Poincaré map

2025-03-01T15:43:42Z

188.192.185.213: add missing index

{{Short description|Type of map used in mathematics, particularly dynamical systems}}
{{inline citations|date=November 2024}}
[[File:Forced Duffing equation Poincaré section.png|300px|thumb|A two-dimensional Poincaré section of the forced [[Duffing equation]]]]
In [[mathematics]], particularly in [[dynamical systems]], a '''first recurrence map''' or '''Poincaré map''', named after [[Henri Poincaré]], is the intersection of a [[periodic orbit]] in the [[state space]] of a [[continuous dynamical system]] with a certain lower-dimensional subspace, called the '''Poincaré section''', [[Transversality (mathematics)|transversal]] to the [[Flow (mathematics)|flow]] of the system. More precisely, one considers a periodic orbit with initial conditions within a section of the space, which leaves that section afterwards, and observes the point at which this orbit first returns to the section. One then creates a [[map (mathematics)|map]] to send the first point to the second, hence the name ''first recurrence map''. The transversality of the Poincaré section means that periodic orbits starting on the subspace flow through it and not parallel to it.

A Poincaré map can be interpreted as a [[discrete dynamical system]] with a state space that is one dimension smaller than the original continuous dynamical system. Because it preserves many properties of periodic and quasiperiodic orbits of the original system and has a lower-dimensional state space, it is often used for analyzing the original system in a simpler way.{{cn|date=April 2020}} In practice this is not always possible as there is no general method to construct a Poincaré map.

A Poincaré map differs from a [[recurrence plot]] in that space, not time, determines when to plot a point. For instance, the locus of the Moon when the Earth is at [[perihelion]] is a recurrence plot; the locus of the Moon when it passes through the plane perpendicular to the Earth's orbit and passing through the Sun and the Earth at perihelion is a Poincaré map.{{cn|date=April 2020}} It was used by [[Michel Hénon]] to study the motion of stars in a [[galaxy]], because the path of a star projected onto a plane looks like a tangled mess, while the Poincaré map shows the structure more clearly.

== Definition ==

[[Image:Poincare map.svg|thumb|230px|In the Poincaré section ''S'', the Poincaré map ''P'' projects a point ''x'' onto the point ''P''(''x'').]]
Let ('''R''', ''M'', ''φ'') be a [[global dynamical system]], with '''R''' the [[real number]]s, ''M'' the [[phase space]] and ''φ'' the [[evolution function]]. Let γ be a [[periodic orbit]] through a point ''p'' and ''S'' be a local differentiable and transversal section of ''φ'' through ''p'', called a '''Poincaré section''' through ''p''.

Given an open and connected [[neighborhood (mathematics)|neighborhood]] <math>U \subset S</math> of ''p'', a [[Function (mathematics)|function]]
:<math>P: U \to S</math>
is called '''Poincaré map''' for the orbit γ on the '''Poincaré section''' ''S'' through the point ''p'' if
* ''P''(''p'') = ''p''
* ''P''(''U'') is a neighborhood of ''p'' and ''P'':''U'' → ''P''(''U'') is a [[diffeomorphism]]
* for every point ''x'' in ''U'', the [[positive semi-orbit]] of ''x'' intersects ''S'' for the first time at ''P''(''x'')
==Example==
Consider the following system of differential equations in polar coordinates, <math>(\theta, r)\in \mathbb{S}^1\times \mathbb{R}^+ </math>:
:<math>
\begin{cases}
\dot{\theta} = 1\\
\dot{r} = (1-r^2)r
\end{cases}
</math>
The flow of the system can be obtained by integrating the equation: for the <math>\theta</math> component we simply have
<math>
\theta(t) = \theta_0 + t
</math>
while for the <math>r</math> component we need to separate the variables and integrate:
:<math>
\int \frac{1}{(1-r^2)r} dr = \int dt \Longrightarrow \log\left(\frac{r}{\sqrt{1-r^2}}\right) = t+c
</math>
Inverting last expression gives
:<math>
r(t) = \sqrt{\frac{e^{2(t+c)}}{1+e^{2(t+c)}}}
</math>
and since
:<math>
r(0)=\sqrt{\frac{e^{2c}}{1+e^{2c}}}
</math>
we find
:<math>
r(t) = \sqrt{\frac{e^{2t}r_0^2}{1+r_0^2(e^{2t}-1)}} = \sqrt{\frac{1}{1+e^{-2t}\left(\frac{1}{r_0^2}-1\right)}}
</math>
The flow of the system is therefore
:<math>
\Phi_t(\theta, r) = \left(\theta_0 + t, \sqrt{\frac{1}{1+e^{-2t}\left(\frac{1}{r_0^2}-1\right)}}\right)
</math>
The behaviour of the flow is the following:
* The angle <math>\theta</math> increases monotonically and at constant rate.
* The radius <math>r</math> tends to the equilibrium <math>\bar{r}=1</math> for every value.
Therefore, the solution with initial data <math>(\theta_0, r_0\neq 1)</math> draws a spiral that tends towards the radius 1 circle.

