The stability of boundary equilibria of three-dimensional Filippov systems

TL;DR

The paper addresses the stability of boundary equilibria in three-dimensional Filippov systems with sliding motion on a switching surface Σ. It shows that stability is governed either by the local linear dynamics encoded in $D f^L(x^*) = A$ and the sliding-field Jacobian $D f^S(x^*) = B$, or, in regimes with persistent switching, by a reinjection map defined by a four-parameter piecewise-linear system whose first-return map has a scalar multiplier $\Lambda$. The main result (Theorem th:main) provides explicit criteria: instability if $A$ (or $B$) has a positive eigenvalue; asymptotic stability when $A$ has three negative eigenvalues and $B$’s nonzero eigenvalues are complex or negative; or otherwise stability is determined by $\Lambda$ with $\Lambda$ undefined or $<1$ implying stability and $\Lambda > 1$ implying instability. The authors develop a reduction chain to a four-parameter normal form, justify the reductions, and perform a thorough numerical exploration of how $\Lambda$ depends on the parameters, illustrating rich dynamics and tipping-point behavior. These results yield practical diagnostics for boundary bifurcations in 3D Filippov models and suggest pathways to generalize the approach to higher dimensions via folding maps on visible folds.

Abstract

For three-dimensional piecewise-smooth systems of ordinary differential equations, this paper characterises the stability of points that belong to a switching surface and are equilibria of exactly one of the two neighbouring pieces of the system. Stability is challenging to characterise when nearby orbits repeatedly switch between regular motion on one side of the switching surface, and sliding motion on the switching surface, as defined via Filippov's convention. We prove that in this case stability is governed by the behaviour of a global reinjection mechanism of a four-parameter family of piecewise-linear hybrid systems, and perform a detailed numerical study of this family.

Figures (5)

• Figure 1: A phase portrait of a three-dimensional Filippov system of the form \ref{['eq:f']}. The switching surface $\Sigma$ (where $H(x) = 0$) consists of an attracting sliding region (grey), a crossing region (green), and the boundary $\Gamma$ between these regions (pink). Orbits are coloured black where they evolve under $f^L$, red where they evolve under $f^R$, and orange where they evolve under the sliding vector field $f^S$. The boundary equilibrium $x^* \in \Gamma$ obeys $f^L(x^*) = {\bf 0}$ and $f^R(x^*) \ne {\bf 0}$. Below $x^*$, points on $\Gamma$ are visible folds (see §\ref{['sub:pointsOfTangency']}); above $x^*$, points on $\Gamma$ are invisible folds.
• Figure 2: A phase portrait of the piecewise-linear system \ref{['eq:pwl']}--\ref{['eq:gLgS']} with $(a,b,c,d) = (-0.2,5,-0.2,3)$. Orbits evolve under $g^L$ and are coloured black until reaching $\tilde{\Sigma}$ (where $y_1 = 0$). Here they switch to evolution under $g^S$ and are coloured orange until reaching $\tilde{\Gamma}$ (where $y_1 = y_2 = 0$). Given $z < 0$, we write $(0,0,\zeta(z))$ for the next point at which the forward orbit of $(0,0,z)$ intersects $\tilde{\Gamma}$.
• Figure 3: The twelve pairs of values $(a,b)$ used in Fig. \ref{['fig:D']}. Each pair uses $a \in \{ -1.2,-0.2,0.2,1.2 \}$ and $b \in \{ 0.5, 2, 5 \}$.
• Figure 4: Regions of the $(c,d)$-plane where $\Lambda$ is undefined or $\Lambda < 1$ (blue) and $\Lambda > 1$ (red) for the values of $a$ and $b$ indicated in Fig. \ref{['fig:C']}. Points where the algorithm described in text failed to obtain a value for $\Lambda$ are coloured blue where the norm of the orbit at some time fell below $10^{-6}$, and red where the norm exceeded $10^6$. In the region $c > 0$ and $d < \frac{c^2}{4}$ (white) $\Lambda$ does not apply (refer instead to case (i) of Theorem \ref{['th:main']}).
• Figure 5: Sample orbits of \ref{['eq:pwl']}--\ref{['eq:gLgS']} with $(a,b,c,d) = (0.2,5,0.2,1)$ in (a) and $(a,b,c,d) = (-0.2,0.5,-0.5,8)$ in (b).

Theorems & Definitions (13)

• Theorem 2.1
• Remark 2.1
• Remark 2.2
• Definition 3.1
• Definition 3.2
• Remark 3.1
• Proposition 4.1
• Proposition 4.2
• proof : Proof of Theorem \ref{['th:main']}
• Lemma A.1