The leading-order approximation to a Filippov system $ f $ about a generic boundary equilibrium $ x^* $ is a system $ F $ that is affine one side of the boundary and constant on the other side. We prove $ x^* $ is exponentially stable for $ f $ if and only if it is exponentially stable for $ F $ when the constant component of $ F $ is not tangent to the boundary. We then show exponential stability and asymptotic stability are in fact equivalent for $ F $. We also show exponential stability is preserved under small perturbations to the pieces of $ F $. Such results are well known for homogeneous systems. To prove the results here additional techniques are required because the two components of $ F $ have different degrees of homogeneity. The primary function of the results is to reduce the problem of the stability of $ x^* $ from the general Filippov system $ f $ to the simpler system $ F $. Yet in general this problem remains difficult. We provide a four-dimensional example of $ F $ for which orbits appear to converge to $ x^* $ in a chaotic fashion. By utilising the presence of both homogeneity and sliding motion the dynamics of $ F $ can in this case be reduced to the combination of a one-dimensional return map and a scalar function.
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Figure 1.
A phase portrait of a two-dimensional Filippov system with an exponentially stable boundary equilibrium
Figure 2.
The left plot is a phase portrait of (2.2) with (2.6) and
Figure 6.
The return map for Filippov solutions of (8.1) projected onto
Figure 7.
Implications between the three types of stability listed in Definition 2.3 for a boundary equilibrium of a Filippov system
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