# Measuring the transition between nonhyperbolic and hyperbolic regimes in open Hamiltonian systems

@article{Nieto2020MeasuringTT, title={Measuring the transition between nonhyperbolic and hyperbolic regimes in open Hamiltonian systems}, author={Alexandre R. Nieto and Euaggelos E. Zotos and Jes{\'u}s M. Seoane and Miguel A. F. Sanju'an}, journal={Nonlinear Dynamics}, year={2020}, volume={99}, pages={3029-3039} }

We show that the presence of KAM islands in nonhyperbolic chaotic scattering has deep implications on the unpredictability of open Hamiltonian systems. When the energy of the system increases, the particles escape faster. For this reason, the boundary of the exit basins becomes thinner and less fractal. Hence, we could expect a monotonous decrease in the unpredictability as well as in the fractal dimension. However, within the nonhyperbolic regime, fluctuations in the basin entropy have been…

## 12 Citations

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## References

SHOWING 1-10 OF 36 REFERENCES

Limit of small exits in open Hamiltonian systems.

- MathematicsPhysical review. E, Statistical, nonlinear, and soft matter physics
- 2003

It is claimed that in the limit of small exits, the invariant sets tend to fill up the whole phase space with the strong consequence that a new kind of basin appears, where the unpredictability grows indefinitely.

To Escape or not to Escape, that is the Question - perturbing the HéNon-Heiles Hamiltonian

- PhysicsInt. J. Bifurc. Chaos
- 2012

This work study the Henon–Heiles Hamiltonian, as a paradigm of open Hamiltonian systems, in the presence of different kinds of perturbations as dissipation, noise and periodic forcing finds an exponential-like decay law for the survival probability of the particles in the scattering region where the frequency of the forcing plays a crucial role.

Uncertainty dimension and basin entropy in relativistic chaotic scattering.

- PhysicsPhysical review. E
- 2018

This work focuses on the study of some relevant characteristics of the exit basin topology in the relativistic Hénon-Heiles system: the uncertainty dimension, the Wada property, and the basin entropy.

Wada basins and chaotic invariant sets in the Hénon-Heiles system.

- PhysicsPhysical review. E, Statistical, nonlinear, and soft matter physics
- 2001

The main goal of this paper is to show, by using various computational methods, that the corresponding exit basins of this open Hamiltonian are not only fractal, but they also verify the more restrictive property of Wada.

Basin entropy: a new tool to analyze uncertainty in dynamical systems

- Computer ScienceScientific reports
- 2016

The basin entropy provides an efficient method to probe the behavior of a system when different parameters are varied and provides a sufficient condition for the existence of fractal basin boundaries: when the basin entropy of the boundaries is larger than log2, the basin is fractal.

Elucidating the escape dynamics of the four hill potential

- Physics
- 2017

The escape mechanism of the four hill potential is explored. A thorough numerical investigation takes place in several types of two-dimensional planes and also in a three-dimensional subspace of the…

Fractal Control Boundaries of Driven Oscillators and Their Relevance to Safe Engineering Design

- Physics
- 1991

When a metastable, damped oscillator is driven by strong periodic forcing, the catchment basin of constrained motions in the space of the starting conditions {x(0),ẋ(0)} develops a fractal boundary…

Fractal structures in the Hénon-Heiles Hamiltonian

- Physics
- 2008

During the past few years, several papers (Aguirre J., Vallejo J. C. and Sanjuán M. A. F., Phys. Rev. E, 64 (2001) 066208; de Moura A. P. S. and Letelier P. S., Phys. Lett. A, 256 (1999) 362; Seoane…

Weakly noisy chaotic scattering.

- PhysicsPhysical review. E, Statistical, nonlinear, and soft matter physics
- 2013

This work investigates how a weak source of additive uncorrelated Gaussian noise affects both the dynamics and the topology of a paradigmatic chaotic scattering problem as the one taking place in the open nonhyperbolic regime of the Hénon-Heiles Hamiltonian system.

Chaotic Phenomena Triggering the Escape from a Potential Well

- Physics
- 1991

This paper explores the manner in which a driven mechanical oscillator escapes from the cubic potential well typical of a metastable system close to a fold. The aim is to show how the well-known…