Talk:State space
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From Reviewer A:
I think it would be nice to start out the article with some discussion of modelling--how is a phase space selected. For example: what are the varables ( \dot{x} and \dot{y} as opposed to \dotdot{x})? Is the phase space a manifold (not necessarily, some of the variables might be discrete in a model, like the coin toss or a cellular automata)?
As writing the article jumps right into the phase plane abstractly. It doesn't say anything about dimension or number of degrees-of-freedom (for a Hamiltonian system). Moreover it assumes that the only appropriate dynamics is an ode, when we cold have maps or pdes and these certainly have phase spaces--at lest I think so! Of course my "dynamical systems" article discusses these issues in its "state space" section. So it is perhaps not necessary here.
So: an alternative is to retitle the article "phase plane" which is usually used in the context of 2D odes.
Then the section on Higher dimensions, together with stuff I've suggested above could be in a more general article on phase space.
Otherwise, I think the article is very nice. You might add references to "node" etc for the "equilibrium" article that is being written. Also to "Hamiltonian systems" for the pendulum example.
--Jim Meiss
Review of the new version (February 18, 2008)
I think the article is now fine. I agree with some of the points of Reviewer B about the "PPOV"--and some of this should be taken into account in my main "dynamical systems" article. I plan to do this. (My only concern there is not to make things too abstract, since the article should be understandable by a broad readership).
--Jim Meiss
Answer to reviewer's comments to authors' reply the reviewer criticism of Feb.08, 2008
I removed those points where we (authors and the reviewer) agree. Also, I agree with Reviewer A (James Meiss) that we do not want this article to be too abstract. It should be understandable to physicists and engineers.
- Abstract definition
- The abstract definition of a dynamical systems (provided in reviewer B comments) is more suitable for the main article dynamical systems. Though, it is implicitely defines the state space, and hence might be useful in this article too. I have included it into the text. However, I changed it to the one that was provided to me (user:Eugene) by my teacher Dr. Sheldon Newhouse. A dynamical system is a homomorphism of an Abelian group to a group of all automorphisms of a space \(X\).. Actually, we can use a semigroup here, as you indicated. The state/phase space \(X\) does not have to be topological at all - any space would suffice. I added a section devoted to "abstract state spaces".
- If one admits a semigroup (which is important), then "automorphisms" should be replaced by "endomorphisms". "Abelian" may be disputable, but indeed in most cases it is Abelian. The requirement that the space is topological (although richer structure is also allowable) is necessary to distinguish Dynamical Systems from Iteration Theory (where there is no structure) and Ergodic Theory (where the space is a probability space), unless of course one wants to include those areas in the Dynamical Systems.
- Done
- The abstract definition of a dynamical systems (provided in reviewer B comments) is more suitable for the main article dynamical systems. Though, it is implicitely defines the state space, and hence might be useful in this article too. I have included it into the text. However, I changed it to the one that was provided to me (user:Eugene) by my teacher Dr. Sheldon Newhouse. A dynamical system is a homomorphism of an Abelian group to a group of all automorphisms of a space \(X\).. Actually, we can use a semigroup here, as you indicated. The state/phase space \(X\) does not have to be topological at all - any space would suffice. I added a section devoted to "abstract state spaces".
- In the second paragraph, in cellular automata the phase space is infinite. There is something that is usually called "the set of states" which is finite, but I would not call it "state space" in the dynamical sense.
- I added "... as in finite-state cellular automata...". Most popular cellular automata are finite, though continuous state automata also possible.
- This is a confusion caused by two meanings of the word "state". For instance, for the Game of Life, each cell can be in two states: alive or dead. However, the state space is not the set consisting of two elements. The state space is the set of all functions from \(\mathbb{Z}^2\) to {alive, dead}, for which only finite number of cells are alive. This space is uncountable.
- References to Game of Life deleted; no confusion in the new version (thanks for pointing this out).
- I added "... as in finite-state cellular automata...". Most popular cellular automata are finite, though continuous state automata also possible.
- PPOV: In the second paragraph, the phase space (= state space) can be any topological space (usually compact). It does not have to be a manifold for mappings.
- I added the phrase "... as usually the case in ...". However, it makes most sence to talk about manifolds when we introduce the dimension of the spate space. It is typically \(\R^n\) in most applications.
- In this way you are excluding in particular symbolic dynamics, which constitutes a sizable portion of Dynamical Systems and is extremely useful not only in abstract, but also in applied problems.
- Could you modify the article's text so that symbolic dynamics is not exluded.
- I added the phrase "... as usually the case in ...". However, it makes most sence to talk about manifolds when we introduce the dimension of the spate space. It is typically \(\R^n\) in most applications.