We can take as Poincaré section for this flow the positive horizontal axis, namely
<math>
\Sigma = \{(\theta, r) \ : \ \theta =0 \}
</math>: obviously we can use <math>r</math> as coordinate on the section. Every point in <math>\Sigma</math> returns to the section after a time <math>t=2\pi</math> (this can be understood by looking at the evolution of the angle): we can take as Poincaré map the restriction of <math>\Phi</math> to the section <math>\Sigma</math> computed at the time <math>2\pi</math>, <math>\Phi_{2\pi}|_{\Sigma}</math>.
The Poincaré map is therefore :<math>\Psi(r) = \sqrt{\frac{1}{1+e^{-4\pi}\left(\frac{1}{r^2}-1\right)}}</math>

The behaviour of the orbits of the discrete dynamical system <math> (\Sigma, \mathbb{Z}, \Psi) </math> is the following:
* The point <math>r=1</math> is fixed, so <math>\Psi^n(1)=1</math> for every <math>n</math>.
* Every other point tends monotonically to the equilibrium, <math>\Psi^n(z) \to 1</math> for <math>n\to \pm \infty</math>.

== Poincaré maps and stability analysis ==

Poincaré maps can be interpreted as a [[discrete dynamical system]]. The [[stability theory|stability]] of a periodic orbit of the original system is closely related to the stability of the fixed point of the corresponding Poincaré map.

Let ('''R''', ''M'', ''φ'') be a [[differentiable dynamical system]] with periodic orbit γ through ''p''. Let
:<math>P: U \to S</math>
be the corresponding Poincaré map through ''p''. We define

:<math>P^{0} := \operatorname{id}_{U}</math>
:<math>P^{n+1} := P \circ P^n</math>
:<math>P^{-n-1} := P^{-1} \circ P^{-n}</math>

and

:<math>P(n, x) := P^n(x)</math>

then ('''Z''', ''U'', ''P'') is a discrete dynamical system with state space ''U'' and evolution function

:<math>P: \mathbb{Z} \times U \to U.</math>

Per definition this system has a fixed point at ''p''.

The periodic orbit γ of the continuous dynamical system is [[Stability theory|stable]] if and only if the fixed point ''p'' of the discrete dynamical system is stable.

The periodic orbit γ of the continuous dynamical system is [[asymptotically stable]] if and only if the fixed point ''p'' of the discrete dynamical system is asymptotically stable.

== See also ==
* [[Poincaré recurrence]]
* [[Hénon map]]
* [[Recurrence plot]]
* [[Mironenko reflecting function]]
* [[Invariant measure]]

== References ==
* {{cite book
| last = Teschl
| given = Gerald
|authorlink=Gerald Teschl | title = Ordinary Differential Equations and Dynamical Systems
| publisher=[[American Mathematical Society]]
| place = [[Providence, Rhode Island|Providence]]
| year =
| url = https://www.mat.univie.ac.at/~gerald/ftp/book-ode/}}

==External links==
* Shivakumar Jolad, ''[https://web.archive.org/web/20060520213802/http://www.personal.psu.edu/users/s/a/saj169/Poincaremap/Htmlfiles/PoincareMapintro.html Poincare Map and its application to 'Spinning Magnet' problem]'', (2005)

{{DEFAULTSORT:Poincare map}}
[[Category:Dynamical systems]]
[[Category:Henri Poincaré|Map]]

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Poincaré map