Valeri V. Smirnov

Localization of Low-Frequency Oscillations in Single-Walled Carbon Nanotubes

In the framework of the continuum shell theory, we analytically predict a new phenomenon: the weak localization of optical low-frequency oscillations in carbon nanotubes. We clarify the origin of the localization by means of the concept of the limiting phase trajectory and confirm the obtained analytical results by molecular dynamics simulations of simply supported carbon nanotubes. The performed analysis contributes to the new universal approach to the treatment of nonstationary resonant processes.

CISM Courses and Lectures: Resonant energy exchange in nonlinear oscillatory chains and Limiting Phase Trajectories: from small to large systems

We present an adequate analytical approach to the description of nonlinear vibration with strong energy exchange between weakly coupled oscillators and oscillatory chains. The fundamental notion of the limiting phase trajectory (LPT) corresponding to complete energy exchange is introduced. In certain sense this is an alternative to the nonlinear normal mode (NNM) characterized by complete energy conservation. Well-known approximations based on NNMs turn out to be valid for the case of weak energy exchange, and the proposed approach can be used for the description of nonlinear processes with strong energy exchange between weakly coupled oscillators or oscillatory chains. Such a description is formally similar to that of a vibro-impact process and can be considered as starting approximation when dealing with other processes with intensive energy transfer. At first we propose a simple analytical description of vibrations of nonlinear oscillators. We show that two dynamical transitions occur in the system. First of them corresponds to the bifurcation of anti-phase vibrations of oscillators. And the second one is caused by coincidence of LPT with separatrix dividing two stable stationary states and leads to qualitative change in both phase and temporal behavior of the LPT (in particular, temporal dependence of the amplitude becomes resembling that for vibro-impact vibrations). Next problem under consideration relates to intensive inter modal exchange in the periodic nonlinear systems with finite (n>2) number of degrees of freedom. We consider two limiting cases. If the number of particles is not large enough, the energy exchange between nonlinear normal modes in two-dimensional integral manifolds is considered. When the number of the particles increases the energy exchange between neighbor integral manifolds becomes important that leads to formation of the localized excitations resembling the breathers in the one-dimensional continuum media. Finally, the breathers in the infinite systems of complex (helix) structure are presented.

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

AbstractA finite (periodic) FPU chain is chosen as a convenient model for investigating thenenergy exchange phenomenon in nonlinear oscillatory systems. As we have recently shown,nthis phenomenon may occur as a consequence of the resonant interaction betweennhigh-frequency nonlinear normal modes. This interaction determines both the completenenergy exchange between different parts of the chain and the transition to energynlocalization in an excited group of particles. In the paper, we demonstrate that thisnmechanism can exist in realistic (asymmetric) models of atomic or molecular oscillatorynchains. Also, we study the resonant interaction of conjugated nonlinear normal modes andnprove a possibility of linearization of the equations of motion. The theoreticalnconstructions developed in this paper are based on the concepts of “effective particles”nand Limiting Phase Trajectories. In particular, an analytical description of energynexchange between the “effective particles” in the terms of non-smooth functions isnpresented. The analytical results are confirmed with numerical simulations.n

Large-amplitude nonlinear normal modes of the discrete sine lattices

We present an analytical description of the large-amplitude stationary oscillations of the finite discrete system of harmonically coupled pendulums without any restrictions on their amplitudes (excluding a vicinity of π). Although this model has numerous applications in different fields of physics, it was studied earlier in the infinite limit only. The discrete chain with a finite length can be considered as a well analytical analog of the coarse-grain models of flexible polymers in the molecular dynamics simulations. The developed approach allows to find the dispersion relations for arbitrary amplitudes of the nonlinear normal modes. We emphasize that the long-wavelength approximation, which is described by well-known sine-Gordon equation, leads to an inadequate zone structure for the amplitudes of about π/2 even if the chain is long enough. An extremely complex zone structure at the large amplitudes corresponds to multiple resonances between nonlinear normal modes even with strongly different wave numbers. Due to the complexity of the dispersion relations the modes with shorter wavelengths may have smaller frequencies. The stability of the nonlinear normal modes under condition of the resonant interaction are discussed. It is shown that this interaction of the modes in the vicinity of the long wavelength edge of the spectrum leads to the localization of the oscillations. The thresholds of instability and localization are determined explicitly. The numerical simulation of the dynamics of a finite-length chain is in a good agreement with obtained analytical predictions.

Revolution of Pendula: Rotational Dynamics of the Coupled Pendula

The analysis of the rotational dynamics of two coupled pendula is presented. The description of the oscillations of the pendulum on the background of the rotation with the average velocity was performed by the asymptotic method for the single pendulum. The source and the significance of the formation of the Limiting Phase Trajectory is clarified. The stability analysis of the rotation of two coupled pendula shows a qualitative difference between in-phase and out-of-phase rotational modes. It is shown that the origin of the in-phase rotation instability is its parametric excitation by the out-of-phase perturbations. The domain of in-phase rotation instability has been determined in the space of the system parameters. The analytic results are confirmed by the numerical simulation data.