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

  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},
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… 
Distribution of stable islands within chaotic areas in the non-hyperbolic and hyperbolic regimes in the Hénon–Heiles system
We provide rigorous computer-assisted proofs of the existence of different dynamical objects, like stable families of periodic orbits, bifurcations and stable invariant tori around them, in the
Unpredictability, Uncertainty and Fractal Structures in Physics
In Physics, we have laws that determine the time evolution of a given physical system, depending on its parameters and its initial conditions. When we have multi-stable systems, many attractors
Transient chaos enforces uncertainty in the British power grid
Multistability is a common phenomenon which naturally occurs in complex networks. If coexisting attractors are numerous and their basins of attraction are complexly interwoven, the long-term response
Influence of the gravitational radius on asymptotic behavior of the relativistic Sitnikov problem.
Two relevant issues on the influence of the gravitational radius in the context of chaotic scattering phenomena are added: first, a metamorphosis of the KAM islands for which the escape regions change insofar as λ increases, and there are two inflection points in the unpredictability of the final state of the system when λ≃0.02 and λ=0.028.
Analysis and chaos control of a four-dimensional magnetohydrodynamic model with hyperchaotic solutions
In this paper, the dynamical behavior of a four-dimensional magnetohydrodynamic model, consisting of a generalized Lorenz model, is investigated. A nonlinear dynamical analysis is performed using
The numerical search for the internal dynamics of NHIMs and their pictorial representation
A mechanism explaining the metamorphoses of KAM islands in nonhyperbolic chaotic scattering
In the context of nonhyperbolic chaotic scattering, it has been shown that the evolution of the KAM islands exhibits four abrupt metamorphoses that strongly affect the predictability of Hamiltonian


Resonant behavior and unpredictability in forced chaotic scattering
Chaotic scattering in open Hamiltonian systems is a topic of fundamental interest in physics, which has been mainly studied in the purely conservative case. However, the effect of weak perturbations
Limit of small exits in open Hamiltonian systems.
  • J. Aguirre, M. Sanjuán
  • Mathematics
    Physical 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
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.
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.
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
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
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
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
During the past few years, several papers (Aguirre J., Vallejo J. C. and Sanjuan 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