective.8,41–46 Rather than rehash those debates, this article seeks to synthesize and summarize recent BOX 1 progress and current themes in dynamical cognitive science.