Journal of Nonlinear Science

, Volume 29, Issue 6, pp 2845–2875 | Cite as

Two-Parameter Boundary Equilibrium Bifurcations in 3D-Filippov Systems

  • Rony Cristiano
  • Daniel J. PaganoEmail author


This work concerns the analysis of boundary equilibrium bifurcations (BEBs) in 3D piecewise-smooth systems of Filippov type. In particular, we consider a family whose vector fields are linear on both sides of the switching boundary and which exhibit two parallel tangency lines each containing a cusp point. This configuration is observed in piecewise-linear control systems in which the control action is discontinuous such as the sliding mode control. We consider a general system of this class and then derive a canonical form to reduce the number of system parameters. The general objective in this work is, from the canonical form, to perform an analysis of the equilibria, stability, sliding dynamics and BEBs. The main result is the classification of the BEBs and its unfoldings in the sliding vector field. This and others results obtained on the existence and stability of equilibria are applied in a practical example involving the control of DC–DC buck power converter.


Filippov systems Boundary equilibrium Pseudo-equilibrium Sliding vector field Bifurcations Buck converter Sliding mode control 

Mathematics Subject Classification

34A36 34C20 34C23 34C60 34H05 



Daniel J. Pagano acknowledges CNPq/BRAZIL for partially funding its work under Project 302229/2018-3. Rony Cristiano acknowledges the CAPES/Brazil for funding its work.


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© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of Automation and SystemsFederal University of Santa CatarinaFlorianópolisBrazil

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