On the stability of boundary equilibria in Filippov systems

Figure 1.  A phase portrait of a two-dimensional Filippov system with an exponentially stable boundary equilibrium $ x^* $. To the left [right] of the discontinuity surface $ \Sigma $, the dynamics is governed by $ \dot{x} = f^L(x) $ [$ \dot{x} = f^R(x) $]. The central point $ x^* \in \Sigma $ is a zero of $ f^L $ but not of $ f^R $. On $ \Sigma $ orbits above $ x^* $ slide towards $ x^* $

Figure 2.  The left plot is a phase portrait of (2.2) with (2.6) and $ \nu = 0.2 $. Here the origin is exponentially stable. The right plot is a phase portrait of the corresponding reduced system (2.5) (given by replacing $ x_2 $ in $ f^R(x) $ with $ 0 $). Here the origin is unstable. This example does not contradict Theorem 2.1 because $ c_1 = 0 $

Figure 3.  A phase portrait of a two-dimensional Filippov system of the form (2.2). This system has two tangency points (triangles) that divide the discontinuity surface $ \Sigma $ into a crossing region and attracting and repelling sliding regions

Figure 4.  A typical Filippov solution of (8.1). The solution slides on $ \Sigma $ until reaching $ \Sigma_\text{tang} $. Note that this figure shows only three of the four variables

Figure 5.  The Filippov solution of Fig. 4 projected onto the unit sphere $ \mathbb{S}^3 $. The projected solution repeatedly intersects the one-dimensional manifold $ \Gamma $ (8.2) which is used to define the one-dimensional return map shown in Fig. 6

Figure 6.  The return map for Filippov solutions of (8.1) projected onto $ \mathbb{S}^3 $ using the one-dimensional manifold $ \Gamma $ as the domain of the map. The map has three fixed points (black circles) that correspond to periodic orbits of the projected dynamics (these correspond to Filippov solutions of (8.1) that spiral into the origin in a simple fashion). We also show, as a cobweb diagram, the orbit of $ G $ corresponding to the solution shown in Fig. 5

Figure 7.  Implications between the three types of stability listed in Definition 2.3 for a boundary equilibrium of a Filippov system $ f $ of the form (2.2) and its corresponding local approximation $ F $ (2.5). For any system exponential stability implies asymptotic stability and asymptotic stability implies Lyapunov stability. Theorems 2.1 and 2.2 and Conjecture 9.1 (if true) provide additional implications as shown. Adjoining implications can be composed, for example to see that asymptotic stability for $ F $ implies asymptotic stability for $ f $ (but the converse is not necessarily true)