Example 2.1 - Stability analysis of retarded DDE

We consider the following retarded delay differential equation (RDDE) from [Verheyden et al., 2008] Section 6.1:

\[\begin{split}\dot{x}(t) = \begin{bmatrix} -1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & -10 & -4 \\ 0 & 0 & 4 & -10 \end{bmatrix} x(t) + \begin{bmatrix} 3 & 3 & 3 & 3\\ 0 & -1.5 & 0 & 0 \\ 0 & 0 & 3 & -5 \\ 0 & 5 & 5 & 5 \end{bmatrix} x(t-1).\end{split}\]

We will follow the steps from [Appeltans and Michiels, 2023] Section 2.2 to achive the same results as presented there, i.e. we will

1. create the RDDE matrix representation

2. compute characteristic roots via the tdcpy.roots function

3. plot the computed characteristic roots using tdcpy.plot.eigen_plot function

import matplotlib.pyplot as plt
import numpy as np
import tdcpy
import tdcpy.plot

# Create RDDE matrix representation
A0 = np.array([[-1, 0, 0, 0],
               [0, 1, 0, 0],
               [0, 0, -10, -4],
               [0, 0, 4, -10]])
A1 = np.array([[3, 3, 3, 3],
               [0, -1.5, 0, 0],
               [0, 0, 3, -5],
               [0, 5, 5, 5]])

rdde = tdcpy.RDDE(A=[A0, A1], hA=[0, 1])
cr, info = tdcpy.roots(rdde, r=-2.5)

tdcpy.plot.eigen_plot(cr)
plt.show()