Complex Neuro-Cognitive Systems - Foundations on Natural and Artificial Computation - page 4

the set of all its possible states, s , determined by the values of all the variables
that describe the system at a particular moment in time. If s is described by
m variables, it can be represented by a point in m -dimensional space, s R m .

The system dynamics is given by the set of equations or rules that control the
system evolution over time. In many cases the dynamics consists of a system
of m coupled differential equations, one for each system variable. Most natural
systems are continuous systems and therefore s is a function of time, s ( t ). As
the system evolves in time its states trace a trajectory in the state space.
An attractor in state space may be defined as a state ( point attractor) or
set of states, toward which the system settles (relaxes) over time. Besides point
attractors, three more attractor types can occur: (i) limit cycles; (ii) torus at-
tractors; (iii) chaotic attractors. A limit cycle is a closed trajectory (an orbit )in
state space that the system performs cyclically; when a system evolves towards
a periodic attractor, it will oscillate endless through the same sequence of states
(unless perturbed). A torus attractor has a 'donut like' shape, and corresponds
to quasi periodic dynamics. A chaotic attractor is a non-repeating orbit in state
space, i.e. the system dynamics, although deterministic, will never repeat the
same state; it is called deterministic chaos .
Several measures are used to characterize the properties of attractors, and thus
of the corresponding dynamics more exactly. Correlation dimension is a measure
of the complexity of the deterministic dynamics. A point attractor has dimension
zero, a limit cycle dimension one, a torus has an integer dimension corresponding
to the number of superimposed periodic oscillations, and a chaotic attractor has
a fractal dimension. Lyapunov exponents indicate the exponential divergence
(positive exponents) or convergence (negative exponents) of nearby trajectories
on the attractor, thus giving information about the systems dependence on initial
conditions. A positive Lyapunov exponent is a strong indicator of chaos.

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4.2 An Example

Complex behaviour ( dynamical complexity ) can arise even in simple systems
with low compositional complexity. The damped, periodically driven non-linear

Fig. 1. The damped, periodically driven non-linear pendulum. Displayed is the chaotic
attractor in the ( θ, ω )-phase space that occurs for large values of the driving force g .