We consider sliding motion, in the sense of Filippov, on a discontinuity surface Σ of co-dimension 2. We characterize, and restrict to, the case of Σ being attractive through sliding. In this situation, we show that a certain Filippov sliding vector field f_F (suggested in Alexander and Seidman, 1998 [2], di Bernardo et al., 2008 [6], Dieci and Lopez, 2011 [10]) exists and is unique. We also propose a characterization of first order exit conditions, clarify its relation to generic co-dimension 1 losses of attractivity for Σ, and examine what happens to the dynamics on Σ for the aforementioned vector field f_F . Examples illustrate our results.

We propose a system of first-order ordinary differential equations to describe and understand the physiological mechanisms of the interplay between plasma glucose and insulin and their behaviors in diabetes. The proposed model is based on Hill and step functions which are used to simulate the switch-like behavior that occurs in metabolic regulatory variables when some of the threshold parameters are approached. A simplified piece-wise linear system is also proposed to study the possible equilibria and solutions and used to introduce simple theoretical control mechanisms representing the action of an artificial pancreas and regulating exogenous insulin.

This work is dedicated to the memory of Donato Trigiante who has been the first teacher of Numerical Analysis of the second author. We remember Donato as a generous teacher, always ready to discuss with his students, able to give them profound and interesting suggestions. Here, we present a survey of numerical methods for differential systems with discontinuous right hand side. In particular, we will review methods where the discontinuities are detected by using an event function (so-called event driven methods) and methods where the discontinuities are located by controlling the local errors (so-called time-stepping methods). Particular attention will be devoted to discontinuous systems of Filippov’s type where sliding behavior on the discontinuity surface is allowed.

In this paper we consider numerical techniques to locate the event points of the differential system x′=f(x), where f is a discontinuous vector field along an event surface splitting the state space into two different regions R1 and R2 and f(x)=fi(x) when x∈Ri, for i=1,2 while f1(x)≠f2(x) when x∈Σ. Methods based on Adams multistep schemes which approach the event surface Σ from one side only and in a finite number of steps are proposed. Particularly, these techniques do not require the evaluation of the vector field f1 (respectively, f2) in the region R2 (respectively R1) and are based on the computation–at each step– of a new time step

We consider the fundamental matrix solution associated to piecewise smooth differential systems of Filippov type, in which the vector field varies discontinuously as solution trajectories reach one or more surfaces. We review the cases of transversal intersection and of sliding motion on one surface. We also consider the case when sliding motion takes place on the intersection of two or more surfaces. Numerical results are also given.

We consider the numerical integration of discontinuous differential systems of ODEs of the type: x' = f_1(x) when h(x) < 0 and x'= f_2(x) when h(x) > 0, and with f1 \neq f2 for x ∈ Σ, where Σ := {x: h(x) = 0} is a smooth co-dimension one discontinuity surface. Often, f1 and f2 are defined on the whole space, but there are applications where f1 is not defined above Σ and f2 is not defined below Σ. For this reason, we consider explicit Runge–Kutta methods which do not evaluate f1 above Σ (respectively, f2 below Σ). We exemplify our approach with subdiagonal explicit Runge–Kutta methods of order up to 4. We restrict attention only to integration up to the point where a trajectory reaches Σ.

In this paper we study the numerical solution of singularly perturbed systems with a discontinuous right hand side. We will avoid to consider the associate reduced differential system because often this study leads to wrong conclusions. To handle either the stiffness, due to different scales, or the discontinuity of the vector field we will consider numerical method which are semi-implicit and of low order of accuracy.

The interpolation polynomial in the k-step Adams-Bashforth method may be used to compute the numerical solution at off grid points. We show that such a numerical solution is equivalent to the one obtained by the Nordsieck technique for changing the step size. We also provide an application of this technique to the event location in discontinuous differential systems

In this paper we consider the issue of sliding motion in Filippov systems on the intersection of two or more surfaces. To this end, we propose an extension of the Filippov sliding vector field on manifolds of co-dimension p, with p a parts per thousand yen 2. Our model passes through the use of a multivalued sign function reformulation. To justify our proposal, we will restrict to cases where the sliding manifold is attractive. For the case of co-dimension p = 2, we will distinguish between two types of attractive sliding manifold: "node-like" and "spiral-like". The case of node-like attractive manifold will be further extended to the case of p a parts per thousand yen 3. Finally, we compare our model to other existing methodologies on some examples.

The Mimetic Finite Difference (MFD) methods for PDEs mimic crucial properties of mathematical systems: duality and self-adjointness of differential operators, conservation laws and properties of the solution on general polytopal meshes. In this article the structure and the spectral properties of the linear systems derived by the spatial discretization of diffusion problem are analysed. In addition, the numerical approximation of parabolic equations is discussed where the MFD approach is used in the space discretization while implicit #-method and explicit Runge Kutta Chebyshev schemes are used in time discretization. Moreover, we will show how the numerical solution preserves certain conservation laws of the theoretical solution.

The infiltration process into the soil is generally modeled by the Richards’ partial differential equation (PDE). In this paper a new approach for modeling the infiltration process through the interface of two different soils is proposed, where the interface is seen as a discontinuity surface defined by suitable state variables. Thus, the original 1D Richards’ PDE, enriched by a particular choice of the boundary conditions, is first approximated by means of a time semidiscretization, that is by means of the transversal method of lines (TMOL). In such a way a sequence of discontinuous initial value problems, described by a sequence of second order differential systems in the space variable, is derived. Then, Filippov theory on discontinuous dynamical systems may be applied in order to study the relevant dynamics of the problem. The numerical integration of the semidiscretized differential system will be performed by using a one-step method, which employs an event driven procedure to locate the discontinuity surface and to adequately change the vector field.

In this paper, we consider numerical methods for the location of events of ordinary differential equations. These methods are based on particular changes of the independent variable, called time-transformations. Such a time-transformation reduces the integration of an equation up to the unknown point, where an event occurs, to the integration of another equation up to a known point. This known point corresponds to the unknown point by means of the time-transformation. This approach extends the one proposed by Dieci and Lopez [BIT 55 (2015), no. 4, 987-1003], but our generalization permits, amongst other things, to deal with situations where the solution approaches the event in a tangential way. Moreover, we also propose to use this approach in a different manner with respect to that of Dieci and Lopez.

In this paper, we consider the class of sliding Filippov vector fields in R^3 on the intersection of two smooth surfaces: S= 1 \ 2, where S_i = {x : h_i(x) = 0}, and h_i : R^3 -> R, i = 1, 2, are smooth functions with linearly independent normals. Although, in general, there is no unique Filippov sliding vector field on , here we prove that –under natural conditions– all Filippov sliding vector fields determine the same solution trajectory on . In other words, the aforementioned ambiguity has no meaningful impact. We also examine the implication of this result in the important case of a periodic orbit a portion of which slides on S.