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We explore the chaotic dynamics of a large one-dimensional lattice of coupled maps with diffusive coupling of varying strength using the covariant Lyapunov vectors (CLVs). Using a lattice of diffusively coupled quadratic maps, we quantify the growth of spatial structures in the chaotic dynamics as the strength of diffusion is increased. When the diffusion strength is increased from zero, we find that the leading Lyapunov exponent decreases rapidly from a positive value to zero to yield a small window of periodic dynamics which is then followed by chaotic dynamics. For values of the diffusion strength beyond the window of periodic dynamics, the leading Lyapunov exponent does not vary significantly with the strength of diffusion with the exception of a small variation for the largest diffusion strengths we explore. The Lyapunov spectrum and fractal dimension are described analytically as a function of the diffusion strength using the eigenvalues of the coupling operator. The spatial features of the CLVs are quantified and compared with the eigenvectors of the coupling operator. The chaotic dynamics are composed entirely of physical modes for all of the conditions we explore. The leading CLV is highly localized and localization decreases with increasing strength of the spatial coupling. The violation of the dominance of Oseledets splitting indicates that the entanglement of pairs of CLVs becomes more significant between neighboring CLVs as the strength of diffusion is increased. 

Even a relatively weak drive force is enough to push a typical nanomechanical resonator into the nonlinear regime. Consequently, nonlinearities are widespread in nanomechanics and determine the critical characteristics of nanoelectromechanical systems' (NEMSs) resonators. A thorough understanding of the nonlinear dynamics of higher eigenmodes of NEMS resonators would be beneficial for progress, given their use in applications and fundamental studies. Here, we characterize the nonlinearity and the linear dynamic range (LDR) of each eigenmode of two nanomechanical beam resonators with different intrinsic tension values up to eigenmode n = 11. We find that the modal Duffing constant increases as n4, while the critical amplitude for the onset of nonlinearity decreases as 1/n. The LDR, determined from the ratio of the critical amplitude to the thermal noise amplitude, increases weakly with n. Our findings are consistent with our theory treating the beam as a string, with the nonlinearity emerging from stretching at high amplitudes. These scaling laws, observed in experiments and validated theoretically, can be leveraged for pushing the limits of NEMS-based sensing even further. 

We explore the dynamics of a nanoscale doubly clamped beam that is under high tension, immersed in a viscous fluid, and driven externally by a spatially varying drive force. We develop a theoretical description that is valid for all possible values of tension, includes the motion of the higher modes of the beam, and accounts for a harmonic force that is applied over a limited spatial region of the beam near its ends. We compare our theoretical predictions with experimental measurements for a nanoscale beam that is driven electrothermally and immersed in air and water. The theoretical predictions show good agreement with experiments, and the validity of a simplified string approximation is demonstrated.