Symmetry-breaking bifurcation for the one-dimensional Hénon equation

We show the existence of a symmetry-breaking bifurcation point for the one-dimensional Hénon equation [Formula: see text] where [Formula: see text] and [Formula: see text]. Moreover, employing a variant of Rabinowitz’s global bifurcation, we obtain the unbounded connected set (the first of the alternatives about Rabinowitz’s global bifurcation), which emanates from the symmetry-breaking bifurcation point. Finally, we give an example of a bounded branch connecting two symmetry-breaking bifurcation points (the second of the alternatives about Rabinowitz’s global bifurcation) for the problem [Formula: see text] where [Formula: see text] is a specified continuous parametrization function.

A Study on the Symmetry-Breaking Bifurcation Phenomenon in Transportation of Low-Strength Composites Sheets by Rolls

In the manufacturing of low-strength composites sheets, the symmetry-breaking bifurcation phenomenon may happen when the sheets are transported by rolls, which can lead to vibration and crack in the sheets. However, the mechanism and the conditions for this phenomenon has not been studied so far. In this paper, we propose a theoretical model to study the formation mechanism of the bifurcation phenomenon in a driven-idle-driven rolls system, and the critical length of the thin-sheets between rolls is obtained, above which the bifurcation will happen. Results show that the bifurcation occurs because the relationship between the lengths of the sheet between rolls and the tension caused by gravity in sheets is not monotonic, and the critical length is about 2.5 times the distance between the rolls. We also carry out experiments to verify the validity of the proposed model. This study is of great importance for the design of the multi-roll system to transport the low strength sheet materials, such as paper and composite sheets.

MIXED FOURIER-LEGENDRE SPECTRAL METHODS FOR THE MULTIPLE SOLUTIONS OF THE SCHRODINGER EQUATION ON THE UNIT DISK

In this paper, we first compute the multiple non-trivial solutions of the Schrodinger equation on a unit disk, by using the Liapunov-Schmidt reduction and symmetry-breaking bifurcation theory, combined with the mixed Fourier-Legendre spectral and pseudospectral methods. After that, we propose the extended systems, which can detect the symmetry-breaking bifurcation points on the branch of the O(2) symmetric positive solutions. We also compute the multiple positive solutions with various symmetries of the Schrodinger equation by the branch switching method based on the Liapunov-Schmidt reduction. Finally, the bifurcation diagrams are constructed, showing the symmetry/peak breaking phenomena of the Schr¨odinger equation. Numerical results demonstrate the effectiveness of these approaches.