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Nonlinear Processes in Geophysics An interactive open-access journal of the European Geosciences Union

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© Author(s) 2018. This work is distributed under
the Creative Commons Attribution 4.0 License.
Research article
11 Apr 2018
Review status
This discussion paper is a preprint. It is a manuscript under review for the journal Nonlinear Processes in Geophysics (NPG).
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 1Universita’ 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. The transition to chaos induced by periodic forcing in systems that are not chaotic in the autonomous limit is studied with a four-dimensional nonlinear spectral ocean model. The analysis is based on the systematic construction of the system’s pullback attractors (PBAs) through ensemble simulations derived from a large number of initial states in the remote past. A preliminary analysis of the autonomous system is carried out by constructing its bifurcation diagram, as well as by calculating a metric measuring 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; 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.
Citation: Pierini, S., Chekroun, M. D., and Ghil, M.: The onset of chaos in nonautonomous dissipative dynamical systems: A low-order ocean–model case study, Nonlin. Processes Geophys. Discuss.,, in review, 2018.
Stefano Pierini et al.
Stefano Pierini et al.
Stefano Pierini et al.


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