Limit cycles bifurcated from a focus-fold singularity in general piecewise smooth planar systems.

• A normal form is derived for codimension-two focus-fold bifurcation. • Observations of focus-fold bifurcation are extended from piecewise linear systems to general piecewise smooth systems. • Two types of sliding limit cycles are bifurcated from the focus-fold singularity for general piecewise smooth systems. • Two nested crossing limit cycles are bifurcated from the focus-fold singularity for general piecewise smooth systems. • Convergence in finite time is proved for sliding limit cycles. We consider codimension-two focus-fold bifurcation for general piecewise smooth (PWS) planar systems. Some changes of variables and parameters are employed to derive a normal form, which is topologically equivalent to the original system. Making use of the normal form, we verify that observations of focus-fold bifurcation in some special piecewise linear systems also hold for general PWS systems, including the existence of two types of sliding limit cycles and coexistence of crossing and sliding limit cycles. It is shown that two nested crossing limit cycles can be bifurcated from the focus-fold singularity. This result is the improvement of the work [Han and Zhang (2010) [29] ], in which only one bifurcated limit cycle is obtained. In addition, the sliding limit cycles are proved to be convergent in finite time. [ABSTRACT FROM AUTHOR]