Low-dimensional paradigms for high-dimensional hetero-chaos.

@article{Saiki2018LowdimensionalPF,
  title={Low-dimensional paradigms for high-dimensional hetero-chaos.},
  author={Yoshitaka Saiki and Miguel A. F. Sanju{\'a}n and James A. Yorke},
  journal={Chaos},
  year={2018},
  volume={28 10},
  pages={
          103110
        }
}
The dynamics on a chaotic attractor can be quite heterogeneous, being much more unstable in some regions than others. Some regions of a chaotic attractor can be expanding in more dimensions than other regions. Imagine a situation where two such regions and each contains trajectories that stay in the region for all time-while typical trajectories wander throughout the attractor. Furthermore, if arbitrarily close to each point of the attractor there are points on periodic orbits that have… Expand
5 Citations
Piecewise linear maps with heterogeneous chaos
Chaotic dynamics can be quite heterogeneous in the sense that in some regions the dynamics are unstable in more directions than in other regions. When trajectories wander between these regions, theExpand
Lagrangian descriptors for open maps.
TLDR
The concept of Lagrangian descriptors, which have been recently introduced as efficient indicators of phase space structures in chaotic systems, are applied to the open tribaker map, a paradigmatic example not only in classical but also in quantum chaos. Expand
Forecasting and Chaos
TLDR
A chaotic orbit can be chaotic and still be predictable, in the sense that the chaotic orbit is followed, or shadowed, by a real orbit, thus making its predictions physically valid, and the presence of chaos does not always imply a low predictability. Expand
Dynamical Regimes and Time Scales
The key factor to build the finite-time distributions is finding the most adequate interval length, to be large enough to ensure a satisfactory reduction of the local fluctuations, but small enoughExpand
Artificial Intelligence, Chaos, Prediction and Understanding in Science
TLDR
The main thesis here is that prediction and understanding are two very different and important ideas that should guide us about the progress of science. Expand

References

SHOWING 1-10 OF 68 REFERENCES
Bifurcation to High-Dimensional Chaos
TLDR
A heuristic theory and numerical examples are presented to illustrate one possible route to high-dimensional chaos, by which a subsystem becomes chaotic with one positive Lyapunov exponent via one of the known routes to low-dimensional Chaos, after which the complementary subsystem become chaotic, leading to additional positive LyAPunov exponents for the whole system. Expand
Multichaos from Quasiperiodicity
TLDR
A simple 2-dimensional paradigm for multichaos is presented, in which a quasiperiodic orbit plays the key role, replacing the large hyperbolic set. Expand
Crossing bifurcations and unstable dimension variability.
TLDR
A global bifurcation in three dimensions which can result in a crisis is described and a new scaling law is derived describing the density of the new portion of the attractor formed in the crisis. Expand
Unstable dimension variability: a source of nonhyperbolicity in chaotic systems
The hyperbolicity or nonhyperbolicity of a chaotic set has profound implications for the dynamics on the set. A familiar mechanism causing nonhyperbolicity is the tangency of the stable and unstableExpand
Dimension of chaotic attractors
Dimension is perhaps the most basic property of an attractor. In this paper we discuss a variety of different definitions of dimension, compute their values for a typical example, and review previousExpand
Blowout bifurcations: the occurrence of riddled basins and on-off intermittency
Abstract We consider situations where a nonlinear dynamical system possesses a smooth invariant manifold. For parameter values p less than a critical value pc, the invariant manifold has within it aExpand
Periodic orbit analysis at the onset of the unstable dimension variability and at the blowout bifurcation.
TLDR
A new mechanism is presented, whereby the period of the unstable orbits losing transversal stability tends to infinity as the authors approach the onset of UDV. Expand
Unstable periodic trajectories of a barotropic model of the atmosphere
Abstract Unstable periodic trajectories of a chaotic dissipative system belong to the attractor of the system and are its important characteristics. Many chaotic systems have an infinite number ofExpand
Statistical characteristics, circulation regimes and unstable periodic orbits of a barotropic atmospheric model
  • A. Gritsun
  • Mathematics, Medicine
  • Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences
  • 2013
TLDR
This study will try to apply the idea of UPO expansion to the simple atmospheric system based on the barotropic vorticity equation of the sphere to check how well orbits approximate the system attractor, its statistical characteristics and PDF. Expand
Bubbling bifurcation: Loss of synchronization and shadowing breakdown in complex systems
Abstract Complex dynamical systems with many degrees of freedom may exhibit a wealth of collective phenomena related to high-dimensional chaos. This paper focuses on a lattice of coupled logisticExpand
...
1
2
3
4
5
...