# Poincaré plot

A Poincaré plot, named after Henri Poincaré, is used to quantify self-similarity in processes, usually periodic functions. It is also known as a return map.[1][2] Poincaré plots can be used to distinguish chaos from randomness by embedding a data set into a higher-dimensional state space.

Given a time series of the form

${\displaystyle x_{t},x_{t+1},x_{t+2},\ldots ,\,}$

a return map in its simplest form first plots (xtxt+1), then plots (xt+1xt+2), then (xt+2xt+3), and so on.

## Applications in Electrocardiography

An electrocardiogram (ECG) is a tracing of the voltage changes in the chest generated by the heart, whose contraction in the normal person is triggered by an electrical impulse that originates from its "pacemaker", the sinoatrial node. The ECG normally consists of a series of waves, labeled the P, Q, R, S and T waves. The P wave is due to the depolarization of the atria, the Q-R-S series of waves due to depolarization of the ventricles, and the T wave due to the repolarization (the opposite of depolarization) of the ventricles. The interval between two successive R waves (the RR interval) is a measure of the heart rate.

The heart rate normally varies slightly: during a deep breath, it speeds up and during a deep exhalation, it slows down. (The RR interval will shorten when the heart speeds up, and lengthen when it slows.) An RR tachograph is a graph of the numerical value of the RR-interval (measured heartbeat-to-heartbeat) versus time.

In the context of RR tachography, a Poincare plot (which is a special case of a recurrence plot) is a graph of RR(n) on the x-axis versus RR(n + 1) - i.e., the succeeding RR interval - on the y-axis. That is, one takes a sequence of intervals and plots each interval against the following interval.[3] The recurrence plot is used as a standard visualizing technique to detect the presence of oscillations in non-linear dynamic systems. In the context of electrocardiography, the rate of the healthy heart is normally tightly controlled by the body's regulatory mechanisms (specifically, by the autonomic nervous system). An increase in the variability of heart rate suggests pathological conditions: it often increases after a myocardial infarction ("heart attack"), either because the sinoatrial node (pacemaker) is damaged, or because the damaged heart tissue produces abnormal electric current that cause the ventricles to beat prematurely or irregularly.