Poincare illustration

Illustration of the return of a trajectory .

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The Poincaré map (also Poincaré map , first return map , after the French mathematician Henri Poincaré ) is a mathematical method for studying the flow of a continuous n-dimensional dynamic system . To do this, one considers the intersections of a trajectory with an (n-1) -dimensional transverse hypersurface , the Poincaré cut . The Poincaré mapping is the mapping that assigns each of these intersection points to the next and is therefore an (n-1) -dimensional discrete dynamic system. ${\ displaystyle \ Sigma}$ ${\ displaystyle x}$ ${\ displaystyle P (x)}$

example

Poincaré cut for a periodic trajectory

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Consider the differential equation and denote with the flow, i.e. the solution to the initial condition . Suppose there is a periodic trajectory, which is a solution that at start and after a certain time returns there . Then one can choose a surface that is transverse to the trajectory and intersects it. All trajectories that start at points near will then intersect the surface again after a certain time. So there is a smallest positive time for which applies. Then the Poincaré map is given by . Specifically, to obtain a trajectory for the periodic checkpoint : . The question of whether the periodic trajectory is stable is now equivalent to the question of whether the corresponding fixed point of the Poincaré mapping is stable. ${\ displaystyle {\ dot {x}} (t) = f (x (t))}$ ${\ displaystyle \ Phi (t, x)}$ ${\ displaystyle \ Phi (0, x) = x}$ ${\ displaystyle \ Phi (t, p)}$ ${\ displaystyle p}$ ${\ displaystyle \ tau}$ ${\ displaystyle \ Phi (\ tau, p) = p}$ ${\ displaystyle \ Sigma}$ ${\ displaystyle \ Phi (t, p)}$ ${\ displaystyle p}$ ${\ displaystyle x \ in \ Sigma}$ ${\ displaystyle p}$ ${\ displaystyle \ tau (x)> 0}$ ${\ displaystyle \ Phi (\ tau (x), x) \ in \ Sigma}$ ${\ displaystyle P (x) = \ Phi (\ tau (x), x)}$ ${\ displaystyle P (p) = p}$

application

The Poincaré mapping is particularly suitable for the investigation of the geometric structures of chaotic attractors , since the temporal discretization represents a significant simplification.

In cardiology , the display is used when evaluating a long-term ECG . By applying it to the intervals between the respective heartbeats, conclusions can be drawn about cardiac arrhythmias such as atrial fibrillation .

Another application can be found in stress research: here, the parasympathetic and sympathetic influences on the heart rate can be read from the Poincaré images with the two orthogonally aligned diameters SD1 and SD2 ( heart rate variability ).

literature

Herbert Amann: Ordinary differential equations . 2nd Edition. de Gruyter, Berlin 1995, ISBN 3-11-014582-0 .
DV Anosov : Poincaré return map . In: Michiel Hazewinkel (Ed.): Encyclopaedia of Mathematics . Springer-Verlag , Berlin 2002, ISBN 978-1-55608-010-4 (English, online ).
Gerald Teschl : Ordinary Differential Equations and Dynamical Systems (= Graduate Studies in Mathematics . Volume 140 ). American Mathematical Society, Providence 2012, ISBN 978-0-8218-8328-0 ( mat.univie.ac.at ).

Individual evidence

↑ Manfred von Ardenne et al .: Effects of physics and their applications . Verlag Harry Deutsch, Frankfurt 2005. ISBN 3-8171-1682-9 , p. 1130

[1] Manfred von Ardenne et al .: Effects of physics and their applications . Verlag Harry Deutsch, Frankfurt 2005. ISBN 3-8171-1682-9 , p. 1130