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I came across the topic of symbolic dynamics when studying about time series analysis. Since I have not formally taken any course on chaotic dynamics, I have some difficulties in understanding some terms and need help. The document which I am following is available here

Link1 : http://london.ucdavis.edu/~reu/REU10/smith.pdf

Link2: Incompatible implementations of physical symbol systems

Confusion 1: In the document, the definition of symbolic dynamics is that the symbolic dynamics is obtained from the "state-space" partitioning. My understanding of state-space is the plot of variables. In case of Tent Map, this will be plot of $x_{n+1}$ vs. $x_n$. A symbol is assigned if a state which is now a 2 d coordinate expressed as, $\mathbf{x} = [x_{n+1}, x_{n}]$ lies within an interval or not. How is this symbolizing procedure different from the symbolizing using the time series which I wrote above? In Link2, figure 2 shows the symbolic sequence from the time series, but figure 1 describes theoretically and pictorially using state space concept. What is the correct way? Is my example /approach correct?

Confusion 2: Is trajectory another name for time series or is it the curve obtained in phase space? In the document, sometimes it is indicated that the trajectory is a time series whereas sometime they say that is a curve / orbit in phase space.

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About 1

Your example is correct, but the tent map is a 1-D system, and its state space is the interval $[0,1]$. State space is classically the same as the phase space: the space of all possible states of the system. Check this question for more.

The plot of $x_{n+1}$ as a function of $x_n$ is simply a way to visualize the equation that defines the 1-D map and its effect on points. It's not the state state of the tent map.

In the paper you link, the author, as far as I can tell, doesn't specify the equations that generate the results in the figures, so it's hard to comment. But, in general you can consider only one variable to define your symbolic dynamics, and that corresponds to rectangular partitions of the state/phase space.

About 2

Trajectory, or orbit: a sequence of positions, or curve in the phase space, $\vec{x}$ parametrized by $t$. In the case of the tent map, or any usual 1-D map, that is a sequence of points $x_n$ over the interval. It doesn't look like what you'd expect from a "curve", but that's what you get from a time-discrete system (though in more dimensions, these points might eventual build "proper" curves in the phase space).

Time series: that's the phase space coordinates as a function of $t$ or $n$ for a given trajectory.

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  • $\begingroup$ Thank you for your reply. But I still have some questions, could you please clarify? The phase space plot is the plot of the variables. So, for 1 D map, what would be the phase space plot or the phase portrait? In an answer to a question asked stackoverflow.com/questions/30020351/matlab-phase-space-plot ,the responder to the answer mentions that the phase space plot is the plot of one variable with another; for Tent map it will be $x_n$ vs $x_{n+1}$. The phase space plot is the same as the graph of the Tent map. So I am confused what the phase space plot or the phase portrait is. $\endgroup$ – Srishti M Sep 30 '17 at 0:22
  • $\begingroup$ As an example, for Lorenz system, the phase space plot is 3 d showing the lobe-like butterfly structure. So why is it different for TEnt map or 1 D map-- One can create a multidimensional system from Tent map by delay embedding and then plot the variables to obtain the phase space plot. Can you plz explain with an example or some other way what phase space is? $\endgroup$ – Srishti M Sep 30 '17 at 0:36
  • $\begingroup$ Lastly, just to confirm In the article, Erik M. Bollt, Theodore Stanford, Ying-Cheng Lai, Karol Życzkowski, What symbolic dynamics do we get with a misplaced partition?: On the validity of threshold crossings analysis of chaotic time-series, In Physica D: Nonlinear Phenomena, Volume 154, Issues 3–4, 2001, Pages 259-286 the generating partition for Tent map is 12. But I had studies that the generating partition is the critical point. A critical point is where the map cannot be differentiated, it is the turning point. For the Tent map, the turning point is 0.5. $\endgroup$ – Srishti M Sep 30 '17 at 0:37
  • $\begingroup$ So how come 12 is the critical point? $\endgroup$ – Srishti M Sep 30 '17 at 0:37
  • $\begingroup$ @SrishtiM, the answerer of the question you link to is mistaken. The usual definitions are the ones I give in my answer (and you'll notice George Datseris simultaneous answer and mine are to the most part identical). 1) Phase space is the same as state space and that's 1-D for the tent map. The phase space must contain all the variables needed to completely describe the state of the system; for a 1-D system, that's one single number $x$. 2) A phase portrait is a figure of the phase space with one or more trajectories plotted, usually to evince its structure. (to be continued...) $\endgroup$ – stafusa Sep 30 '17 at 1:06
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Confusion 1 answer: A state-space is not connected with plotting in any manner. It is a (sometimes abstracted) mathematical space, as many dimensional as the number of dynamical variables of your system. In the case of the tent map, that dimension is 1, which means that the state-space is one dimensional (i.e. a line). An easy way to think about it is that the phase-space is where your system "lives": it contains all possible states you could find your system in. In the case of the tent map, a one dymensional system, a state is simply a number between 0 and 1. Thus the phase space is the line segment [0,1).

