We investigate how generic the onset of chaos in interacting many-body classical systems is in the context of lattices of classical spins with nearest-neighbor anisotropic couplings. Seven large lattices in different spatial dimensions were considered. For each lattice, more than 2000 largest Lyapunov exponents for randomly sampled Hamiltonians were numerically computed. Our results strongly suggest the absence of integrable nearest-neighbor Hamiltonians for the infinite lattices except for the trivial Ising case. In the vicinity of the Ising case, the largest Lyapunov exponents exhibit a power-law growth, while further away they become rather weakly sensitive to the Hamiltonian anisotropy. We also provide an analytical derivation of these results.

Abstract. We numerically investigate Lyapunov instabilities for one-, two-and three-dimensional lattices of interacting classical spins at infinite temperature. We obtain the largest Lyapunov exponents for a very large variety of nearestneighbor spin-spin interactions and complete Lyapunov spectra in a few selected cases. We investigate the dependence of the largest Lyapunov exponents and whole Lyapunov spectra on the lattice size and find that both quickly become size-independent. Finally, we analyze the dependence of the largest Lyapunov exponents on the anisotropy of spin-spin interaction with the particular focus on the difference between bipartite and nonbipartite lattices.

Extracting reliable indicators of chaos from a single experimental time series is a challenging task, in particular, for systems with many degrees of freedom. The techniques available for this purpose often require unachievably long time series. In this paper, we explore a method of discriminating chaotic from multi-periodic integrable motion in many-particle systems. The applicability of this method is supported by our numerical simulations of the dynamics of classical spin lattices at high temperatures. We compared chaotic and nonchaotic regimes of these lattices and investigated the transition between the two. The method is based on analyzing higher-order time derivatives of the time series of a macroscopic observable-the total magnetization of the spin lattice. We exploit the fact that power spectra of the magnetization time series generated by chaotic spin lattices exhibit exponential high-frequency tails, while, for the integrable spin lattices, the power spectra are terminated in a non-exponential way. We have also demonstrated the applicability limits of the above method by investigating the high-frequency tails of the power spectra generated by quantum spin lattices and by the classical Toda lattice.