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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 mathematics, in the study of iterated functions and dynamical systems, a periodic point of a function is a point which the system returns to after a certain number of function iterations or a certain amount of time. Iterated functions Given a ...
in the
state space A state space is the set of all possible configurations of a system. It is a useful abstraction for reasoning about the behavior of a given system and is widely used in the fields of artificial intelligence and game theory. For instance, the to ...
of a continuous dynamical system with a certain lower-dimensional subspace, called the Poincaré section, transversal to the 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 A map is a symbolic depiction emphasizing relationships between elements of some space, such as objects, regions, or themes. Many maps are static, fixed to paper or some other durable medium, while others are dynamic or interactive. Although ...
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 In mathematics, a dynamical system is a system in which a function describes the time dependence of a point in an ambient space. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water in a ...
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. 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 descriptive statistics and chaos theory, a recurrence plot (RP) is a plot showing, for each moment i in time, the times at which the state of a dynamical system returns to the previous state at i, i.e., when the phase space trajectory visits roug ...
in that space, not time, determines when to plot a point. For instance, the locus of the Moon when the Earth is at
perihelion An apsis (; ) is the farthest or nearest point in the orbit of a planetary body about its primary body. For example, the apsides of the Earth are called the aphelion and perihelion. General description There are two apsides in any elli ...
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. 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

Let (R, ''M'', ''φ'') be a global dynamical system, with R the
real number In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...
s, ''M'' the phase space and ''φ'' the
evolution function In mathematics, a dynamical system is a system in which a function describes the time dependence of a point in an ambient space. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water in a ...
. Let γ be a
periodic orbit In mathematics, in the study of iterated functions and dynamical systems, a periodic point of a function is a point which the system returns to after a certain number of function iterations or a certain amount of time. Iterated functions Given a ...
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 U \subset S of ''p'', a
function Function or functionality may refer to: Computing * Function key, a type of key on computer keyboards * Function model, a structured representation of processes in a system * Function object or functor or functionoid, a concept of object-oriente ...
:P: U \to S 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 In mathematics, a diffeomorphism is an isomorphism of smooth manifolds. It is an invertible function that maps one differentiable manifold to another such that both the function and its inverse are differentiable. Definition Given two ...
* for every point ''x'' in ''U'', the
positive semi-orbit In mathematics, specifically in the study of dynamical systems, an orbit is a collection of points related by the evolution function of the dynamical system. It can be understood as the subset of phase space covered by the trajectory of the dynamic ...
of ''x'' intersects ''S'' for the first time at ''P''(''x'')


Example

Consider the following system of differential equations in polar coordinates, (\theta, r)\in \mathbb^1\times \mathbb^+ : : \begin \dot = 1\\ \dot = (1-r^2)r \end The flow of the system can be obtained by integrating the equation: for the \theta component we simply have \theta(t) = \theta_0 + t while for the r component we need to separate the variables and integrate: : \int \frac dr = \int dt \Longrightarrow \log\left(\frac\right) = t+c Inverting last expression gives : r(t) = \sqrt and since : r(0)=\sqrt we find : r(t) = \sqrt = \sqrt The flow of the system is therefore : \Phi_t(\theta, r) = \left(\theta+ t, \sqrt\right) The behaviour of the flow is the following: * The angle \theta increases monotonically and at constant rate. * The radius r tends to the equilibrium \bar=1 for every value. Therefore, the solution with initial data (\theta_0, r_0\neq 1) draws a spiral that tends towards the radius 1 circle. We can take as Poincaré section for this flow the positive horizontal axis, namely \Sigma = \ : obviously we can use r as coordinate on the section. Every point in \Sigma returns to the section after a time t=2\pi (this can be understood by looking at the evolution of the angle): we can take as Poincaré map the restriction of \Phi to the section \Sigma computed at the time 2\pi, \Phi_, _. The Poincaré map is therefore :\Psi(r) = \sqrt The behaviour of the orbits of the discrete dynamical system (\Sigma, \mathbb, \Psi) is the following: * The point r=1 is fixed, so \Psi^n(1)=1 for every n. * Every other point tends monotonically to the equilibrium, \Psi^n(z) \to 1 for n\to \pm \infty.


Poincaré maps and stability analysis

Poincaré maps can be interpreted as a
discrete dynamical system In mathematics, a dynamical system is a system in which a function describes the time dependence of a point in an ambient space. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water in a ...
. The
stability Stability may refer to: Mathematics *Stability theory, the study of the stability of solutions to differential equations and dynamical systems ** Asymptotic stability ** Linear stability ** Lyapunov stability ** Orbital stability ** Structural sta ...
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 In mathematics, a dynamical system is a system in which a function describes the time dependence of a point in an ambient space. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water in a ...
with periodic orbit γ through ''p''. Let :P: U \to S be the corresponding Poincaré map through ''p''. We define :P^ := \operatorname_ :P^ := P \circ P^n :P^ := P^ \circ P^ and :P(n, x) := P^n(x) then (Z, ''U'', ''P'') is a discrete dynamical system with state space ''U'' and evolution function :P: \mathbb \times U \to U. Per definition this system has a fixed point at ''p''. The periodic orbit γ of the continuous dynamical system is 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 Various types of stability may be discussed for the solutions of differential equations or difference equations describing dynamical systems. The most important type is that concerning the stability of solutions near to a point of equilibrium. T ...
if and only if the fixed point ''p'' of the discrete dynamical system is asymptotically stable.


See also

*
Poincaré recurrence Poincaré is a French surname. Notable people with the surname include: * Henri Poincaré (1854–1912), French physicist, mathematician and philosopher of science * Henriette Poincaré (1858-1943), wife of Prime Minister Raymond Poincaré * Luci ...
*
Stroboscopic map Stroboscopic may refer to: * Stroboscopic effect, visual temporal aliasing * Stroboscopic effect (lighting), a temporal light artefact visible if a moving object is lit with modulated light with specific modulation frequencies and amplitudes * Stro ...
* Hénon map *
Recurrence plot In descriptive statistics and chaos theory, a recurrence plot (RP) is a plot showing, for each moment i in time, the times at which the state of a dynamical system returns to the previous state at i, i.e., when the phase space trajectory visits roug ...
* Mironenko reflecting function *
Invariant measure In mathematics, an invariant measure is a measure that is preserved by some function. The function may be a geometric transformation. For examples, circular angle is invariant under rotation, hyperbolic angle is invariant under squeeze mapping, ...


References

*


External links

* Shivakumar Jolad,
Poincare Map and its application to 'Spinning Magnet' problem
', (2005) {{DEFAULTSORT:Poincare map Dynamical systems
Map A map is a symbolic depiction emphasizing relationships between elements of some space, such as objects, regions, or themes. Many maps are static, fixed to paper or some other durable medium, while others are dynamic or interactive. Although ...