Partitioning the phase space literally means: divide it into boxes. In the case of 1 dimension, those boxes are line segments. In your specific case of symbolic dynamics, the phase-space is partitioned into only 2 segments: [0, 0,5) and [0.5, 1).

In general, partitioning is useful in many things: finding patterns, analyzing chaotic behavior, calculating entropies and/or dimensions of attracting sets. The difference with the symbolic representation and the time-series representation simply falls to the user: "What do I want to do with it?"

Comment: Notice that the plot $x_{n+1}$ versus $x_n$ simply plots the function of the equation of motion (in your case, a "tent").

Confusion 2 answer: Unfortunately the term "time-series" is not strictly defined and can be inderchanged to mean strictly one-dimensional time-series or multi-dimensional. A 1D time-series, is simply a series of numbers versus time. In many dimensions you can also refer to as time-series a group of 1D time-series. The main reason for this confusion is that the plural and singular of this word is the same.

A trajectory is a solution to a dynamical system of equations with some initial condition. In my experience this is equivalent to a multi-dimensional time-series that contains all the 1D time-series of all the variables of the system. A trajectory is also equivalent to a curve in the full-dimensional phase-space, in the case of a continuous dynamical system.

The orbit however is something different, at least in my knowledge: It is the projection of the trajectory in the real space coordinates. For example, if your system's variables were $(x, y, p_x, p_y)$, then the orbit would be $(x,y)$ (again as a function of time). An orbit is a curve in real-space which itself is (most of the time) a proper subset of the phase-space.

Extra: Visualizing 1D phase-spaces (edit): Now, about visualizing a 1D phasespace. This is a bit tricky but it is very possible. For example, you might want to make parts of the line segment thicker and thicker, depending on how many orbit points are in this segment. A similar approach would be to use a colorbar, coloring differently line segments with different properties, e.g. more orbit points.

More extra : Another way to visualize a 1D map is to use it's invariant density (also called invariant measure), which when plotted is a curve \rho(x) versus x, the system's "state" (which is simply a number in 1D).

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  • $\begingroup$ Thank you for your answer. But few things are unclear(1) how to obtain the phase-space plot, also known as the phase portrait for a 1 D map? For example, in Lorenz chaotic system, based on your answer the phase space for Lorenz system would be composed of 3 variables (please correct me if wrong). The plot of $x-y-z$ gives the butterfly like structure and the diagram is known as phase portrait. In the same way, if I want to obtain the phase space plot for Tent Map, it would be a plot of $x_n$ vs. $x_{n+1}$. $\endgroup$ – Srishti M Sep 30 '17 at 0:51
  • $\begingroup$ (2) Another thing unclear is that in cases when the system is not 1 D, can symbolic dynamics be obtained from the time series of one variable as I have illustrated in the question? (3) difference between trajectory and time series is not clear. Could you please explain with an example? Thank you for your clarfications. $\endgroup$ – Srishti M Sep 30 '17 at 0:52
  • $\begingroup$ Just like stafusa and me highlighted, the definition of phase-space as it is in our answers is the ``correct'' one. I am answering what plotting $x_{n+1}$ vs. $x_n$ would do in my answer. Now, about visualizing a 1D phasespace. This is a bit tricky but it is very possible. For example, you might want to make parts of the line segment thicker and thicker, depending on how many orbit points are in this segment. A similar approach would be to use a colorbar, coloring differently line segments with different properties, e.g. more orbit points. $\endgroup$ – George Datseris Oct 1 '17 at 8:44
  • $\begingroup$ Can Takens' embedding and phase phase delay reconstruction of 1 D maps be used to visualizing 1D phase-spaces in delay coordinates? For instance, from an experimental 1 D time series, one can delay embed it and visualize it in phase-space to determine the dynamics. Often, the structure of the phase portrait will be similar to the chaotic system from which the time series was obtained. Can this technique be applied to 1 D maps? $\endgroup$ – Srishti M Oct 1 '17 at 21:13
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    $\begingroup$ @stafusa no worries, we are all here for spreading love and knowledge of physics anyway!!! Very nice idea to mention the invariant density: it is one of the prime ways of visualizing 1D maps, I can't believe I didn't think to mention it to my original answer. I am editing it in now! $\endgroup$ – George Datseris Oct 2 '17 at 22:02

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