Spectral Contraction Framework for Planar Filippov Systems: Existence and Stability of Nonsmooth Periodic Orbits

Pascal Stiefenhofer

doi:10.33434/cams.1729696

Spectral Contraction Framework for Planar Filippov Systems: Existence and Stability of Nonsmooth Periodic Orbits

Abstract

We present a contraction-based framework for analyzing periodic behavior in planar nonsmooth dynamical systems governed by Filippov differential inclusions. The approach combines a time- and state-dependent weighted Riemannian metric with Clarke’s generalized Jacobian, together with a uniform jump condition across switching manifolds, to obtain sufficient conditions for exponential contraction on compact forward-invariant sets. These local estimates are assembled into a global stability result via a Banach fixed-point argument, establishing the existence and uniqueness of an exponentially attracting periodic orbit. The theory extends classical contraction results from the scalar setting to two-dimensional systems with discontinuities, sliding motion, and switching phenomena. Beyond clarifying the role of contraction in nonsmooth dynamics, the framework provides a systematic analytic tool for stability verification, with potential applications to hybrid control, nonsmooth mechanical oscillators, and computational methods for piecewise-smooth systems.

Keywords

Clarke generalized Jacobian, Contraction theory, Filippov differential inclusions, Nonsmooth dynamical systems, Periodic solutions

Ethical Statement

no data used

References

[1] M. di Bernardo, C. Budd, A. R. Champneys, P. Kowalczyk, Piecewise-Smooth Dynamical Systems: Theory and Applications, Springer, London, 2008.
[2] P. Giesl, The basin of attraction of periodic orbits in nonsmooth differential equations, ZAMM–J. Appl. Math. Mech., 85(2) (2005), 89–104. https://doi.org/10.1002/zamm.200310164
[3] M. Kunze, Non-Smooth Dynamical Systems, Lecture Notes in Mathematics, Vol. 1744. Springer-Verlag, Berlin, 2000.
[4] M. Kunze, T. K¨upper, Non-smooth dynamical systems: An overview, In B. Fiedler (Ed.), Ergodic Theory, Analysis, and Efficient Simulation of Dynamical Systems, Springer, Berlin, 2001, pp. 431–452. https://doi.org/10.1007/978-3-642-56589-2 19
[5] R. I. Leine, D. H. van Campen, W. A. van de Vrande, Stick-slip vibrations induced by alternate friction models, Nonlinear Dyn., 16(1) (1998), 41–54. https://doi.org/10.1023/A:1008289604683
[6] M. Guardia, T. M. Seara, M. A. Teixeira, Generic bifurcations of low codimension of planar Filippov systems, J. Differential Equations, 250(4) (2011), 1967–2023. https://doi.org/10.1016/j.jde.2010.11.016
[7] P. Stiefenhofer, P. Giesl, Local stability theory of nonsmooth periodic orbits: Example II, Appl. Math. Sci., 13(11) (2019), 521-531. https://doi.org/10.12988/ams.2019.9464
[8] P. Stiefenhofer, P. Giesl, A global stability theory of nonsmooth periodic orbits: Example I, Appl. Math. Sci., 13(11) (2019), 511–520. https://doi.org/10.12988/ams.2019.9463
[9] A. Pavlov, A. Pogromsky, N. van de Wouw, H. Nijmeijer, Convergent dynamics, a tribute to Boris Pavlovich Demidovich, Syst. Control Lett., 52(3–4) (2004), 257–261. https://doi.org/10.1016/j.sysconle.2004.02.003
[10] T. Ito, A Filippov solution of a system of differential equations with discontinuous right-hand side, Econom. Lett., 4(4) (1979), 349–354. https://doi.org/10.1016/0165-1765(79)90183-6

