Nonsmooth bifurcations of equilibria in planar continuous systems

J.J.B. Biemond, N. Wouw, van de, H. Nijmeijer

Research output: Contribution to journalArticleAcademicpeer-review

10 Citations (Scopus)

Abstract

In this paper we present a procedure to find all limit sets near bifurcating equilibria in a class of hybrid systems described by continuous, piecewise smooth differential equations. For this purpose, the dynamics near the bifurcating equilibrium is locally approximated as a piecewise affine systems defined on a conic partition of the plane. To guarantee that all limit sets are identified, conditions for the existence or absence of limit cycles are presented. Combining these results with the study of return maps, a procedure is presented for a local bifurcation analysis of bifurcating equilibria in continuous, piecewise smooth systems. With this procedure, all limit sets that are created or destroyed by the bifurcation are identified in a computationally feasible manner.
Original languageEnglish
Pages (from-to)451-474
JournalNonlinear Analysis: Hybrid Systems
Volume4
Issue number3
DOIs
Publication statusPublished - 2010

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Limit Set
Continuous System
Hybrid systems
Differential equations
Bifurcation
Piecewise continuous
Piecewise Affine Systems
Return Map
Local Bifurcations
Bifurcation Analysis
Hybrid Systems
Limit Cycle
Partition
Differential equation

Cite this

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Nonsmooth bifurcations of equilibria in planar continuous systems. / Biemond, J.J.B.; Wouw, van de, N.; Nijmeijer, H.

In: Nonlinear Analysis: Hybrid Systems, Vol. 4, No. 3, 2010, p. 451-474.

Research output: Contribution to journalArticleAcademicpeer-review

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AU - Biemond, J.J.B.

AU - Wouw, van de, N.

AU - Nijmeijer, H.

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N2 - In this paper we present a procedure to find all limit sets near bifurcating equilibria in a class of hybrid systems described by continuous, piecewise smooth differential equations. For this purpose, the dynamics near the bifurcating equilibrium is locally approximated as a piecewise affine systems defined on a conic partition of the plane. To guarantee that all limit sets are identified, conditions for the existence or absence of limit cycles are presented. Combining these results with the study of return maps, a procedure is presented for a local bifurcation analysis of bifurcating equilibria in continuous, piecewise smooth systems. With this procedure, all limit sets that are created or destroyed by the bifurcation are identified in a computationally feasible manner.

AB - In this paper we present a procedure to find all limit sets near bifurcating equilibria in a class of hybrid systems described by continuous, piecewise smooth differential equations. For this purpose, the dynamics near the bifurcating equilibrium is locally approximated as a piecewise affine systems defined on a conic partition of the plane. To guarantee that all limit sets are identified, conditions for the existence or absence of limit cycles are presented. Combining these results with the study of return maps, a procedure is presented for a local bifurcation analysis of bifurcating equilibria in continuous, piecewise smooth systems. With this procedure, all limit sets that are created or destroyed by the bifurcation are identified in a computationally feasible manner.

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DO - 10.1016/j.nahs.2009.11.003

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