Phase space of a discrete dynamical system

Question

Suppose a dynamical system of one variable $x$ with discrete time-steps. I've seen in some papers a type of graph in which $x(n+1)$ is plotted versus $x(n)$. My questions are : 1/ Can this be considered as the phase portrait of the system ? 2/ Does this method has a specific name ? 3/ Has there been any studies with regard to the topology of this space ?

Thanks for your help!

Answer 1

The phase space dynamics of the discrete dynamical system is just what you describe--- x(n+1) as a function of x(n). The phase space itself is the range of values of the x(n), whatever space they might live on, while the dynamics is the function that specifies the evolution in one step in time.

The connection with mechanical phase space is provided by a Poincare section. The Poincare section describes a dynamical continuous system by its intersections with a given surface in the full phase space. For a 1d motion, you can consider the half-line x=0,p>0, or in canonical action-angle coordinates $\theta$ fixed, J arbitrary. When you have a separable integrable motion, you take any one of the $\theta$ variables and define a surface by setting it to zero. Then the motion will intersect this surface once every period.

In mechanical phase space, the phase-space volume is conserved, but this is not so for maps. The condition of transversal intersection means that the map from the Poincare surface to itself can get The topological properties of maps on

Topological properties

The properties of maps on spaces are as complicated as you like. The question is then which topological properties are you interested in?

The simplest topological theorems on maps is the Brouwer fixed point theorem, which can be restated as follows:

• Link the points x and f(x) by a path. If you draw a contractible sphere, and you find that as you go around the boundary, this x-f(x) map has a nonzero winding, then there is a fixed point inside this sphere.

The winding of a sphere around another sphere is the index of the map--- it is how many times the sphere covers the other sphere in the map. The Brouwer theorem is classical.

Another classical theorem of this sort is Sharkovskii's theorem:

• There is a linear order on periods of periodic cycles in 1d maps, such that each periodic orbit of length l implies that there is a periodic orbit of length l' whenever l' is greater than l.

Some other results are given by symbolic dynamics, the coarse grained position as a function of time. The notions of the entropy of a dynamical system is related to this. These results are not really topological in character, but they are general, and give qualitative insight, so they are similar.

Many further results can be found here, http://elib.tu-darmstadt.de/tocs/35981431.pdf

Answer 2

Yes, given a discrete dynamics q(t-1), q(t), q(t+1) ..., one can view to Poincare plot q(t+1) vs q(t) as a representation of the phase space dynamics of the system. However, considering the variable q as the generalized coordinate of the discrete system, we would get closer to a phase space description if we would define the generalized momentum p(t) := q(t) - q(t-1), and plot the 'momenta' p(t) versus the 'positions' q(t).

An example would be the reversible dynamics based on q(t+1) - 3 q(t) + q(t-1) = 0. Using the momenta as defined above, this dynamics can be re-written in phase-space form:

q(t+1) = 2 q(t) + p(t)
p(t+1) = q(t) + p(t)

in which we recognize Arnold's cat map. This is an area preserving map with chaotic properties.

Usually maps like these are studied by applying periodic boundary conditions. in that case the phase space has the topology of a torus. But one should not assign any specific meaning to that (convenient) choice. Not sure what else to say about the topology of the phase space of discrete dynamical systems. In principle, you can define it the way you want it to be.