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Volume 25, issue 3 | Copyright

Special issue: Numerical modeling, predictability and data assimilation in...

Nonlin. Processes Geophys., 25, 671-692, 2018
https://doi.org/10.5194/npg-25-671-2018
© Author(s) 2018. This work is distributed under
the Creative Commons Attribution 4.0 License.

Research article 10 Sep 2018

Research article | 10 Sep 2018

The onset of chaos in nonautonomous dissipative dynamical systems: a low-order ocean-model case study

Stefano Pierini1,2, Mickaël D. Chekroun3, and Michael Ghil3,4 Stefano Pierini et al.
  • 1Dipartimento di Scienze e Tecnologie, Universita' di Napoli Parthenope, Naples, Italy
  • 2CoNISMa, Rome, Italy
  • 3University of California at Los Angeles, Los Angeles, California, USA
  • 4Ecole Normale Supérieure and PSL Research University, Paris, France

Abstract. A four-dimensional nonlinear spectral ocean model is used to study the transition to chaos induced by periodic forcing in systems that are nonchaotic in the autonomous limit. The analysis relies on the construction of the system's pullback attractors (PBAs) through ensemble simulations, based on a large number of initial states in the remote past. A preliminary analysis of the autonomous system is carried out by investigating its bifurcation diagram, as well as by calculating a metric that measures the mean distance between two initially nearby trajectories, along with the system's entropy. We find that nonchaotic attractors can still exhibit sensitive dependence on initial data over some time interval; this apparent paradox is resolved by noting that the dependence only concerns the phase of the periodic trajectories, and that it disappears once the latter have converged onto the attractor. The periodically forced system, analyzed by the same methods, yields periodic or chaotic PBAs depending on the periodic forcing's amplitude ε. A new diagnostic method – based on the cross-correlation between two initially nearby trajectories – is proposed to characterize the transition between the two types of behavior. Transition to chaos is found to occur abruptly at a critical value εc and begins with the intermittent emergence of periodic oscillations with distinct phases. The same diagnostic method is finally shown to be a useful tool for autonomous and aperiodically forced systems as well.

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A four-dimensional nonlinear spectral ocean model is used to study the transition to chaos induced by periodic forcing in systems that are nonchaotic in the autonomous limit. The analysis makes use of ensemble simulations and of the system's pullback attractors. A new diagnostic method characterizes the transition to chaos: this is found to occur abruptly at a critical value and begins with the intermittent emergence of periodic oscillations with distinct phases.
A four-dimensional nonlinear spectral ocean model is used to study the transition to chaos...
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