Fig 1.
This figure illustrates how well exactly conserved quantities are preserved by our symplectic numerical solvers, compared to the standard Runge-Kutta methods implemented in scipy.integrate.solve_ivp.
For the 3-site Toda lattice, reduced to relative coordinates, the constants of motion are the Hamiltonian H⊥ of Eq (11) and of Eq (12). Here ΔH⊥ = [H⊥(t) − H⊥(0)]/H⊥(0), calculated from the numerical solutions, and similar for ΔC3⊥. Here τ is the fixed timestep of the sympletic solvers.
Fig 2.
See the caption to Fig 1.
Fig 3.
Poincaré sections for an orbit of the reduced 3-site Toda lattice model of Eq (11).
Each panel shows 4000 crossings of the (q1 = 0)-plane, 2000 in the positive direction (p1 > 0, marked blue), and 2000 in the negative direction (p1 < 0, marked red). The dynamics between each crossing is determined by KiMoKi solvers of order 2 (left panel) resp. 4 (right panel), with timestep τ = 0.005. The initial condition is z0 = (0, 1, 9.95073, 10), with H0 = 100, C3⊥ = −15852.72982, for the left panel, and z0 = (0, 1, 19.97541, 10), with H0 = 250, C3⊥ = 16117.70199, for the right panel.
Fig 4.
Poincaré sections for an orbit of the reduced 3-site Toda lattice model of Eq (11).
Each panel shows 4000 crossings of the (q2 = 0)-plane, 2000 in the positive direction (p2 > 0, marked blue), and 2000 in the negative direction (p2 < 0, marked red). The dynamics between each crossing is determined by a KiMoKi solver of order 6, with timestep τ = 0.01. The initial condition is z0 = (1, 0, 22, 5.04993), with H0 = 256, C3⊥ = 72099.45264, for the left panel, and z0 = (0, 0.1, 1.40733), with H0 = 1, C3⊥ = 23.51188 for the right panel.
Fig 5.
Poincaré sections for an orbit of the reduced 3-site Toda lattice model of Eq (11).
The left panel shows 4 000 crossings of the (q2 = 0)-plane, 2 000 in the positive direction (p2 > 0, marked blue) and 2 000 in the negative direction (p2 < 0, marked red). The right panel similarly shows 4 000 crossings of the (q1 = 0)-plane. The dynamics between each crossing is determined by a KiMoKi solver of order 8, with timestep τ = 0.05. The initial condition is z0 = (0.1, 0, 0.1, 1.40709), with H0 = 1, C3⊥ = −6.45412, for the left panel, and z0 = (0, 1, 6.92943, 1), with H0 = 25, C3⊥ = 2597.68431, for the right panel.
Fig 6.
A quasi-periodic orbit z(t) for times 0 ≤ t ≤ 2000, found by the KiMoKi solvers of order N = 2 (left panel) and order N = 4 (right panel) with timestep τ = 0.1, projected to respectively the (q1, q2, p2) (left) and (q1, q2, p1) (right) subspaces.
The initial condition z0 = (0.1, 0, 0.1, 1.40709), with H0 = 1, C3⊥ = −6.45412. The viewing angle is set to (ϑ, φ) = (68, 78) (left), respectively (ϑ, φ) = (−128, −8). Here ϑ is the elevation angle (elev) and φ the azimuth angle (azim).
Fig 7.
A quasi-periodic orbit z(t) for times 0 ≤ t ≤ 2000, found by the KiMoKi solvers of order N = 6 (left panel) and order N = 8 (right panel) with timestep τ = 0.1, projected to respectively the (q1, q2, p2) (left) and (q1, q2, p1) (right) subspaces.
The initial condition z0 = (0.1, 0, 0.1, 1.40709), with H0 = 1, C3⊥ = −6.45412. The viewing angle is set to (ϑ, φ) = (15, −87) (left), respectively (ϑ, φ) = (−128, 133). Here ϑ is the elevation angle (elev) and φ the azimuth angle (azim).
Fig 8.
Long time energy error for solutions of a Toda lattice model computed by the KiMoKi solvers.
An orbit z(t) with initial value z0 = (0.1, 0, 0.1, 1.40709), corrsponding to H0 = 1 and C3⊥ = −6.45412. The solution is computed for times 0 ≤ t ≤ 5 000 with timestep τ = 0.1; for better visibility only the last hundred time units are plotted.
Fig 9.
Scaled energy errors for some higher order symplectic integrators, when applied to the reduced 3-site Toda lattice Hamiltonian of Eq (11).
The plots are for an orbit z(t) with initial value (0.1, 0, 0.1, 1.40709), corresponding to H0 = 1, C3⊥ = −6.45412, computed with KiMoKi solvers of orders N = 2, 4, 6, 8, and timesteps τ = 0.025, 0.05, and 0.1.