nonlinearTseries (version 0.3.1)

poincareMap: Poincare map

Description

Computes the Poincare map of the reconstructed trajectories in the phase-space.

The Poincare map is a classical dynamical system technique that replaces the n-th dimensional trajectory in the phase space with an (n-1)-th order discrete-time called the Poincare map. The points of the Poincare map are the intersection of the trajectories in the phase-space with a certain Hyper-plane.

Usage

poincareMap(
  time.series = NULL,
  embedding.dim = 2,
  time.lag = 1,
  takens = NULL,
  normal.hiperplane.vector = NULL,
  hiperplane.point
)

Value

Since there are three different Poincare maps, an R list is returned storing all the information related which all of these maps:

• The positive Poincare map is formed by all the intersections with the hyperplane in positive direction (defined by the normal vector). The pm.pos returns the points of the map whereas that pm.pos.time returns the number of time steps since the beginning where the intersections occurred. Note that these time steps probably won't be integers since the algorithm uses an interpolation procedure for calculating the intersection with the hyperplane.

• Similarly we define a negative Poincare map (pm.neg and pm.neg.time).

• Finally, we may define a two-side Poincare map that stores all the intersections (no matter the direction of the intersection) (pm and pm.time).

Arguments

time.series

The original time series from which the phase-space reconstruction is done.

embedding.dim

Integer denoting the dimension in which we shall embed the time.series.

time.lag

Integer denoting the number of time steps that will be use to construct the Takens' vectors.

takens

Instead of specifying the time.series, the embedding.dim and the time.lag, the user may specify directly the Takens' vectors.

normal.hiperplane.vector

The normal vector of the hyperplane that will be used to compute the Poincare map. If the vector is not specifyed the program choses the vector (0,0,...,1).

hiperplane.point

A point on the hyperplane (an hyperplane is defined with a point and a normal vector).

Author

Constantino A. Garcia

Details

This function computes the Poincare map taking the Takens' vectors as the continuous trajectory in the phase space. The takens param has been included so that the user may specify the real phase-space instead of using the phase-space reconstruction (see examples).

References

Parker, T. S., L. O. Chua, and T. S. Parker (1989). Practical numerical algorithms for chaotic systems. Springer New York

Examples

if (FALSE) {
r=rossler(a = 0.2, b = 0.2, w = 5.7, start=c(-2, -10, 0.2),
time=seq(0,300,by = 0.01), do.plot=FALSE)
takens=cbind(r$x,r$y,r$z)
# calculate poincare sections
pm=poincareMap(takens = takens,normal.hiperplane.vector = c(0,1,0), 
 hiperplane.point=c(0,0,0) )
if (requireNamespace("rgl", quietly = TRUE)) {
   rgl::plot3d(takens,size=0.7)
   rgl::points3d(pm$pm,col="red")
}
}