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. A four-dimensional nonlinear spectral ocean model is usedto study the transition to chaos induced by periodic forcing in systems thatare nonchaotic in the autonomous limit. The analysis relies on theconstruction of the system's pullback attractors (PBAs) through ensemblesimulations, based on a large number of initial states in the remote past. Apreliminary analysis of the autonomous system is carried out by investigatingits bifurcation diagram, as well as by calculating a metric that measures themean distance between two initially nearby trajectories, along with thesystem's entropy. We find that nonchaotic attractors can still exhibitsensitive dependence on initial data over some time interval; this apparentparadox is resolved by noting that the dependence only concerns the phase ofthe periodic trajectories, and that it disappears once the latter haveconverged onto the attractor. The periodically forced system, analyzed by thesame methods, yields periodic or chaotic PBAs depending on the periodicforcing's amplitude ε. A new diagnostic method – based on thecross-correlation between two initially nearby trajectories – is proposedto characterize the transition between the two types of behavior. Transitionto chaos is found to occur abruptly at a critical value εcand begins with the intermittent emergence of periodic oscillations withdistinct phases. The same diagnostic method is finally shown to be a usefultool for autonomous and aperiodically forced systems as well.

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