International Encyclopedia of Systems and Cybernetics

The reference space in which all of the possible future states of an object submitted to various parameters are included.

It is also defined as "the set of all positions and velocities compatible with the system's invariances" (M. FARGE, 1992, p.226).

Or, in J. HOLLAND terms: "… for the process X, the set of all possible combinations of values for (the intervening) variables". (Adapted from J. HOLLAND, 1992, p.54)

I. STEWART explains: "Each state is determined by a set of numbers. Thinking of these as coordinates, we obtain a multidimensional space whose points correspond to all possible states" (1989, p.43).

Obviously:

1° – the phase space is an abstract one and

2° – it can be used to describe future possible states of the object or system, of which, of course, one and one only is to materialize at any future instant.

However:

1°- in most cases, it becomes impossible to pinpoint and predict some specific future state at some precise moment (as shown by chaos theory, or topological dynamics)

2°-… which does not implied that all the future states are totally indetermined, since periodic solutions may appear, implying the existence of steady states.

Moreover M. DUBOIS, P. ATTEN and P. BERGÉ state: "… the phase space… must contain the complete information about the dynamics of the considered system. Thus the precisely necessary and sufficient parameters should be selected in order to determine these dynamics. Moreover these parameters must be independent if each one is to contribute its own information. This implies that, if one of them is not determined, a degree of freedom will subsist in the determination of the considered states…

"… the number of these degrees of freedom is of paramount importance in the appearance and for the characteristic of chaotic behaviour" (1987, p.194).

The conservative (or Hamiltonian) systems are those in which the phase space is closely restricted, while still able to contain different trajectories.

An interesting comment by P. BERGÉ, Y. POMEAU and Ch. VIDAL is that: "If we arbitrarily increase the number of degrees of freedom, we can always include a dissipative system within more extended Hamiltonian system, of which it thus becomes a 'part'" (1984, p.23).

This comment reflects in a new way the unavoidable dependence of any system from some meta- or supra-system, which obeys the 2d. Principle of Thermodynamics, but altogether, the growing degrees of freedom obtained from complexity.

D. BOHM and F.D. PEAT describe phases space in relation to a whirl (or vortex): "Imagine an irregular and changing whirlpool that fluctuates in a very complex way, but always remains within a certain region of the river… As the velocity of the river increases, this variation in space may grow. But in addition, there will also be an inward growth of subvortices of ever finer nature. Therefore, a measure of the overall range of variation of the whirlpool should include both of these factors – the inward and the outward growth… As a matter of fact, in classical mechanics, a natural measure of this kind has already been worked out. Its technical name is phases space and its measure is determined by multiplying the range of variation of position and the range of variation in momentum. The former… corresponds roughly to the changes in location of the vortex as it spreads out into the river… (and) the latter… to the extent to which the whirlpool is excited internally so that it breaks into finer and finer vortices" (1987, p.138).

In this way, the phases space becomes partitioned into "Markovian partitions" (I. SINAI, 1992, p.85)

It becomes obvious that incipient chaos conditions result of a growing energizing of the vortex and begins with the emergence of a finer structuration of the phases space.

This is consistent with Ch. LAVILLE's theory of vortices (1950).