Robust Devaney chaos in the two-dimensional border-collision normal form.

@article{Ghosh2022RobustDC,
  title={Robust Devaney chaos in the two-dimensional border-collision normal form.},
  author={Indranil Ghosh and David J. W. Simpson},
  journal={Chaos},
  year={2022},
  volume={32 4},
  pages={
          043120
        }
}
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 that throughout an open region of parameter space, this family has an attractor satisfying Devaney's definition of chaos. This strengthens the existing results on the robustness of chaos in piecewise-linear maps. We further show that the stable manifold of a saddle fixed point, despite being a one… 

Figures from this paper

References

SHOWING 1-10 OF 33 REFERENCES
Renormalisation of the two-dimensional border-collision normal form
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.
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
Equilibrium-torus bifurcation in nonsmooth systems
Robust chaos and the continuity of attractors
As the parameters of a map are varied an attractor may vary continuously in the Hausdorff metric. The purpose of this paper is to explore the continuation of chaotic attractors. We argue that this is
Border-Collision Bifurcations in ℝN
TLDR
This article reviews border-collision bifurcations with a general emphasis on results that apply to maps of any number of dimensions.
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.
Invariant polygons in systems with grazing-sliding.
TLDR
Generic three-dimensional nonsmooth systems with a periodic orbit near grazing-sliding that consists of forward sliding orbits are investigated and the number of lines involved in forming the attractor is classified as a function of the parameters.
Elements of Contemporary Theory of Dynamical Chaos: A Tutorial. Part I. Pseudohyperbolic Attractors
The paper is devoted to topical issues of modern mathematical theory of dynamical chaos and its applications. At present, it is customary to assume that dynamical chaos in finite-dimensional smooth
Boundary crisis bifurcation in two parameters
The boundary crisis bifurcation is well known as a mechanism for destroying (or creating) a strange attractor by variation of one parameter: at the moment of the boundary crisis bifurcation the
...
1
2
3
4
...