Multi-site breathers in Klein-Gordon lattices: stability, resonances, and bifurcations

We prove the most general theorem about spectral stability of multi-site breathers in the discrete Klein-Gordon equation with a small coupling constant. In the anti-continuum limit, multi-site breathers represent excited oscillations at different sites of the lattice separated by a number of "holes" (sites at rest). The theorem describes how the stability or instability of a multi-site breather depends on the phase difference and distance between the excited oscillators. Previously, only multi-site breathers with adjacent excited sites were considered within the first-order perturbation theory. We show that the stability of multi-site breathers with one-site holes change for large-amplitude oscillations in soft nonlinear potentials. We also discover and study a symmetry-breaking (pitchfork) bifurcation of one-site and multi-site breathers in soft quartic potentials near the points of 1:3 resonance.Comment: 34 pages, 12 figure

Energy dissipation in a nonlinearly damped Duffing oscillator

International audienceIn this paper, we study the effect of including a nonlinear damping term proportional to the power of the velocity in the dynamics of a double-well Duffing oscillator. In particular, we focus our attention in understanding how the energy dissipates over a cycle and along the time, by the use of different tools of analysis. Analytical and numerical results for different damping terms are shown, and the presence of a discontinuity and an inversion of behavior depending on the initial energy are discussed. An averaged power loss in a period is defined, showing similar characteristics as the energy dissipation over a cycle, although no discontinuity is present. The discontinuity gap which appears for the energy dissipation at a certain value of the initial energy decreases as the power of the damping term increases and an associated scaling law is found

Investigation of a coupled duffing oscillator system in a varying potential field

Nonlinear systems are known to exhibit widely differing steady-state behaviors based on small modifications to the control parameters within the equations. These small modifications may be the difference between a chaotic output and a periodic output. Many investigators choose to study the varying behaviors through varying forcing conditions, specifically the forcing amplitude or frequency. However, from a linear vibration theory standpoint, systems are often tuned to minimize system response to a known forcing input by varying the strength of the damping and stiffness elements within the system. It may also be the case that the parameters governing the strengths of these elements are constant, but uncertain within a specific range. In these cases, it is more advantageous to understand how the response will vary based on these design parameters or uncertain constants. The parameters defining the potential field for a nonlinearly coupled Duffing oscillator system were used as the control parameters in this study. The steady-state system response was investigated through the techniques of Poincaré maps, bifurcation diagrams, Lyapunov exponents and spectra, power spectra estimates, and phase portrait projections. The system of equations was integrated through a semi-discrete algorithm based on continuous transformation group theory, which improved the accuracy of the integrated trajectories and the accuracy of the Lyapunov exponents. Additionally the Poincaré maps, power spectra estimates, and phase portrait projections were animated to simplify the analysis of the varying parameters. The use of these animations saved countless hours of analysis time, and revealed details of the parameter-based variations that would not have been observed otherwise. This technique has not been used in any of the known literature. Two separate forcing conditions were considered; synchronous sinusoidal forcing on each oscillator and nonsynchronous forcing. Each system exhibits a wide variety of nonlinear phenomena including period-doubling sequences, quasiperiodicity, and chaos. The existence of the merging of chaotic attractors and hysteresis was also confirmed. This study also suggests the hyperchaotic attractors and chaotic tori may be present under certain parameter combinations

A moment-equation-copula-closure method for nonlinear vibrational systems subjected to correlated noise

We develop a moment equation closure minimization method for the inexpensive approximation of the steady state statistical structure of nonlinear systems whose potential functions have bimodal shapes and which are subjected to correlated excitations. Our approach relies on the derivation of moment equations that describe the dynamics governing the two-time statistics. These are combined with a non-Gaussian pdf representation for the joint response-excitation statistics that has i) single time statistical structure consistent with the analytical solutions of the Fokker-Planck equation, and ii) two-time statistical structure with Gaussian characteristics. Through the adopted pdf representation, we derive a closure scheme which we formulate in terms of a consistency condition involving the second order statistics of the response, the closure constraint. A similar condition, the dynamics constraint, is also derived directly through the moment equations. These two constraints are formulated as a low-dimensional minimization problem with respect to unknown parameters of the representation, the minimization of which imposes an interplay between the dynamics and the adopted closure. The new method allows for the semi-analytical representation of the two-time, non-Gaussian structure of the solution as well as the joint statistical structure of the response-excitation over different time instants. We demonstrate its effectiveness through the application on bistable nonlinear single-degree-of-freedom energy harvesters with mechanical and electromagnetic damping, and we show that the results compare favorably with direct Monte-Carlo Simulations