# A constructive approach to robust chaos using invariant manifolds and expanding cones

@article{Glendinning2021ACA, title={A constructive approach to robust chaos using invariant manifolds and expanding cones}, author={Paul Glendinning and David J. W. Simpson}, journal={Discrete \& Continuous Dynamical Systems}, year={2021} }

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 rigorously in some parameter regimes. Here we provide formal results for robust chaos in the original parameter regime of [S. Banerjee, J.A. Yorke, C. Grebogi, Robust Chaos, Phys. Rev. Lett. 80(14):3049-3052, 1998]. We first construct a trapping region in phase space to prove the existence of a topological…

## 7 Citations

Chaos in the border-collision normal form: A computer-assisted proof using induced maps and invariant expanding cones

- MathematicsApplied Mathematics and Computation
- 2022

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

- MathematicsChaos
- 2022

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…

Renormalisation of the two-dimensional border-collision normal form

- Physics
- 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.…

Detecting invariant expanding cones for generating word sets to identify chaos in piecewise-linear maps

- MathematicsJournal of Difference Equations and Applications
- 2022

We show how the existence of three objects, $\Omega_{\rm trap}$, ${\bf W}$, and $C$, for a continuous piecewise-linear map $f$ on $\mathbb{R}^N$, implies that $f$ has a topological attractor with a…

On the difficulty of learning chaotic dynamics with RNNs

- Computer Science
- 2021

It is mathematically proved that RNNs producing stable equilibrium or cyclic behavior have bounded gradients, whereas the gradients of RNN's with chaotic dynamics always diverge, and ways of how to optimize the training process on chaotic data according to the system’s Lyapunov spectrum, regardless of the employed RNN architecture are suggested.

How to train RNNs on chaotic data?

- Computer ScienceArXiv
- 2021

It is mathematically prove that RNNs producing stable equilibrium or cyclic behavior have bounded gradients, whereas the gradients of RNN's with chaotic dynamics always diverge, and is offered an effective yet simple training technique for chaotic data and guidance on how to choose relevant hyperparameters according to the Lyapunov spectrum.

H OW TO TRAIN RNN S ON CHAOTIC DATA ?

- Computer Science
- 2021

It is mathematically prove that RNNs producing stable equilibrium or cyclic behavior have bounded gradients, whereas the gradients of RNN's with chaotic dynamics always diverge, and is offered an effective yet simple training technique for chaotic data and guidance on how to choose relevant hyperparameters according to the Lyapunov spectrum.

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