Piecewise linear LLL-system

A Shiny app that builds 3D phase portraits for the following piecewise linear LLL-system:

$$ \begin{cases} \dot{x}_1 = L(a_1, k_1, x_3) - k_1 x_1,\\ \dot{x}_2 = L(a_2, k_2, x_1) - k_2 x_2,\\ \dot{x}_3 = L(a_3, k_3, x_2) - k_3 x_3, \end{cases} $$

where $L$ is a piecewise linear function:

$$ L(a, k, x) = \begin{cases} ak & \quad \textrm{for}\ \ \ 0 \leq x \leq 1,\\ 0 & \quad \textrm{for}\ \ \ 1 < x. \end{cases} $$

Parameters

• $a_1$, $a_2$, $a_3$, $k_1$, $k_2$, $k_3$: the dynamical system parameter.
• $\textrm{TotalTime}$: the total time simulation.
• $\textrm{SkipTime}$: the initial time interval that will not be presented on the plot.
• $N$: the number of simulated trajectories.
• $\textrm{Seed}$: the randomization seed that controls the initial positions of all the trajectories.

Initial points

We uniformly generate the initial point for each simulated trajectory from the following area:

$$ [0.5; (2+a_1)/3] \times [0.5; (2+a_2)/3] \times [0.5; (2+a_3)/3]. $$