Lyapunov:
We consider the system of differential equations below (see [1]):
\begin{cases}
\overset{.}{x_1} = x_1 * (1 - x_1^2 - x_2^2) * (x_1 + 1/2) - x_2 \\
\overset{.}{x_2} = x_2 * (1 - x_1^2 - x_2^2) * (x_1 + 1/2) + x_1 \\
\end{cases}
The initial condition is
\(x(0) = (0.8, 0.8) \)
Results:
We consider a system with uncertainty \(w=0.01\), set of initial conditions \(B(x_0,\varepsilon)\) with \(\varepsilon=0.15\), the time-step used in Euler's method is \(\tau = 10^{-4}\), and we take \(T=k\tau\) with \(k=62833\) as an approximate period.
Using the figures shown below, we check that:
- \(B((i_0+1)T)\subset B(i_0T)\) for \(i_0=0\)
- and \(\Sigma_{i=1}^k\lambda_i\approx\) \(-63368 < 0\).
Then, we can conclude that the system
converges towards an attractive LC
contained in
\([B(0),B(T)]\).
The figures below show respectively the simulation of \(x_1(t), x_2(t)\) and \(\delta_{{\cal W}}(t)\) with perturbation (w=0.01) over 2 periods (2T=12.5667) for dt=0.0001.
References:
[1]GIESL, Peter. Construction of Lyapunov functions and Contraction Metrics to determine the Basin of Attraction. In : Workshop on Algorithms for Dynamical Systems. 2013.
