7 July, 2026

Also available as a slide deck.

To do!

• plot selected trajectories: \((y_0, p_{y0})_i \in \{(0.28984,0.0), (-0.079,0.0), (-0.06485,0.0), \dots\}\)

---

What I am doing

• I am reading Hamiltonian Systems: Chaos and Quantization by Alfredo M. Ozorio de Almeida (Ozorio de Almeida 1988)
• I am trying to study the basics of chaos theory to

Poincaré sections & trajectories

Figure 1: Combined Poincaré section (\(x = 0\), \(p_x > 0\)) at \(E = 0.1127\) for all sampled initial conditions.

(a) Section: \((y_0, p_{y0}) = (0.28984, 0.0)\)

(b) Trajectory: \((y_0, p_{y0}) = (0.28984, 0.0)\)

(c) Section: \((y_0, p_{y0}) = (-0.079, 0.0)\)

(d) Trajectory: \((y_0, p_{y0}) = (-0.079, 0.0)\)

(e) Section: \((y_0, p_{y0}) = (0.0, 0.115)\)

(f) Trajectory: \((y_0, p_{y0}) = (0.0, 0.115)\)

(g) Section: \((y_0, p_{y0}) = (0.185, 0.26)\)

(h) Trajectory: \((y_0, p_{y0}) = (0.185, 0.26)\)

(i) Section: \((y_0, p_{y0}) = (0.38, 0.288)\)

(j) Trajectory: \((y_0, p_{y0}) = (0.38, 0.288)\)

(k) Section: \((y_0, p_{y0}) = (0.15, 0.382)\)

(l) Trajectory: \((y_0, p_{y0}) = (0.15, 0.382)\)

(m) Section: \((y_0, p_{y0}) = (0.282, 0.373)\)

(n) Trajectory: \((y_0, p_{y0}) = (0.282, 0.373)\)

(o) Section: \((y_0, p_{y0}) = (0.0, 0.2375)\)

(p) Trajectory: \((y_0, p_{y0}) = (0.0, 0.2375)\)

(q) Section: \((y_0, p_{y0}) = (-0.352, 0.0)\)

(r) Trajectory: \((y_0, p_{y0}) = (-0.352, 0.0)\)

(s) Section: \((y_0, p_{y0}) = (-0.077018, 0.12995522)\)

(t) Trajectory: \((y_0, p_{y0}) = (-0.077018, 0.12995522)\)

(u) Section: \((y_0, p_{y0}) = (-0.0935, 0.0)\)

(v) Trajectory: \((y_0, p_{y0}) = (-0.0935, 0.0)\)

Figure 2: Individual sections (left) and configuration-space trajectories (right) at \(E = 0.1127\).

See Figure 1 for the full section; individual orbits in Figure 2.

References

Ozorio de Almeida, Alfredo M. 1988. Hamiltonian Systems: Chaos and Quantization. Cambridge Monographs on Mathematical Physics. Cambridge University Press.