D J W Simpson and J D Meiss 2009 Nonlinearity 22 1123 doi:10.1088/0951-7715/22/5/009
Shrinking point bifurcations of resonance tongues for piecewise-smooth, continuous maps
D J W Simpson and J D Meiss
Show affiliationsRecommended by A Chenciner
Resonance tongues are mode-locking regions of parameter space in which stable periodic solutions occur; they commonly occur, for example, near Neimark–Sacker bifurcations. For piecewise-smooth, continuous maps these tongues typically have a distinctive lens-chain (or sausage) shape in two-parameter bifurcation diagrams. We give a symbolic description of a class of 'rotational' periodic solutions that display lens-chain structures for a general N-dimensional map. We then unfold the codimension-two, shrinking point bifurcation, where the tongues have zero width. A number of codimension-one bifurcation curves emanate from shrinking points and we determine those that form tongue boundaries.
- PACS
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45.10.Db Variational and optimization methods
05.45.Ac Low-dimensional chaos
45.20.Jj Lagrangian and Hamiltonian mechanics
- MSC
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37G15 Bifurcations of limit cycles and periodic orbits
37E45 Rotation numbers and vectors
65H17 Eigenvalues, eigenvectors (See also 47Hxx, 47Jxx, 58C40, 58E07, 90C30)
37E30 Homeomorphisms and diffeomorphisms of planes and surfaces
- Subjects
- Dates
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Issue 5 (May 2009)
Received 20 September 2008, in final form 19 March 2009
Published 16 April 2009
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Shrinking point bifurcations of resonance tongues for piecewise-smooth, continuous maps
D J W Simpson and J D Meiss 2009 Nonlinearity 22 1123
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