Some aspects of the West African monsoon circulation As deduced from a geostationary satellite

This paper focusses on some aspects or the West African monsoonal circulation observed during the period 15 July-l0 August 1979 of the PGGE, as derived from the satellite cloud windvectors. Temporal averages of the computed winsfields reveal that the flow at the low level is southerly (monsoonal), and Its line of discontinuity with the continental northeasterly was found at approximately 16°-18°N, lying about 300 km south of the accepted mean position. At both the middle and upper tropospheres the flow is easterly with axis about 12o-14,N and, latitude 8 No respectively, such that it is a circulation south of the axis and northwards, it is anticyclonic. The satellite-observed tropospheric circulation IS then discussed in relation to the, weather manifestations over the sub-region typical of the July / August period. The mass fields obtained from our gridded satellite-winds indicate that inflow into the land area occur mainly at the lowest layer (1000:850 hPa), whereas at the upper, levels (that is, above 850 hPa) it is predominantly an outflow, The tropospheric average gives a net mass for divergence from within the area, The significance of this result in relation to the observed weather phenomenology of a temporary cessation of the monsoon precipitations occurring about the peak of the season IS also discussed.

Limit Cycles of Planar Piecewise Differential Systems with Linear Hamiltonian Saddles

We provide the maximum number of limit cycles for continuous and discontinuous planar piecewise differential systems formed by linear Hamiltonian saddles and separated either by one or two parallel straight lines. We show that when these piecewise differential systems are either continuous or discontinuous and are separated by one straight line, or are continuous and are separated by two parallel straight lines, they do not have limit cycles. On the other hand, when these systems are discontinuous and separated by two parallel straight lines, we prove that the maximum number of limit cycles that they can have is one and that this maximum is reached by providing an example of such a system with one limit cycle. When the line of discontinuity of the piecewise differential system is formed by one straight line, the symmetry of the problem allows to take this straight line without loss of generality as the line x=0. Similarly, when the line of discontinuity of the piecewise differential system is formed by two parallel straight lines due to the symmetry of the problem, we can assume without loss of generality that these two straight lines are x=±1.

A sphere packing model for shear bands in dense soils

The rhombic sphere packing can be used to model the biaxial test on granular soils in a very simple way. According to the angle of assemblage, the packing is dilatant or contractive. Correspondingly, overall stresses are transmitted as chains of forces or oblique forces of contact. The connection of the soil stress-strain behaviour and the packing void ratio is achieved by mapping both of the plots. The mapping shows that dense soils are dilatant and loose soils are contractive, separated by the critical state. It also shows that the bifurcation point and the peak strength are features only of dense soils. The band of strain localization is analysed in the elastic regime, and its inclination is found maximizing the intensity of the mobilized stress ratio. The stresses within the shear band are obtained by assuming a partially coaxial packing rotated to reach the full plastic state. The equilibrium of the overall stress at the line of discontinuity reveals a relationship between the peak friction angle and the coefficient of lateral pressure at rest. As long as these parameters are obtained independently of each other, they allow the validation of the theory.

Limit Cycles and Bifurcations in a Class of Discontinuous Piecewise Linear Systems

In this paper, we consider planar piecewise linear differential systems with a line of discontinuity sharing a linear part. We study not only the number of crossing limit cycles, but also the number of sliding ones, and the coexistence of two configurations of limit cycles. In particular, we proved that both numbers of crossing limit cycles and sliding ones are at most [Formula: see text], but the total number of limit cycles is at most [Formula: see text]. Finally, by complete analysis on the number of limit cycles, we show some bifurcations which exist in generic Filippov systems, revealing also two nongeneric bifurcations.

Numerical constructions of optimal feedback in models of chemotherapy of a malignant tumor

In this paper, a problem of chemotherapy of a malignant tumor is considered. Dynamics is piecewise monotone and a therapy function has two maxima. The aim of therapy is to minimize the number of tumor cells at the given final instance. The main result of this work is the construction of optimal feedbacks in the chemotherapy task. The construction of optimal feedback is based on the value function in the corresponding problem of optimal control (therapy). The value function is represented as a minimax generalized solution of the Hamilton–Jacobi–Bellman equation. It is proved that optimal feedback is a discontinuous function and the line of discontinuity satisfies the Rankin–Hugoniot conditions. Other results of the work are illustrative numerical examples of the construction of optimal feedbacks and Rankin–Hugoniot lines.

On Singular "Semifocus" Type Point Bifurcations of Piecewise Smooth Dynamical System

For the processes described by dynamical systems, closed trajectories of dynamical systems are in line with periodic oscillations. Therefore, there is a considerable interest in describing the bifurcations of the generation of closed trajectories from equilibrium when the parameters change. In typical one-parameter and two-parameter families of smooth dynamical systems on a plane, closed trajectories can be generated only from equilibrium – weak focus. In mathematical modeling in the theory of automatic control, in mechanics and in other applications, piecewise smooth dynamical systems are often used. For them, there are other bifurcations of the generation of closed trajectories from equilibrium. The paper describes one of them, which is a typical family of dynamical systems specified by a piecewise smooth vector field on a two-dimensional manifold depending on two small parameters. It is assumed that for zero values of the parameters the vector field has a singular point O on the line of discontinuity of the field, and the point O is stable; in one half-neighborhood of the point O the field coincides with a smooth vector field for which the point O is a weak focus with positive (negative) first Lyapunov value, and in the other half-neighborhood it coincides with a smooth vector field directed at the points of the line of discontinuity inside the first of the semi-neighborhoods. The paper describes bifurcations in the neighborhood of the point O as the parameters change, in particular, indicating the regions of the parameters for which the vector field has a stable closed trajectory.

Bifurcation Analysis of Planar Piecewise Smooth Systems with a Line of Discontinuity

In this paper, we consider bifurcations of a class of planar piecewise smooth differential systems constituted by a general linear system and a quadratic Hamiltonian system. The linear system has four parameters. When the parameters vary in different regions, the left linear system can have a saddle, a node or a focus. For each case, we provide a completely qualitative analysis of the dynamical behavior for this piecewise smooth system. Our results generalize and improve the results in this direction.