Laub-Loomis Model

The model was initially proposed in a paper from Laub and Loomis [1], and then studied by several papers in reachability computation such as [2], [3]. The dynamics of the model is defined by

 \left\{ \begin{array}{lcl} \dot{x}_1 & = & 1.4 x_3 - 0.9 x_1 \\ \dot{x}_2 & = & 2.5 x_5 - 1.5 x_2 \\ \dot{x}_3 & = & 0.6 x_7 - 0.8 x_2 x_3 \\ \dot{x}_4 & = & 2 - 1.3 x_3 x_4 \\ \dot{x}_5 & = & 0.7 x_1 - x_4 x_5 \\ \dot{x}_6 & = & 0.3 x_1 - 3.1 x_6 \\ \dot{x}_7 & = & 1.8 x_6 - 1.5 x_2 x_7 \end{array} \right.

We firstly consider a small initial set which is given by
 x1 \in [1.1,1.3], x2 \in [0.95,1.15], x3 \in [1.4,1.6], x4 \in [2.3,2.5],
 x5 \in [0.9,1.1], x6 \in [0,0.2], x7 \in [0.35,0.55]
The following figures shows the interval overapproximations of the flowpipes in the time horizon [0,20] computed by Flow* 2.1.0. The total time cost is 18 seconds.

laubloomis_small_x2_x6

laubloomis_small_x5_x7

[Model file]

For a larger initial set
 x1 \in [1,1.4], x2 \in [0.85,1.25], x3 \in [1.3,1.7], x4 \in [2.2,2.6],
 x5 \in [0.8,1.2], x6 \in [-0.1,0.3], x7 \in [0.25,0.65]
over the same time horizon, Flow* spends 60 seconds and generates the following result.

laubloomis_x2_x6

laubloomis_x5_x7

[Model file]

References

[1] M. T. Laub and W. F. Loomis.
A molecular network that produces spontaneous oscillations in excitable cells of dictyostelium.
Molecular Biology of the Cell, 9:3521–3532, 1998.

[2] R. Testylier and T. Dang.
NLTOOLBOX: A library for reachability computation of nonlinear dynamical systems.
In Proc. of ATVA’13, volume 8172 of LNCS, pages 469–473. Springer, 2013.

[3] X. Chen and S. Sankaranarayanan.
Decomposed Reachability Analysis for Nonlinear Systems.
In IEEE Real Time Systems Symposium (RTSS), pp. 13-24, 2016.