We study how global system parameters and local realizations of quenched disorder shape the dynamics of a classical many-body spin system and how classical indicators of chaos, Lyapunov's exponents, relate to quantum signatures of chaos. In particular, we focus on the classical analogue of the quantum spin glass shards model in a random transverse magnetic field studied by Georgeot and Shepelyansky. The classical spin phase space is constructed as a symplectic manifold and we evolve trajectories with a second-order Suzuki–Trotter integrator. This symplectic structure-preserving scheme enables reliable computation of Lyapunov's exponents via standard repeated QR-based orthogonalization of tangent vectors, yielding accurate finite-time Lyapunov spectra for trajectories in different regions of phase space.

Using these tools, we examine the dynamics of trajectories sampled from different regions of phase space while varying two global system parameters: the relative strength of the spin-spin coupling and the transverse magnetic field. We find that both the strong spin-spin coupling and strong magnetic field limits are nearly integrable, with maximal chaos emerging at intermediate coupling–field ratios. When initial spin configurations are sampled uniformly over the Bloch sphere, different disorder realizations do not qualitatively change whether dynamics are chaotic or integrable, while configurations concentrated near the $x$- or $z$-axis are highly sensitive to the specific disorder realization and can exhibit either almost fully integrable or strongly chaotic behavior under identical global parameters.

For \(17\) spins, all observed trajectories in the classical system are chaotic when the spin-spin coupling is approximately three times stronger than the transverse field. A comparison with the quantum level-spacing statistics of the corresponding quantum model shows qualitative agreement regarding which choices of global parameters lead to integrable or chaotic dynamics. However, there is a quantitative mismatch in which global parameter values produce the {strongest} chaotic dynamics. This demonstrates that the relationship between classical Lyapunov exponents and quantum energy-level spacing statistics is complex and non-trivial.