In this paper, we have described the simplest mathematical features of continuous deterministic Brownian motion, i.e. unlimited walk driven by deterministic chaotic force, and have shown that the complicated patterns of spiral wave meandering observed in numerical experiments may be interpreted as such a motion.
This sort of motion results from two main features of the mathematical problem. The first is the difference between symmetry groups of the dynamical system and particular solutions, so that the factor group, if it is non-compact, gives birth to a noncompact set of congruent solutions and thus creates the possibility of unlimited drift along this set. The other is chaos in the quotient systems, which makes this drift similar to stochastic Brownian motion. We have considered the case of continuous factor group, and predicted two types of deterministic Brownian motion, with or without directed component. Which type occurs in a particular case depends on whether or not the maximal attractor of the semi-reduced system is uniquely ergodic.
Non-symmetric solutions of symmetric systems are ubiquitous, and in all
such cases, noncompact factor group and chaotic dynamics in the
quotient system can lead to deterministic Brownian motion or
deterministic diffusion. The Theorem describes such motion in case
when the factor group is the group of translations of the straight
line, as in [11]. The hypermeander of spiral waves
considered in this paper is a more complicated motion as the symmetry
group involved, 
, is non-commutative, the corresponding
three-dimensional manifold is curved and the Theorem is not applicable
directly. We were able to overcome this difficulty by noting that
can be decomposed onto two Abelian subgroups, those of
translations and rotations, and reduction by the subgroup of
translations yields the required result. Deterministic chaos-driven
walk along more complicated groups presents an interesting mathematical
problem for future study.