Transient and chaotic low-energy transfers in a system with bistable nonlinearity

The low-energy dynamics of a two-dof system composed of a grounded linear oscillator coupled to a lightweight mass by means of a spring with both cubic nonlinear and negative linear components is investigated. The mechanisms leading to intense energy exchanges between the linear oscillator, excited by a low-energy impulse, and the nonlinear attachment are addressed. For lightly damped systems, it is shown that two main mechanisms arise: Aperiodic alternating in-well and cross-well oscillations of the nonlinear attachment, and secondary nonlinear beats occurring once the dynamics evolves solely in-well. The description of the former dissipative phenomenon is provided in a two-dimensional projection of the phase space, where transitions between in-well and cross-well oscillations are associated with sequences of crossings across a pseudo-separatrix. Whereas the second mechanism is described in terms of secondary limiting phase trajectories of the nonlinear attachment under certain resonance conditions. The analytical treatment of the two aformentioned low-energy transfer mechanisms relies on the reduction of the nonlinear dynamics and consequent analysis of the reduced dynamics by asymptotic techniques. Direct numerical simulations fully validate our analytical predictions

Limiting phase trajectories and the origin of energy localization in nonlinear oscillatory chains

We demonstrate that the modulation instability of the zone boundary mode in a finite (periodic) Fermi-Pasta-Ulam chain is the necessary but not sufficient condition for the efficient energy transfer by localized excitations. This transfer results from the exclusion of complete energy exchange between spatially different parts of the chain, and the excitation level corresponding to that turns out to be twice more than threshold of zone boundary mode's instability. To obtain this result one needs in far going extension of the beating concept to a wide class of finite oscillatory chains. In turn, such an extension leads to description of energy exchange and transition to energy localization and transfer in terms of 'effective particles' and Limiting Phase Trajectories. The 'effective particles' appear naturally when the frequency spectrum crowding ensures the resonance interaction between zone boundary and two nearby nonlinear normal modes, but there are no additional resonances. We show that the Limiting Phase Trajectories corresponding to the most intensive energy exchange between 'effective particles' can be considered as an alternative to Nonlinear Normal Modes, which describe the stationary process

Energy exchange and transition to localization in the asymmetric Fermi-Pasta-Ulam oscilliatory chain

A finite (periodic) FPU chain is chosen as a convenient point for investigating the energy exchange phenomenon in nonlinear oscillatory systems. As we have recently shown, this phenomenon may occur as a consequence of the resonant interaction between high-frequency nonlinear normal modes. This interaction determines both the complete energy exchange between different parts of the chain and the transition to energy localization in an excited group of particles. In the paper, we demonstrate that this mechanism can exist in realistic (asymmetric) models of atomic or molecular oscillatory chains. Also, we study the resonant interaction of conjugated nonlinear normal modes and prove a possibility of linearization of the equations of motion. The theoretical constructions developed in this paper are based on the concepts of "effective particles" and Limiting Phase Trajectories. In particular, an analytical description of energy exchange between the "effective particles" in the terms of non-smooth functions is presented. The analytical results are confirmed with numerical simulations.Comment: 15 pages, 8 figure

Discrete breathers in polyethylene chain

The existence of discrete breathers (DBs), or intrinsic localized modes (localized periodic oscillations of transzigzag) is shown. In the localization region periodic contraction-extension of valence C-C bonds occurs which is accompanied by decrease-increase of valence angles. It is shown that the breathers present in thermalized chain and their contribution dependent on temperature has been revealed.Comment: 5 pages, 6 figure

Nonlinear optical vibrations of single-walled carbon nanotubes.

We demonstrate a new specific phenomenon of the long-time resonant energy exchange in carbon nanotubes (CNTs), which is realized by two types of optical vibrations, the Circumferential Flexure Mode (CFM) and the Radial Breathing Mode (RBM). We show that the modified nonlinear Schrdinger equation, obtained in the framework of the nonlinear theory of elastic thin shells, allows us to describe the nonlinear dynamics of CNTs for specified frequency bands. Comparative analysis of the oscillations of the CFM and RBM branches shows the qualitative difference of nonlinear effects for these branches. While the nonlinear resonant interaction of the low-frequency modes in the CFM branch leads to energy capture in some domains of the CNT, the same interaction in the RBM branch does not demonstrate any tendency for energy localization. The reason lies in the distinction in the nonlinear terms in the equations of motion. While CFMs are characterized by soft polynomial nonlinearity, RBM dynamics is characterized by hard gradient nonlinearity. Moreover, in contrast to the CFM, the importance of nonlinearity in the case of RBM oscillations decreases as the length to radius ratio increases. Numerical integration of the equations of thin shell theory confirms the results of the analytical study

Nonlinear optical vibrations of single-walled carbon nanotubes.

We demonstrate a new specific phenomenon of the long-time resonant energy exchange in carbon nanotubes (CNTs), which is realized by two types of optical vibrations, the Circumferential Flexure Mode (CFM) and the Radial Breathing Mode (RBM). We show that the modified nonlinear Schrdinger equation, obtained in the framework of the nonlinear theory of elastic thin shells, allows us to describe the nonlinear dynamics of CNTs for specified frequency bands. Comparative analysis of the oscillations of the CFM and RBM branches shows the qualitative difference of nonlinear effects for these branches. While the nonlinear resonant interaction of the low-frequency modes in the CFM branch leads to energy capture in some domains of the CNT, the same interaction in the RBM branch does not demonstrate any tendency for energy localization. The reason lies in the distinction in the nonlinear terms in the equations of motion. While CFMs are characterized by soft polynomial nonlinearity, RBM dynamics is characterized by hard gradient nonlinearity. Moreover, in contrast to the CFM, the importance of nonlinearity in the case of RBM oscillations decreases as the length to radius ratio increases. Numerical integration of the equations of thin shell theory confirms the results of the analytical study