tdcpy.gamma module

implementation of tds_gamma_r

tdcpy.gamma.gamma(tds, r, **kwargs)

Computes gamma(r) of given time-delay system

Parameters:

• tds (TDS) – time-delay system

• r (float) – point from complex plane

• **kwargs – TODO

Returns:

• gamma_r (float): gamma(r) of given TDS

• metadata (GammaInfo): metadata, containing:

  TODO

Return type:

tuple containning

Computation consists of the follwing steps:

1. obtain associated delay difference equation (DIFF)

2. normalize DIFF, where DIFF takes form:

   0 = x(t) + DD[0] * x(t-hDD[0]) + … + DD[m-1]*x(t-hDD[m-1]), (1)

3. solve optimzation problem, gamma(r) of (1) is then given by maximum of:

   rho( SUM for all k DD[k]*exp(-r*hDD[k])*exp(1j*theta[k]) )

   where rho(.) is spectral radius of its matrix argument, and theta is from [0, 2*pi)^m.

   The optimization problem is solved via Dekker-Brent method (aka predictor - corrector), see [1].

[1] Atkinson, Kendall. An introduction to numerical analysis.

John wiley & sons, 1991.

Notes

1. this problem scales very badly with number of delays as the predictor

   searches through the grid whichs dimensionality corresponds to number of delays of normalized DIFF