Add hybrid lotka-volterra model

[mforets]

No description provided.

[mforets]

Note : this branch requires to checkout LazySets#mforets/union

[mforets]

This plot captures the overapproximation of a Taylor model reach-set using a box (blue) and a zonotope (red). I was interesecting these with B_ext in order to prove disjointness. As it is seen in the plot, the flowpipe actually has left much earlier, which we can prove either using the zonotope overapprox, or computing a tighter continuous solution eg. by splitting.

[mforets]

Here is a sample result: (abs_tol=1e-12)

Playing with the parameter choices one can get smaller final states; there are mainly 2 options:

• abs_tol the initial abs tolerance for the adaptive TMJets algorithm: smaller => more accurate
• nsplit the number of sets in which the initial state X0 is split: if it's larger => wrapping effects are smaller so the result is more accurate

For instance the following plot was obtained with abs_tol=1e-16 and nsplit=10:

Runtime is 3.3sec and 12.5sec respectively.

[lbenet]

Beautiful!

[mforets]

The intersection time can be obtained with the following function:

solz = overapproximate(sol, Zonotope);
intersecting_reachsets = []
for (i, Fi) in enumerate(solz)
    for (j, Xj) in enumerate(Fi)
        !isdisjoint(Xj, B_ext) && push!(intersecting_reachsets, (i, j))
    end
end

times = [tspan(solz[ind[1]][ind[2]]) for ind in intersecting_reachsets];
tmin = minimum(tstart, times)
tmax = maximum(tend, times)
tmax - tmin

Here we detect, for each flowpipe, those reach-sets that intersect the outer approximation of the ball. Then takes the minimum (resp. maximum) times so we have a proof for an upper bound on the intersection time.

The results obtained for the settings in my previous comment are 0.21554 and 0.17176 respectively.

[mforets]

We can also see that there are flowpipes which never intersected the guard:

[isdisjoint(s, B_ext) for s in solz]
12-element Array{Bool,1}:
 1
 1
 1
 1
 1
 0
 0
 0
 0
 0
 0
 0

Here you see 12 flowpipes because:

• The waiting list initially had 10 sets (since i used nsplit=10)
• For the first jump we convexify => add a new element to the waiting list => a new Flowpipe is computed
• For the second jump (from inside to outside the ball) we convexify a gain and add a new flowpipe