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September  2021, 20(9): 3093-3111. doi: 10.3934/cpaa.2021097

## On the stability of boundary equilibria in Filippov systems

 School of Fundamental Sciences, Massey University, Colombo Road, Palmerston North, 4410, New Zealand

Received  January 2021 Revised  May 2021 Published  September 2021 Early access  June 2021

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.

Citation: D. J. W. Simpson. On the stability of boundary equilibria in Filippov systems. Communications on Pure &amp; Applied Analysis, 2021, 20 (9) : 3093-3111. doi: 10.3934/cpaa.2021097
##### References:
  M. di Bernardo, C. J. Budd, A. R. Champneys and P. Kowalczyk, Piecewise-smooth Dynamical Systems. Theory and Applications, Springer-Verlag, New York, 2008. Google Scholar  M. di Bernardo, A. Nordmark and G. Olivar, Discontinuity-induced bifurcations of equilibria in piecewise-smooth and impacting dynamical systems, Phys. D, 237 (2008), 119-136.  doi: 10.1016/j.physd.2007.08.008.  Google Scholar  V. I. Bogachev, Measure Theory. Volume I, Springer, New York, 2007. doi: 10.1007/978-3-540-34514-5.  Google Scholar  T. Dezuo, L. Rodrigues and A. Trofino, Stability analysis of piecewise affine systems with sliding modes, in Proceedings of the 2014 American Control Conference (ACC)., (2014), 2005–2010. Google Scholar  A. F. Filippov., Differential equations with discontinuous right-hand side, Mat. Sb., 51(93): 99–128, 1960. Google Scholar  A. F. Filippov, Differential Equations with Discontinuous Righthand Sides, Kluwer Academic Publishers., Norwell, 1988. doi: 10.1007/978-94-015-7793-9.  Google Scholar  P. A. Glendinning, Shilnikov chaos, Filippov sliding and boundary equilibrium bifurcations, Euro. J. Appl. Math., 29 (2018), 757-777.  doi: 10.1017/S0956792518000335.  Google Scholar  P. Glendinning and M. R. Jeffrey, Grazing-sliding bifurcations, border collision maps and the curse of dimensionality for piecewise smooth bifurcation theory, Nonlinearity, 28 (2015), 263-283.  doi: 10.1088/0951-7715/28/1/263.  Google Scholar  R. Iervolino, F. Vasca and L. Iannelli, Stability analysis of conewise linear systems with sliding modes, In IEEE 54th Annual Conference on Decision and Control (CDC)., (2016), 1174–1179. Google Scholar  M. R. Jeffrey, Hidden Dynamics. The Mathematics of Switches, Decisions and Other Discontinuous Behaviour, Springer, New York, 2018. doi: 10.1007/978-3-030-02107-8.  Google Scholar  M. Johansson, Piecewise Linear Control Systems., volume 284 of Lecture Notes in Control and Information Sciences, Springer-Verlag, New York, 2003. doi: 10.1007/3-540-36801-9.  Google Scholar  A. Lasota and A. Strauss, Asymptotic behavior for differential equations which cannot be locally linearized, J. Differ. Equ., 10 (1971), 152-172.  doi: 10.1016/0022-0396(71)90103-3.  Google Scholar  A. P. Molchanov and Ye. S. Pyatnitskiy, Criteria of asymptotic stability of differential and difference inclusions encountered in control theory, Syst. & Contr. Lett., 13 (1989), 59-64.  doi: 10.1016/0167-6911(89)90021-2.  Google Scholar  H. L. Royden and P. M. Fitzpatrick, Real Analysis, Prentice Hall, New York, 4th edition, 2010. Google Scholar  D. J. W. Simpson, A general framework for boundary equilibrium bifurcations of Filippov systems, Chaos, 28 (2018), 103114. doi: 10.1063/1.5037947.  Google Scholar  D. J. W. Simpson, The stability of fixed points on switching manifolds of piecewise-smooth continuous maps, J. Dyn. Differ. Equ., 32 (2020), 1527-1552.  doi: 10.1007/s10884-019-09803-9.  Google Scholar  H. H. Sohrab, Basic Real Analysis, Springer, New York, 2nd edition, 2014. doi: 10.1007/978-1-4939-1841-6.  Google Scholar

