Conservative integrators for many–body problems

Conservative integrators for piecewise smooth systems with transversal dynamics

“…We will be using conservative integrators based on DMM [24,22,25]. These integrators preserve artbitrary forms of conserved quantities up to machine precision and have been recently applied to many body problems [23,9] and Hamiltonian Monte Carlo [18] and geodesics in Schwarzchild geometry [21].…”

“…Three species Lotka-Volterra system. Our second example is a three species Lotka-Volterra system studied in [22,23] given by, ẋ(t) = diag(x(t))A ± (x(t)− ξ), where x(t) = (x 1 (t), x 2 (t), x 3 (t)) T ∈ R 3 + are the relative population of three species at time t, A ± is a constant 3 × 3 interaction matrix and ξ ∈ R 3 + is a fixed point. If A ± is skew-symmetric, then [20] showed a conserved quantity of the form ψ(x) = ξ T log(x)−1 T x.…”

“…Thus the two parts of the phase space are U ± = {x ∈ R 3 + | g(x) ≷ 0}. Using the DMM discretization of this system from [23], the discrete PWS vector fields from DMM is given by f…”