Bifurcation of Gap Solitons in Coupled Mode Equations in d Dimensions

We consider a system of first order coupled mode equations in $${\mathbb {R}}^d$$ R d describing the envelopes of wavepackets in nonlinear periodic media. Under the assumptions of a spectral gap and a generic assumption on the dispersion relation at the spectral edge, we prove the bifurcation of standing gap solitons of the coupled mode equations from the zero solution. The proof is based on a Lyapunov–Schmidt decomposition in Fourier variables and a nested Banach fixed point argument. The reduced bifurcation equation is a perturbed stationary nonlinear Schrödinger equation. The existence of solitary waves follows in a symmetric subspace thanks to a spectral stability result. A numerical example of gap solitons in $${\mathbb {R}}^2$$ R 2 is provided.

Study of rigidity of a first-order algebro-differential system with perturbation in the right-hand side

The rigidity of a dynamical system described by a first-order differential equationwith an irreversible operator at the highest derivative is investigated. The system is perturbed by an operator addition of the order of the second power of a small parameter. Conditions under which the system is robust with respect to these disturbances are determined as well as conditions under which the influence of disturbances is significant. For this, the bifurcation equation is derived. It is used to set the type of boundary layer functions. As an example, we investigate the initial boundary value problem for a system of partial differential equations with a mixed second partial derivative which occurs in the study of the processes of sorption anddesorption of gases, drying processes, etc.

Bifurcations of Double Homoclinic Loops in Reversible Systems

This paper is devoted to the study of bifurcation phenomena of double homoclinic loops in reversible systems. With the aid of a suitable local coordinate system, the Poincaré map is constructed. By means of the bifurcation equation, we perform a detailed study to obtain fruitful results, and demonstrate the existence of the R-symmetric large homoclinic orbit of new type near the primary double homoclinic loops, the existence of infinitely many R-symmetric periodic orbits accumulating onto the R-symmetric large homoclinic orbit, and the coexistence of R-symmetric large homoclinic orbit and the double homoclinic loops. The homoclinic bellow can also be found under suitable perturbation. The relevant bifurcation surfaces and the existence regions are located.

Nonlinear interaction between self- and parametrically excited wind-induced vibrations

The aeroelastic behavior of a planar prismatic visco-elastic structure, subject to a turbulent wind, flowing orthogonally to its plane, is studied in the nonlinear field. The steady component of wind is responsible for a Hopf bifurcation occurring at a threshold critical value; the turbulent component, which is assumed to be a small harmonic perturbation of the former, is responsible for parametric excitation. The interaction between the two bifurcations is studied in a three-dimensional parameter space, made of the two wind amplitudes and the frequency of the turbulence. Aeroelastic forces are computed by the quasi-static theory. A one-D.O.F dynamical system, drawn by a Galerkin projection of the continuous model, is adopted. The multiple scale method is applied, to get a two-dimensional bifurcation equation. A linear stability analysis is carried out to determine the loci of periodic and quasi-periodic bifurcations. Limit cycles and tori are computed by exact, asymptotic, and numerical solutions of the bifurcation equations. Numerical results are obtained for a sample structure, and compared with finite-difference solutions of the original partial differential equation of motion.

The effect of an exterior electric field on the instability of dielectric plates

We investigate the theoretical nonlinear response, Hessian stability, and possible wrinkling behaviour of a voltage-activated dielectric plate immersed in a tank filled with silicone oil. Fixed rigid electrodes are placed on the top and bottom of the tank, and an electric field is generated by a potential difference between the electrodes. We solve the associated incremental boundary value problem of superimposed, inhomogeneous small-amplitude wrinkles, signalling the onset of instability. We decouple the resulting bifurcation equation into symmetric and antisymmetric modes. For a neo-Hookean dielectric plate, we show that a potential difference between the electrodes can induce a thinning of the plate and thus an increase of its planar area, similar to the scenarios encountered when there is no silicone oil. However, we also find that, depending on the material and geometric parameters, an increasing applied voltage can also lead to a thickening of the plate, and thus a shrinking of its area. In that scenario, Hessian instability and wrinkling bifurcation may then occur spontaneously once some critical voltages are reached.

Nonlinear Dynamics Analysis for the Taut Inclined Cable Excited by Deck Vibration

A simple mechanical model is proposed to describe the dynamics system of the taut inclined cable excited by the deck vibration. Using Galerkin method, the dynamics system is simplified into a one-degree of freedom nonlinear system. Average method is used to obtain the average equations and bifurcation equation. Bifurcation maps are obtained and used to reveal the evolution mechanism of period-1 motion to period-2 motion. Several motion forms are discussed and evolution progress between different motions is studied. Several kinds of parametric resonance are revealed in various parameter regions. The conclusions indicate that the parametric resonance excited by the frequency ratio 3:2 is significant should be paid more attention.

Bifurcations in Delay Differential Equations: An Algorithmic Approach in Frequency Domain

In this work we study local oscillations in delay differential equations with a frequency domain methodology. The main result is a bifurcation equation from which the existence and expressions of local periodic solutions can be determined. We present an iterative method to obtain the bifurcation equation up to a fixed arbitrary order. It is shown how this method can be implemented in symbolic math programs.

Analysis of the High Order Terms on Periodic Solution Bifurcations of the Generalized Duffing Systems

This paper focuses on the effectsof the high order term in Duffing equation. Firstly the averaging equation andthe bifurcation equation are deduced through the multiple scale method.Secondly, the transition sets and several different bifurcation diagrams areobtained based on the singularity theory. The result shows that the high order term induces richer bifurcationcharacteristics.

Resonant Homoclinic Flips Bifurcation in Principal Eigendirections

A codimension-4 homoclinic bifurcation with one orbit flip and one inclination flip at principal eigenvalue direction resonance is considered. By introducing a local active coordinate system in some small neighborhood of homoclinic orbit, we get the Poincaré return map and the bifurcation equation. A detailed investigation produces the number and the existence of 1-homoclinic orbit, 1-periodic orbit, and double 1-periodic orbits. We also locate their bifurcation surfaces in certain regions.