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##### References:
  M. di Bernardo, C. J. Budd, A. R. Champneys and P. Kowalczyk, Piecewise-smooth Dynamical Systems. Theory and Applications, Springer-Verlag, New York, 2008. Google Scholar  M. di Bernardo, A. Nordmark and G. Olivar, Discontinuity-induced bifurcations of equilibria in piecewise-smooth and impacting dynamical systems, Phys. D, 237 (2008), 119-136.  doi: 10.1016/j.physd.2007.08.008.  Google Scholar  V. I. Bogachev, Measure Theory. Volume I, Springer, New York, 2007. doi: 10.1007/978-3-540-34514-5.  Google Scholar  T. Dezuo, L. Rodrigues and A. Trofino, Stability analysis of piecewise affine systems with sliding modes, in Proceedings of the 2014 American Control Conference (ACC)., (2014), 2005–2010. Google Scholar  A. F. Filippov., Differential equations with discontinuous right-hand side, Mat. Sb., 51(93): 99–128, 1960. Google Scholar  A. F. Filippov, Differential Equations with Discontinuous Righthand Sides, Kluwer Academic Publishers., Norwell, 1988. doi: 10.1007/978-94-015-7793-9.  Google Scholar  P. A. Glendinning, Shilnikov chaos, Filippov sliding and boundary equilibrium bifurcations, Euro. J. Appl. Math., 29 (2018), 757-777.  doi: 10.1017/S0956792518000335.  Google Scholar  P. Glendinning and M. R. Jeffrey, Grazing-sliding bifurcations, border collision maps and the curse of dimensionality for piecewise smooth bifurcation theory, Nonlinearity, 28 (2015), 263-283.  doi: 10.1088/0951-7715/28/1/263.  Google Scholar  R. Iervolino, F. Vasca and L. Iannelli, Stability analysis of conewise linear systems with sliding modes, In IEEE 54th Annual Conference on Decision and Control (CDC)., (2016), 1174–1179. Google Scholar  M. R. Jeffrey, Hidden Dynamics. The Mathematics of Switches, Decisions and Other Discontinuous Behaviour, Springer, New York, 2018. doi: 10.1007/978-3-030-02107-8.  Google Scholar  M. Johansson, Piecewise Linear Control Systems., volume 284 of Lecture Notes in Control and Information Sciences, Springer-Verlag, New York, 2003. doi: 10.1007/3-540-36801-9.  Google Scholar  A. Lasota and A. Strauss, Asymptotic behavior for differential equations which cannot be locally linearized, J. Differ. Equ., 10 (1971), 152-172.  doi: 10.1016/0022-0396(71)90103-3.  Google Scholar  A. P. Molchanov and Ye. S. Pyatnitskiy, Criteria of asymptotic stability of differential and difference inclusions encountered in control theory, Syst. & Contr. Lett., 13 (1989), 59-64.  doi: 10.1016/0167-6911(89)90021-2.  Google Scholar  H. L. Royden and P. M. Fitzpatrick, Real Analysis, Prentice Hall, New York, 4th edition, 2010. Google Scholar  D. J. W. Simpson, A general framework for boundary equilibrium bifurcations of Filippov systems, Chaos, 28 (2018), 103114. doi: 10.1063/1.5037947.  Google Scholar  D. J. W. Simpson, The stability of fixed points on switching manifolds of piecewise-smooth continuous maps, J. Dyn. Differ. Equ., 32 (2020), 1527-1552.  doi: 10.1007/s10884-019-09803-9.  Google Scholar  H. H. Sohrab, Basic Real Analysis, Springer, New York, 2nd edition, 2014. doi: 10.1007/978-1-4939-1841-6.  Google Scholar 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 $x^*$ 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 $c_1 = 0$ 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 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 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 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 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)
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