The onset of chaos in nonautonomous dissipative dynamical systems: a low-order ocean-model case study
<p>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 simulat...
| Main Authors: | , , |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
Copernicus Publications
2018-09-01
|
| Series: | Nonlinear Processes in Geophysics |
| Online Access: | https://www.nonlin-processes-geophys.net/25/671/2018/npg-25-671-2018.pdf |
| _version_ | 1818993050660110336 |
|---|---|
| author | S. Pierini S. Pierini M. D. Chekroun M. Ghil M. Ghil |
| author_facet | S. Pierini S. Pierini M. D. Chekroun M. Ghil M. Ghil |
| author_sort | S. Pierini |
| collection | DOAJ |
| description | <p>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 <i>ε</i>. 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 <i>ε</i><sub>c</sub>
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.</p> |
| first_indexed | 2024-12-20T20:35:54Z |
| format | Article |
| id | doaj.art-aad6909003f64b25a2022cd663bb7aa4 |
| institution | Directory Open Access Journal |
| issn | 1023-5809 1607-7946 |
| language | English |
| last_indexed | 2024-12-20T20:35:54Z |
| publishDate | 2018-09-01 |
| publisher | Copernicus Publications |
| record_format | Article |
| series | Nonlinear Processes in Geophysics |
| spelling | doaj.art-aad6909003f64b25a2022cd663bb7aa42022-12-21T19:27:14ZengCopernicus PublicationsNonlinear Processes in Geophysics1023-58091607-79462018-09-012567169210.5194/npg-25-671-2018The onset of chaos in nonautonomous dissipative dynamical systems: a low-order ocean-model case studyS. Pierini0S. Pierini1M. D. Chekroun2M. Ghil3M. Ghil4Dipartimento di Scienze e Tecnologie, Universita' di Napoli Parthenope, Naples, ItalyCoNISMa, Rome, ItalyUniversity of California at Los Angeles, Los Angeles, California, USAUniversity of California at Los Angeles, Los Angeles, California, USAEcole Normale Supérieure and PSL Research University, Paris, France<p>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 <i>ε</i>. 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 <i>ε</i><sub>c</sub> 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.</p>https://www.nonlin-processes-geophys.net/25/671/2018/npg-25-671-2018.pdf |
| spellingShingle | S. Pierini S. Pierini M. D. Chekroun M. Ghil M. Ghil The onset of chaos in nonautonomous dissipative dynamical systems: a low-order ocean-model case study Nonlinear Processes in Geophysics |
| title | The onset of chaos in nonautonomous dissipative dynamical systems: a low-order ocean-model case study |
| title_full | The onset of chaos in nonautonomous dissipative dynamical systems: a low-order ocean-model case study |
| title_fullStr | The onset of chaos in nonautonomous dissipative dynamical systems: a low-order ocean-model case study |
| title_full_unstemmed | The onset of chaos in nonautonomous dissipative dynamical systems: a low-order ocean-model case study |
| title_short | The onset of chaos in nonautonomous dissipative dynamical systems: a low-order ocean-model case study |
| title_sort | onset of chaos in nonautonomous dissipative dynamical systems a low order ocean model case study |
| url | https://www.nonlin-processes-geophys.net/25/671/2018/npg-25-671-2018.pdf |
| work_keys_str_mv | AT spierini theonsetofchaosinnonautonomousdissipativedynamicalsystemsaloworderoceanmodelcasestudy AT spierini theonsetofchaosinnonautonomousdissipativedynamicalsystemsaloworderoceanmodelcasestudy AT mdchekroun theonsetofchaosinnonautonomousdissipativedynamicalsystemsaloworderoceanmodelcasestudy AT mghil theonsetofchaosinnonautonomousdissipativedynamicalsystemsaloworderoceanmodelcasestudy AT mghil theonsetofchaosinnonautonomousdissipativedynamicalsystemsaloworderoceanmodelcasestudy AT spierini onsetofchaosinnonautonomousdissipativedynamicalsystemsaloworderoceanmodelcasestudy AT spierini onsetofchaosinnonautonomousdissipativedynamicalsystemsaloworderoceanmodelcasestudy AT mdchekroun onsetofchaosinnonautonomousdissipativedynamicalsystemsaloworderoceanmodelcasestudy AT mghil onsetofchaosinnonautonomousdissipativedynamicalsystemsaloworderoceanmodelcasestudy AT mghil onsetofchaosinnonautonomousdissipativedynamicalsystemsaloworderoceanmodelcasestudy |