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In this paper, we analyze the dynamic effect of a reservoir computer (RC) on its performance. Modified Kuramoto's coupled oscillators are used to model the RC, and synchronization, Lyapunov spectrum (and dimension), Shannon entropy, and the upper bound of the Kolmogorov-Sinai entropy are employed to characterize the dynamics of the RC. The performance of the RC is analyzed by reproducing the distribution of random, Gaussian, and quantum jumps series (shelved states) since a replica of the time evolution of a completely random series is not possible to generate. We demonstrate that hyperchaotic motion, moderate Shannon entropy, and a higher degree of synchronization of Kuramoto's oscillators lead to the best performance of the RC. Therefore, an appropriate balance of irregularity and order in the oscillator's dynamics leads to better performances., (© 2024 Author(s). Published under an exclusive license by AIP Publishing.)

The phase space dynamics of higher dimensional nonintegrable conservative systems is characterized via the effect of "sticky" motion on the finite time Lyapunov exponents (FTLEs) distribution. Since a chaotic trajectory suffers the sticky effect when chaotic motion is mixed to the regular one, it offers a way to separate the mixed from the totally chaotic regimes. To detect stickiness, four different measures are used, related to the distributions of the positive FTLEs, and provide conditions to characterize the dynamics. Conservative maps are systematically studied from the uncoupled two-dimensional case up to coupled maps of dimension 20. Sticky motion is detected in all unstable directions above a threshold K(d) of the nonlinearity parameter K for the high dimensional cases d = 10, 20. Moreover, as K increases we can clearly identify the transition from mixed to totally chaotic motion which occurs simultaneously in all unstable directions. Results show that all four statistical measures sensitively characterize the motion in high dimensional systems.

The fragmentation of diatomic molecules under a stochastic force is investigated both classically and quantum mechanically, focusing on their dissociation probabilities. It is found that the quantum system is more robust than the classical one in the limit of a large number of kicks. The opposite behavior emerges for a small number of kicks. Quantum and classical dissociation probabilities do not coincide for any parameter combinations of the force. This can be attributed to a scaling property in the classical system which is broken quantum mechanically.

We numerically compare the semiclassical "frozen Gaussian" Herman-Kluk propagator [Chem. Phys. 91, 27 (1984)] and the "thawed Gaussian" propagator put forward recently by Baranger et al. [J. Phys. A 34, 7227 (2001)] by studying the quantum dynamics in some nonlinear one-dimensional potentials. The reasons for the lack of long-time accuracy and norm conservation in the latter method are uncovered. We amend the thawed Gaussian propagator with a global harmonic approximation for the stability of the trajectories and demonstrate that this revised propagator is a true alternative to the Herman-Kluk propagator with similar accuracy., ((c) 2004 American Institute of Physics)