Mode-locking in Piecewise Smooth Maps

Bifurcation and Continuation


Resonance (mode-locking) tongues are commonly observed in smooth circle maps, and Neimark-Sacker bifurcations. The tongues take a different form for piecewise smooth maps as shown here for a planar, continuous map for an equilibrium that crosses a curve where the Jacobian is discontinuous and eigenvalues jump from inside to outside the unit circle.

The characteristic "sausage" shape of the tongues has also been observed for piecewise smooth circle maps.

Two parameter slices through the resonance tongues for a two-dimensional, piecewise affine, continuous map.


  • Yang, W.-M. and B.-L. Ho (1987). “How the Arnol'd Tongues Become Sausages in a Piecewise Linear Circle Map.” Comm. Theor. Phys 8: 1-15.
  • Simpson, D. J. W. and J. D. Meiss (2008). “Neimark-Sacker Bifurcations in Planar, Piecewise-Smooth, Continuous Maps.” SIAM J. Dyn. Sys. 7(3) 795-824.
Author Institutional AffiliationUniversity of Colorado
Author Email
KeywordsHopf, Neimark-Sacker, Arnold Tongue, Mode-locking

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