Dynamical systems, and more generally, complex
systems, is an approach underpinning many techniques
and tools for analyzing, modeling, and understanding the
complex patterns of measurable phenomena in physics,
biology, psychology, and sociology, among others. In
sport psychology, this approach provides tools for investigating
the dynamics of behavior resulting from interactions
among the nervous system, the body, and its
surrounding environment (including other individuals)
during a sport-related task (Davids et al., 2014). When
applied properly, these tools are more sensitive than traditional
statistical methods, and allow the formulation of
problems differently. While sport psychologists do not
require any particular theoretical commitment for utilizing
these tools, an understanding of the dynamical systems
approach is valuable for interpreting the data
(Gernigon, Vallacher, Nowak, & Conroy, 2015).
Furthermore, as dynamical systems is not a theory of
behavior, a compatible theory of behavior such as
‘Ecological dynamics’ may also be informative for data
interpretation.
A system is a collection of related parts that are conceived
by researchers as a single entity. In a dynamical system model, differential or difference equations
describe a system evolving over time by defining a
state (or configuration space) and a rule for changes in
trajectories starting at various initial conditions.
Dynamic models may be generally classified as linear
or nonlinear, as well as deterministic or stochastic. A
defining feature of dynamic models is that the values of
the state variables are time dependent. In contrast, traditional
statistical models in sport psychology are based
on the general linear model of zeroth order, whereby
state variables are typically time independent.
Furthermore, dynamical models are rich enough to
incorporate systems with different structures, including
continuous and discrete structures such as lattices and
networks. These important properties can be captured
in a system behavior representation such as a phase
(state) or configuration space.
By definition, nonlinear dynamical systems are modeled
as feedback models. When a system establishes a
state based only on the dynamical interactions between
the individual elements within that system, the state is
said to be self-organized. External processes do not
cause self-organization. The patterns that emerge from
a system are different from its components, and cannot
be predicted from the characteristics of those individual
elements. Self-organization can be identified in a system
by capturing certain properties, such as control and
order parameters, multistability, metastability, and hysteresis.
Continuous changes in the control parameter, as
manipulated by the experimenter, should move the system
through critical points (phase transitions or bifurcations),
causing discontinuous changes in the order
parameter, the variable that captures collective organization
of behavior. In a multistable system, there are
multiple stable order parameter values for a single
value of the control parameter. Thus, multistability
implies that the noise in the system is sufficient to
cause it to switch between its multiple stable states or
attractors. There is also a purely dynamical and deterministic
type of metastability, whereby the system
switches states independently of noise (Kelso, 1995).
Hysteresis occurs when the value of a critical point
(when the control parameter causes a bifurcation in the
order parameter) depends on the direction of change of
the control parameter.
Dynamical systems can therefore provide formal
models to identify low-dimensional descriptors, or order
parameters. Furthermore, this approach describes the
nonlinear dependence of order parameters on control
parameters, as well as their evolution over time.