[11] P. Stiefenhofer, Economic stability of non-smooth periodic orbits in the plane: Omega limit sets part I, Nonlinear Anal. Differ. Equ., 8(1) (2020), 41–52. https://doi.org/10.12988/nade.2020.91121
[12] P. Stiefenhofer, Economic stability of non-smooth periodic orbits in the plane: Omega limit sets part II, Nonlinear Anal. Differ. Equ., 8(1) (2020), 43–65. https://doi.org/10.12988/nade.2020.91122
[13] P. Giesl, S. Hafstein, C. Kawan, Review on contraction analysis and computation of contraction metrics, J. Comput. Dyn., 10(1) (2023), 1–47. https://doi.org/10.3934/jcd.2022018
[14] A. F. Filippov, Differential Equations with Discontinuous Righthand Sides, Kluwer Academic Publishers, London, 1988.
[15] P. Giesl, Necessary condition for the basin of attraction of a periodic orbit in non-smooth periodic systems, Discrete Contin. Dyn. Syst., 18(2-3) (2007), 355–373. https://doi.org/10.3934/dcds.2007.18.355
[16] J. Boote, P. Giesl, S. Suhr, Construction of a contraction metric for non-smooth time-periodic systems, Discrete Contin. Dyn. Syst. Ser. B, 30(6) (2025), 1903–1929. https://doi.org/10.3934/dcdsb.2024147
[17] F. Bullo, Contraction Theory for Dynamical Systems, Kindle Direct Publishing, 2024.
[18] W. Lohmiller, J. J. E. Slotine, Nonlinear process control using contraction theory, AIChE J., 46(3) (2000), 588–596. https://doi.org/10.1002/aic.690460317
[19] E. D. Sontag, Contractive systems with inputs, In J. C. Willems, S. Hara, Y. Ohta, H. Fujioka (Eds.), Perspectives in Mathematical System Theory, Control, and Signal Processing, Lecture Notes in Control and Information Sciences, vol. 398, Springer-Verlag, Berlin, 2010, pp. 217-228. https://doi.org/10.1007/978-3-540-93918-4 20
[20] F. H. Clarke, Optimization and Nonsmooth Analysis, John Wiley & Sons, New York, 1983.
[21] L. Dieci, C. Elia, L. Lopez, Networks of piecewise-smooth Filippov systems and stability of synchronous periodic orbits, Discrete Contin. Dyn. Syst. Ser. B, 30(6) ( 2025), 1996-2026. https://doi.org/10.3934/dcdsb.2024146
[22] D. D. Novaes, T. M. Seara, M. A. Teixeira, I. O. Zeli, Study of periodic orbits in periodic perturbations of planar reversible Filippov systems having a twofold cycle, SIAM J. Appl. Dyn. Syst., 19(2) (2020), 1343-1371. https://doi.org/10.1137/19M1289959

Details

Primary Language

English

Subjects

Ordinary Differential Equations, Difference Equations and Dynamical Systems

Journal Section

Research Article

Authors

Pascal Stiefenhofer ^*
0000-0002-3399-4377
United Kingdom

Early Pub Date

December 3, 2025

Publication Date

December 8, 2025

Submission Date

June 29, 2025

Acceptance Date

November 28, 2025

Published in Issue

Year 2025 Volume: 8 Number: 4

DOI

https://doi.org/10.33434/cams.1729696

IZ

https://izlik.org/JA27PK73JZ

APA

Stiefenhofer, P. (2025). Spectral Contraction Framework for Planar Filippov Systems: Existence and Stability of Nonsmooth Periodic Orbits. Communications in Advanced Mathematical Sciences, 8(4), 189-209. https://doi.org/10.33434/cams.1729696

AMA

1.Stiefenhofer P. Spectral Contraction Framework for Planar Filippov Systems: Existence and Stability of Nonsmooth Periodic Orbits. Communications in Advanced Mathematical Sciences. 2025;8(4):189-209. doi:10.33434/cams.1729696

Chicago

Stiefenhofer, Pascal. 2025. “Spectral Contraction Framework for Planar Filippov Systems: Existence and Stability of Nonsmooth Periodic Orbits”. Communications in Advanced Mathematical Sciences 8 (4): 189-209. https://doi.org/10.33434/cams.1729696.

EndNote

Stiefenhofer P (December 1, 2025) Spectral Contraction Framework for Planar Filippov Systems: Existence and Stability of Nonsmooth Periodic Orbits. Communications in Advanced Mathematical Sciences 8 4 189–209.

IEEE

[1]P. Stiefenhofer, “Spectral Contraction Framework for Planar Filippov Systems: Existence and Stability of Nonsmooth Periodic Orbits”, Communications in Advanced Mathematical Sciences, vol. 8, no. 4, pp. 189–209, Dec. 2025, doi: 10.33434/cams.1729696.

ISNAD

Stiefenhofer, Pascal. “Spectral Contraction Framework for Planar Filippov Systems: Existence and Stability of Nonsmooth Periodic Orbits”. Communications in Advanced Mathematical Sciences 8/4 (December 1, 2025): 189-209. https://doi.org/10.33434/cams.1729696.

JAMA

1.Stiefenhofer P. Spectral Contraction Framework for Planar Filippov Systems: Existence and Stability of Nonsmooth Periodic Orbits. Communications in Advanced Mathematical Sciences. 2025;8:189–209.

MLA

Stiefenhofer, Pascal. “Spectral Contraction Framework for Planar Filippov Systems: Existence and Stability of Nonsmooth Periodic Orbits”. Communications in Advanced Mathematical Sciences, vol. 8, no. 4, Dec. 2025, pp. 189-0, doi:10.33434/cams.1729696.

Vancouver

1.Pascal Stiefenhofer. Spectral Contraction Framework for Planar Filippov Systems: Existence and Stability of Nonsmooth Periodic Orbits. Communications in Advanced Mathematical Sciences. 2025 Dec. 1;8(4):189-20. doi:10.33434/cams.1729696

Cited By

Contraction and Periodic Orbits in Time-Periodic Filippov Systems

Annals of Mathematics and Physics

https://doi.org/10.17352/amp.000161

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