Nonlinear energy transfer in classical and quantum systems

We investigate the effect of slowly-varying parameter on the energy transfer in a system of weakly coupled nonlinear oscillators, with special attention to a mathematical analogy between the classical energy transfer and quantum transitions. For definiteness, we consider a system of two weakly coupled oscillators with cubic nonlinearity, in which the oscillator with constant parameters is excited by an initial impulse, while a coupled oscillator with slowly-varying parameters is initially at rest. It is proved that the equations of the slow passage through resonance in this system are identical to equations of the nonlinear Landau-Zener (LZ) tunneling. Three types of dynamical behavior are distinguished, namely, quasi-linear, moderately nonlinear and strongly nonlinear. The quasi-linear systems exhibit a gradual energy transfer from the excited to the attached oscillator, while the moderately nonlinear systems are characterized by an abrupt transition from the energy localization on the excited oscillator to the localization on the attached oscillator. In the strongly nonlinear systems, the transition from the energy localization to strong energy exchange between the oscillators is revealed. A special case of the rapid irreversible energy transfer in the strongly nonlinear system with slowly-varying parameters is also investigated. The conditions providing different types of the dynamical behavior are derived. Explicit approximate solutions describing the transient processes in moderately and strongly nonlinear systems are suggested. Correctness of the constructed approximations is confirmed by numerical results

Non-stationary resonance dynamics of weakly coupled pendula

In this paper we fill the gap in understanding the non-stationary resonance dynamics of the weakly coupled pendula model, having significant applications in numerous fields of physics such as super- conducting Josephson junctions, Bose-Einstein condensates, DNA, etc.. While common knowledge of the problem is based on two alternative limiting asymptotics, namely the quasi-linear approach and the approximation of independent pendula, we present a unified description in the framework of new concept of Limiting Phase Trajectories (LPT), without any restriction on the amplitudes of oscillation. As a result the conditions of intense energy exchange between the pendula and transition to energy localization are revealed in all possible diapason of initial conditions. By doing so, the roots and the domain of chaotic behavior are clarified as they are associated with this transition while simultaneously approaching the pendulum separatrix. The analytical findings are corrobo- rated by numerical simulations. By considering the simplest case of two weakly coupled pendula, we pave the ground for new opening possibilities of significant extensions in both fundamental and applied directions.Comment: 7 pages, 7 figure

Non-stationary resonance dynamics of the harmonically forced pendulum

The stationary and highly non-stationary resonant dynamics of the harmonically forced pendulum are described in the framework of a semi-inverse procedure combined with the Limiting Phase Trajectory concept. This procedure, implying only existence of slow time scale, permits one to avoid any restriction on the oscillation amplitudes. The main results relating to the dynamical bifurcation thresholds are represented in a closed form. The small parameter defining the separation of the time scales is naturally identified in the ana- lytical procedure. Considering the pendulum frequency as the control parameter we reveal two qualitative tran- sitions. One of them corresponding to stationary instability with formation of two additional stationary states, the other, associated with the most intense energy drawing from the source, at which the amplitude of pendulum oscillations abruptly grows. Analytical predictions of both bifurcations are verified by numerical integration of original equation. It is also shown that occurrence of chaotic domains may be strongly connected with the second transition

Non-conventional phase attractors and repellers in weakly coupled autogenerators with hard excitation

In our earlier studies, we found the effect of non-conventional synchronization, which is a specific type of nonlinear stable beating in the system of two weakly coupled autogenerators with hard excitation given by generalized van der Pol-Duffing characteristics. The corresponding synchronized dynamics are due to a new type of attractor in a reduced phase space of the system. In the present work, we show that, as the strength of nonlinear stiffness and dissipation are changing, the phase portrait undergoes a complicated evolution leading to a quite unexpected appearance of difficult to detect repellers separating a stable limit cycle and equilibrium points in the phase plane. In terms of the original coordinates, the limit cycle associates with nonlinear beatings while the stationary points correspond to the stationary synchronous dynamics similar to the so-called nonlinear local modes

Stationary and non-stationary resonance dynamics of the finite chain of weakly coupled pendula

We discuss new phenomena of energy localization and transition to chaos in the finite system of coupled pendula (which is a particular case of the Frenkel-Kontorova model), without any restrictions on the amplitudes of oscillations. The direct significant applications of this fundamental model comprise numerous physical systems. In the infinite and continuum limit the considered model is reduced to integrable sine-Gordon equation or certain non-integrable generalizations of it. In this limit, the chaotization is absent, and the energy localization is indicated by the existence of soliton-like solutions (kinks and breathers). As for more realistic finite models, analytical approaches are lacking, with the exception of cases limited to two and three pendula. We propose a new approach to the problem based on the recently developed Limiting Phase Trajectory (LPT) concept in combination with a semi-inverse method. The analytical predictions of the con-ditions providing transition to energy localization are confirmed by numerical simulation. It is shown that strongly nonlinear effects in finite chains tend to disap- pear in the infinite limit

Semi-Inverse Method in the Nonlinear Dynamics

The semi-inverse method based on using an internal small parameter of the nonlinear problems is discussed on the examples of the chain of coupled pendula and of the forced pendulum. The efficiency of such approach is highly appeared when the non-stationary dynamical problems are considered. In the framework of this method we demonstrate that both the spectrum of nonlinear normal modes and the interaction of them can be analysed successfully

Nonlinear vibrations and energy distribution of Single-Walled Carbon Nanotubes

The nonlinear vibrations of Single-Walled Carbon Nanotubes are analysed. The Sanders-Koiter elastic shell theory is applied in order to obtain the elastic strain energy and kinetic energy. The carbon nanotube deformation is described in terms of longitudinal, circumferential and radial displacement fields. The theory considers geometric nonlinearities due to large amplitude of vibration. The displacement fields are expanded by means of a double series based on harmonic functions for the circumferential variable and Chebyshev polynomials for the longitudinal variable. The Rayleigh-Ritz method is applied in order to obtain approximate natural frequencies and mode shapes. Free boundary conditions are considered. In the nonlinear analysis, the three displacement fields are re-expanded by using approximate eigenfunctions. An energy approach based on the Lagrange equations is considered in order to obtain a set of nonlinear ordinary differential equations. The energy distribution of the system is studied by considering combinations of different vibration modes. The effect of the conjugate modes participation on the energy distribution is analysed