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Symbolic and connectionist approaches to mobile robotics have not been an un-
qualified success (Brooks 2009 ). The computational problem is the navigation of
a dynamic, uncertain environment. An incredible amount of initial data would be
needed in the closed system approach in order to account for all the possible inputs
the robot might receive. In contrast, the open, dynamic framework has proven much
more fruitful; the robot program is open, and modelled more closely on, say, how
an ant might move through a forest. The similarity between the improvisational en-
vironment which will be very dynamic and uncertain, and the passage of an ant, or
mobile robot, through an uncharted environment leads us to expect that the dynam-
ical framework will be advantageous to Live Algorithm research too.
In a dynamical system, a state x evolves according to the application of a
rule, x t + 1 = f(x t ,α) where α stands for any rule parameterisation. The sequence
x t ,x t 1 ,... defines a trajectory in the space H of possible states. A closed dy-
namical system is one whose evolution depends only on a fixed parameter rule and
on the initial state. These dynamical systems are non-interactive because any pa-
rameters are constant. The dynamical systems framework is quite comprehensive,
encompassing ordinary differential equations, iterated maps, finite state machines,
cellular automata and recurrent time neural networks.
Fully specified dynamical systems have a rich and well studied set of behaviours
(Kaplan and Glass 1995 is an introductory text; Beer 2000 provides a very concise
summary). In the long term state trajectories end on a limit set, which might be a
single point or a limit cycle in which the state follows a closed loop. Stable limit sets,
or attractors, have the property that nearby trajectories are drawn towards the limit
set; states that are perturbed from the limit set will return. The set of all converging
points is known as the basin of attraction of the attractor. In contrast, trajectories
near to unstable limit sets will diverge away from the set. Infinite attracting sets with
fractal structure are termed strange; trajectories that are drawn to a strange attractor
will exhibit chaos. The global structure of a dynamical system consists of all limit
sets and their basins of attraction and is known as a phase portrait. Phase portraits
of families of dynamical systems differing only in the values of their parameters α ,
will not in general be identical.
An open dynamical system has time dependent parameters and therefore many
phase portraits. Since smooth variation of parameters can yield topological change
at bifurcation points (a stable equilibrium point can bifurcate into two or more limit
points, or even into a limit cycle), the global properties of open dynamical systems
are highly context dependent and system behaviour can be very rich. In a live al-
gorithmic setting, the open dynamical system parameters derive from the analysis
parameters. If H is chosen to map directly to the control space of Q , system state
can be directly interpreted as a set of synthesiser parameters. Inputs p could be
mapped to attractors, with the advantage that trajectories moving close to p will re-
semble Ψ in (participation). However x may not lie in the basin of attraction of p and
the trajectory might diverge form p , potentially giving rise to novelty and leader-
ship. Small changes in input might lead to bifurcations in the phase portrait, sending
a trajectory into a distant region of H , giving unexpected outputs. The ability of an
open dynamical system to adapt to an unknown input marks it out as a candidate for
autonomy.
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