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Classification of flow behavior near generalized equilibrium points in piecewise smooth systems | ||
| Journal of Mathematical Modeling | ||
| دوره 14، شماره 2، مرداد 2026، صفحه 723-739 اصل مقاله (378.82 K) | ||
| نوع مقاله: Research Article | ||
| شناسه دیجیتال (DOI): 10.22124/jmm.2025.30519.2738 | ||
| نویسندگان | ||
| Zohreh Zarei1؛ Majid Karimi Amaleh* 2 | ||
| 1Department of Math., University of Hormozgan, Bandar Abbas, Iran | ||
| 2Department of Math, University of Hormozgan, Bandar Abbas, Iran | ||
| چکیده | ||
| The aim of this paper is to classify the various states of flow behavior for piecewise smooth systems near generalized equilibrium points. Seven categories are introduced based on the sign of the vector field across the discontinuity boundary, each encompassing distinct dynamical configurations. We investigate how a small perturbation parameter $\mu$ influences the existence, type, and stability of generalized singular points in planar piecewise linear systems. Starting with a one-dimensional example to illustrate core mechanisms, we extend the analysis to two dimensions, providing a detailed classification grounded in the signs of the system’s components. Our results yield a comprehensive framework for understanding how generalized singular points govern local dynamics, including bifurcations induced by parameter variation. This work contributes to the theoretical foundation for analyzing discontinuity-induced phenomena such as sliding modes and non-smooth bifurcations. | ||
| کلیدواژهها | ||
| generalized equilibrium point؛ piecewise smooth systems؛ bifurcation analysis؛ perturbation parameter؛ stability | ||
| مراجع | ||
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[1] B. Coll, A. Gasull, and R. Prohens, Degenerate Hopf bifurcations in discontinuous planar systems, J. Math. Anal. Appl. 253(2) (2001) 671–690. [2] A.F. Filippov, Differential Equations with Discontinuous Righthand Sides: Control Systems, Vol. 18, Springer Science and Business Media, 1988. [3] M. Han, and W. Zhang, On Hopf bifurcation in non-smooth planar systems, J. Differ. Equ. 248 (2010) 2399–2416. [4] G.A. Kriegsmann, The rapid bifurcation of the Wien bridge oscillator, IEEE Trans. Circ. Syst. 34 (1987) 1093–1096. [5] M. Kunze, Non-smooth Dynamical Systems, Vol. 1744, Springer, 2000. [6] Y.A. Kuznetsov, Elements of Applied Bifurcation Theory, Appl. Math. Sci. Springer, New York, 1998. [7] F. Liang, and M. Han, Stability and limit cycle bifurcation for two kinds of generalized double homoclinic loops in planar piecewise smooth systems, Internat. J. Bifur. Chaos 24(12) (2014) 1450153. [8] F. Liang, and M. Han, The stability of some kinds of generalized homoclinic loops in planar piece wise smooth systems, Internat. J. Bifur. Chaos 23(2) (2013) 1350027. [9] J. Llibre,D.D. Novaes, and M.A. Teixeira, Maximum number of limit cycles for certain piecewise linear dynamical systems, Nonlinear Dynam. 82 (2015) 1159–1175. [10] L. Perko, Differential Equations and Dynamical Systems, 3rd ed., Texts Appl. Math. 7, Springer Verlag, New York, 2001. [11] S. Wiggins, Introduction to Applied Nonlinear Dynamical Systems and Chaos, Vol. 2, Springer Verlag, 1990 | ||
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