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.

Let n ∈ ℕ_≥1. Let 1 ≤ p1,…,p_n < ∞ and set the Hölder combination p := (p1; …; p_n) := (∑_jⁿ=1np _j⁻¹)⁻¹. Assume further that 0 < p ≤ 1 and that for the Hölder combinations of p2 to pn and p1 to pn−1, we have 1 ≤ (p2; …; p_n), (p1; …; p_n−1) < ∞. Then there exists a constant C > 0 such that for every (Formula presented) with ∥f⁽ⁿ⁾∥ _∞ < ∞ we have ∥T_f[n] : S_p1 ×⋯ × S_pn → S_p∥ ≤ (Formula presented). Here S_q is the Schatten–von Neumann class, Ḃ_p,q^s the homogeneous Besov space and Tf[n] is the multilinear Schur multiplier of the nth order divided difference function. In particular, our result holds for p = 1 and any 1 ≤ p1,…,p_n < ∞ with p = (p1; …; p_n).

The principle of competitive exclusion states that, in equilibrium, the amount of coexisting species is limited by the amount of limiting resource types in an ecosystem. However, in phytoplankton communities the paradox of plankton appears, amounts of plankton species can coexist that far exceed this
upper limit. A resource competition model is formulated and it is shown that the paradox arises for several systems, which indicates that the bloom in biodiversity is a result of the resource competition and not of any external factors. A proof is given that the principle of competitive exclusion only holds in equilibrium solutions. Therefore, as long as a system does not intersect with an equilibrium solution the biodiversity is not restricted by the amount of limiting resource types. It is concluded that intersecting with an equilibrium solution is avoided when there are only unstable equilibrium solutions present in the system. When a plankton species allows an asymptotically stable equilibrium solution, with a region of convergence equal to the domain of the system, to appear it will be called dominant. It is proven that an asymptotically stable equilibrium solution always exists in a simplified system with less than three limiting resource types. Furthermore, an algorithm is constructed that determines all the new equilibrium solutions, and their respective stabilities, when a new plankton species is introduced to a system.
By applying this algorithm it can be determined whether a species is suitable for an ecosystem, when the goal is to maintain biodiversity. The resource competition model is expanded to include light as an additional resource for all plankton species. It is observed that the coexistence of the plankton species and the total biomass is limited
if there is too little light for the plankton species to consume, or if one plankton species becomes dominant due to it being significantly better at consuming light than the other species.
Additionally, the physical context of a flowing river is introduced, with dispersive and advective mass transfer and finite length. It is observed that while the spatial distribution of the plankton species along the river is strongly influenced by the spatial parameters, the biodiversity of the ecosystem is still primarily determined by the original parameters from the resource competition model, as long as the dispersive mass transfer is the dominant type of mass transfer not too large in comparison to the length of the river.