• Corpus ID: 237571414

Renormalisation of the two-dimensional border-collision normal form

@inproceedings{Ghosh2021RenormalisationOT,
  title={Renormalisation of the two-dimensional border-collision normal form},
  author={Indrani Ghosh and David J. W. Simpson},
  year={2021}
}
We study the two-dimensional border-collision normal form (a four-parameter family of continuous, piecewise-linear maps on R2) in the robust chaos parameter region of [S. Banerjee, J.A. Yorke, C. Grebogi, Robust Chaos, Phys. Rev. Lett. 80(14):3049– 3052, 1998]. We use renormalisation to partition this region by the number of connected components of a chaotic Milnor attractor. This reveals previously undescribed bifurcation structure in a succinct way. 

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Robust Devaney chaos in the two-dimensional border-collision normal form.
The collection of all non-degenerate, continuous, two-piece, piecewise-linear maps on R2 can be reduced to a four-parameter family known as the two-dimensional border-collision normal form. We prove

References

SHOWING 1-10 OF 30 REFERENCES
Renormalization of two-dimensional piecewise linear maps: Abundance of 2-D strange attractors
For a two parameter family of two-dimensional piecewise linear maps and for every natural number \begin{document} $n$ \end{document} , we prove not only the existence of intervals of parameters for
A constructive approach to robust chaos using invariant manifolds and expanding cones
Chaotic attractors in the two-dimensional border-collision normal form (a piecewise-linear map) can persist throughout open regions of parameter space. Such robust chaos has been established
Border collision route to quasiperiodicity: Numerical investigation and experimental confirmation.
TLDR
The present article reports the first experimental verification of the direct transition to quasiperiodicity through a border-collision bifurcation in the two-dimensional piecewise-linear normal form map.
Sequences of Periodic Solutions and Infinitely Many Coexisting Attractors in the Border-Collision Normal Form
TLDR
Several important features of the scenario are shown to be universal, and three examples are given, and infinite coexistence is proved directly by explicitly computing periodic solutions in the infinite sequence.
Neimark-Sacker Bifurcations in Planar, Piecewise-Smooth, Continuous Maps
TLDR
The multipliers of a fixed point of a piecewise-smooth, continuous map may change discontinuously as the fixed point crosses a discontinuity under smooth variation of parameters, similar to the Neimark–Sacker bifurcation of a smooth map.
Persistent two-dimensional strange attractors for a two-parameter family of Expanding Baker Maps
We characterize the attractors for a two-parameter class of two-dimensional piecewise affine maps. These attractors are strange attractors, probably having finitely many pieces, and coincide with the
Iterated maps on the interval as dynamical systems
Motivation and Interpretation.- One-Parameter Families of Maps.- Typical Behavior for One Map.- Parameter Dependence.- Systematics of the Stable Periods.- On the Relative Frequency of Periodic and
Unfolding homoclinic connections formed by corner intersections in piecewise-smooth maps.
TLDR
It is shown that a sequence of border-collision bifurcations limits to a homoclinic corner and that all nearby periodic solutions are unstable.
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