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**, 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 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.^{[citation needed]}

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.^{[citation needed]} 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.^{[citation needed]} 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 numbers, *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 of *p*, a function

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*)

## Poincaré maps and stability analysis

Poincaré maps can be interpreted as a discrete dynamical system. The 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

be the corresponding Poincaré map through *p*. We define

and

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

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 if and only if the fixed point *p* of the discrete dynamical system is asymptotically stable.

## See also

- Poincaré recurrence
- Stroboscopic map
- Hénon map
- Recurrence plot
- Mironenko reflecting function
- Invariant measure

## References

- Teschl, Gerald.
*Ordinary Differential Equations and Dynamical Systems*. Providence: American Mathematical Society.

## External links

- Shivakumar Jolad,
*Poincare Map and its application to 'Spinning Magnet' problem*, (2005)