- PPOV: Phase line, first paragraph: A dynamical system is one-dimensional when the phase space is one-dimensional. This phase space does not have to be the real line, it can be for instance a closed interval, a circle, a tree, a graph, a dendrite...
- All these cases could be mapped into the case \(\R^1\). So, from the PPOV the statement is correct. However, it is often more natural to consider intervals, closed curvers, etc., so I included more explanation into the definition. I do not think graphs (including trees) result in one-dimensional systems, since they have finite or countable number of vertices. Do you mean dendrite as in Julia and Mandelbrot set?
- While an interval is a subset of the real line, a circle is not, and more complicated graphs are also not. Graphs (and trees) by the definition have finitely many vertices. The systems that have them as phase spaces are always considered one-dimensional (because the dimension of those spaces is 1). Dendrites are like trees with infinitely many vertices, but one assumes additionally local connectivity. Indeed, some (but not all!) Julia sets are dendrites.
- I think the confusion here is that a "graph" is treated not as in "graph-theoretic point of view" (as a structure given by the adjecensy matrix), but as a "dendrite" point of view, with one-dimensional segments connective vertices. These structures can still be modeled as mappings of \(R^1\) to itself (the mapping would be peice-wise continuous though). Circles have real line as its covering space. In any case, what would be the most appropriate modification of the definition so that not to confuse the reader?.
- All these cases could be mapped into the case \(\R^1\). So, from the PPOV the statement is correct. However, it is often more natural to consider intervals, closed curvers, etc., so I included more explanation into the definition. I do not think graphs (including trees) result in one-dimensional systems, since they have finite or countable number of vertices. Do you mean dendrite as in Julia and Mandelbrot set?
- PPOV: Phase plane: Again there are many things that are not mentioned. The phase space, if it is two dimensional, can be a subset of the plane, a surface with or without a boundary, etc. The system does not have to be a flow.
- I added a paragraph mentioning the mappings.
- However, there are many systems given by ODE's on two-dimensional manifolds that are not the whole plane (again, think of a pendulum).
- Cylinder has the plane as its covering space, so any flow on the cylinder could be modeled by a flow on the 2-d plane.
- I added a paragraph mentioning the mappings.
- Phase plane, third paragraph: "completely characterizes the solution trajectories" - it characterizes them as subsets of the plane, but quite often trajectories are considered as functions.
- True; please, fix the text as you feel appropriate.
- I would say "completely characterizes the solution trajectories (considered as subsets of the phase space)."
- Done. Thank you.
- True; please, fix the text as you feel appropriate.
- PPOV: An equilibrium point is also called a fixed point of a flow; a periodic solution is also called a periodic orbit.
- It is incorrect to call an equilibrium of a flow a fixed point. Fixed points are for maps, equilibria are for flows. This point is explained in the article equilibrium and will be explained in the article fixed point.
- In "pure" dynamical systems the term "fixed point" is always used for a point whose whole trajectory consists of one point. This is done independently of what group or semigroup is acting. To be on the safe side, I checked two sources that I have on hand: the article of Hasselblatt and Katok in the Handbook of Dynamical Systems and the textbook "Introduction to Dynamical Systems" by Brin and Stuck. Both do the same.
- I make links to both articles Equilibria and fixed points. The articles describe the difference. Actually, the distinction arose, I believe, in the Gorkii school of dynamical systems, and it was maintained by Andronov, Shilnikov, and their students. Though some people use it interchangably, many applied dynamicists distinct the two notions. I try to enforce this throughout the encyclopedia of dynamical systems.
- It is incorrect to call an equilibrium of a flow a fixed point. Fixed points are for maps, equilibria are for flows. This point is explained in the article equilibrium and will be explained in the article fixed point.
- Phase plane, fifth paragraph: There are many notions of stability; it has to be explained which one is used here.
- There is a link to the article stability that explains all the definitions of stability of periodic orbits; there are indeed many definitions, and it would be too cumbersome to list them all here.
- I did not mean listing all of them, but just mentioning which one is used here. What is described a stability of periodic solutions is Lyapunov stability.
- Fixed. Thank you.
- There is a link to the article stability that explains all the definitions of stability of periodic orbits; there are indeed many definitions, and it would be too cumbersome to list them all here.
- PPOV: Higher dimensional systems: Again, other spaces, mappings...
- Mappings mentioned in the new version.
- But again, manifolds are abundant for flows.
- Both, flows and mappings, are mentioned in the current version.
- Mappings mentioned in the new version.
From Reviewer C:
As things stand right now, a dynamical system is defined as "a rule for time evolution on a state space." State space, on the other hand, is defined as "the set of all possible states of a dynamical system." I don't suppose circularity is of any concern for the editors, is it?
--Cristian Cocos