The physics of nonlinear dynamics has been studied in detail in macroscopic mechanical systems like the driven classical pendulum. By now, it is possible to build mechanical devices on the nanometer scale with eigenfrequencies on the order of several 100 MHz. In this work, we want to present how to machine such nanomechanical resonators out of silicon-on-insulator wafers and how to operate them in the nonlinear regime in order to investigate higher-order mechanical mixing at radio frequencies. The nonlinear response then is compared in detail to $n\mathrm{th}\mathrm{-order}$ perturbation theory and nonperturbative numerical calculations.

We conceive a new recurrence quantifier for time series based on the concept of information entropy, in which the probabilities are associated with the presence of microstates defined on the recurrence matrix as small binary submatrices. The new methodology to compute the entropy of a time series has advantages compared to the traditional entropies defined in the literature, namely, a good correlation with the maximum Lyapunov exponent of the system and a weak dependence on the vicinity threshold parameter. Furthermore, the new method works adequately even for small segments of data, bringing consistent results for short and long time series. In a case where long time series are available, the new methodology can be employed to obtain high precision results since it does not demand large computational times related to the analysis of the entire time series or recurrence matrices, as is the case of other traditional entropy quantifiers. The method is applied to discrete and continuous systems.

Sleep plays a crucial role in the regulation of body homeostasis and rhythmicity in mammals. Recently, a specific component of the sleep structure has been proposed as part of its homeostatic mechanism, named micro-arousal. Here, we studied the unique progression of the dynamic behavior of cortical and hippocampal local field potentials (LFPs) during slow-wave sleep-related to motor-bursts (micro-arousals) in mice. Our main results comprised: (i) an abrupt drop in hippocampal LFP amplitude preceding micro-arousals which persisted until the end of motor-bursts (we defined as t interval, around 4s) and a similar, but delayed amplitude reduction in cortical (S1/M1) LFP activity occurring at micro-arousal onset; (ii) two abrupt frequency jumps in hippocampal LFP activity: from Theta (6–12 Hz) to Delta (2–4 Hz), also t seconds before the micro-arousal onset, and followed by another frequency jump from Delta to Theta range (5–7 Hz), now occurring at micro-arousal onset; (iii) a pattern of cortico-hippocampal frequency communication precedes micro-arousals: the analysis between hippocampal and cortical LFP fluctuations reveal high coherence during τ interval in a broader frequency band (2–12 Hz), while at a lower frequency band (0.5–2 Hz) the coherence reaches its maximum after the onset of micro-arousals. In conclusion, these novel findings indicate that oscillatory dynamics pattern of cortical and hippocampal LFPs preceding micro-arousals could be part of the regulatory processes in sleep architecture.

We study the approach to equipartition in the Fermi-Pasta-Ulam oscillator chain with quartic nonlinearity Fermi-Pasta-Ulam–(β system) starting from generic high-frequency-mode initial conditions. Typically 90% of the energy is placed in one high-frequency mode, with 10% in adjacent modes. The mode energy is found to distribute itself into first a number of localized structures which coalesce over time into a single localized structure, a chaotic breather (CB). Over longer times the CB is found to break up, with energy transferred to lower frequency modes which do not have the breather symmetry. A transition with decreasing initial mode frequency is found such that the CB does not form, as expected from the loss of breather symmetry. The scaling of CB formation time with energy density, $E/N,$ is found to be $T_b\propto {\mathrm{}(E/N)\mathrm{}}^{-1},$ and the scaling of equipartition time found to be $T_{\mathrm{eq}}\propto {\mathrm{}(E/N)\mathrm{}}^{-2}.$ The scaling of $T_{\mathrm{eq}}$ can be predicted from an argument which postulates stochastic diffusion from high-frequency-mode chaotic beat oscillations to the low-frequency modes. The theory also predicts that a miminum value of $E/N$ exists below which $T_{\mathrm{eq}}$ should increase more rapidly with $E/N$ than in the power law range, and this transition has been found numerically.

The recurrence analysis of dynamic systems has been studied since Poincaré’s seminal work. Since then, several approaches have been developed to study recurrence properties in nonlinear dynamical systems. In this work, we study the recently developed entropy of recurrence microstates. We propose a new quantifier, the maximum entropy (${\mathrm{S}}_{\text{max}}$ ). The new concept uses the diversity of microstates of the recurrence plot and is able to set automatically the optimum recurrence neighborhood ($ϵ$ —vicinity), turning the analysis free of the vicinity parameter. In addition, $ϵ$ turns out to be a novel quantifier of dynamical properties itself. We apply ${\mathrm{S}}_{\text{max}}$ and the optimum $ϵ$ to deterministic and stochastic systems. The ${\mathrm{S}}_{\text{max}}$ quantifier has a higher correlation with the Lyapunov exponent and, since it is a parameter-free measure, a more useful recurrence quantifier of time series.