Limit cycles of piecewise polynomial differential systems with the discontinuity line xy = 0

<p style='text-indent:20px;'>In this paper we study the maximum number of limit cycles bifurcating from the periodic orbits of the center <inline-formula><tex-math id="M1">\begin{document}$ \dot x = -y((x^2+y^2)/2)^m, \dot y = x((x^2+y^2)/2)^m $\end{document}</tex-math></inline-formula> with <inline-formula><tex-math id="M2">\begin{document}$ m\ge0 $\end{document}</tex-math></inline-formula> under discontinuous piecewise polynomial (resp. polynomial Hamiltonian) perturbations of degree <inline-formula><tex-math id="M3">\begin{document}$ n $\end{document}</tex-math></inline-formula> with the discontinuity set <inline-formula><tex-math id="M4">\begin{document}$ \{(x, y)\in\mathbb{R}^2: xy = 0\} $\end{document}</tex-math></inline-formula>. Using the averaging theory up to any order <inline-formula><tex-math id="M5">\begin{document}$ N $\end{document}</tex-math></inline-formula>, we give upper bounds for the maximum number of limit cycles in the function of <inline-formula><tex-math id="M6">\begin{document}$ m, n, N $\end{document}</tex-math></inline-formula>. More importantly, employing the higher order averaging method we provide new lower bounds of the maximum number of limit cycles for several types of piecewise polynomial systems, which improve the results of the previous works. Besides, we explore the effect of 4-star-symmetry on the maximum number of limit cycles bifurcating from the unperturbed periodic orbits. Our result implies that 4-star-symmetry almost halves the maximum number.</p>

Rigorous Solution of Slopes’ Stability considering Hydrostatic Pressure

According to characteristics of soils in failure, a sliding mechanism of slopes in limit state is divided into five parts, for building a slip line field satisfying all possible boundary conditions. An algorithm is built to obtain the rigorous solution approaching upper and lower bound values simultaneously, which satisfies the static boundary and the kinematical boundary based on the slip line field, while stress discontinuity line and velocity discontinuity line are key points. This algorithm is copared with the Spencer method to prove its feasibility with a special example. The variation of rigorous solution, including an ultimate load and a sliding belt the rigid body sliding along rather than a single slip surface for friction-type soils, is achieved considering hydrostatic pressure with soil parameters changing.

Analyse numérique du régime THD dans un patin échelon

A lack of thermohydrodynamic (THD) models adapted to discontinuous fluid film geometries can be noted in the lubrication-related literature. The finite differences method, largely used in THD analysis, cannot be easily applied to discontinuous flow domains. This article proposes two finite volumes based models, suitable for discontinuous domains. The first model, based on Reynolds generalized equation, lacks in accuracy near the discontinuity line, but gains in computing speed. The second model, based on Navier-Stokes equations, is more accurate but also significantly more demanding in terms of computing requirements. The two models are applied and compared on the simple geometry of a bi-dimensional Rayleigh step bearing. Different configurations are analyzed, allowing the validation of the Reynolds model for most of the cases encountered in fluid film lubrication.

6.—Smooth Fronted Waves in the Shallow Water Approximation

SynopsisThis paper examines the mathematical problem of the propagation of a smooth fronted wavein the context of shallow water theory. Here, a smooth fronted wave will be taken to be one in which the surface slope is continuous across a line in the free-surface, while the second derivative of the surface slope is discontinuous across that same line. This discontinuity line in the surface then plays the role of the wavefront. After establishing that such wavefronts propagate along the characteristics, and deriving the appropriate transport equations, the explicit form is found for the acceleration with respect to distance of the horizontal component of the water velocity of the surface immediately behind the wavefront as a function of position and seabed profile when the wave propagates into still water. The result is then used to prove that in this approximation such a wave cannever break immediately behind the wavefront before the shore line is reached.