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Chenciner bubbles and torus break-up in a periodically forced delay differential equation. (English) Zbl 1393.37094

The authors consider the periodically forced delay differential equation \[ \dot h=-b\tanh{[\kappa h(t-\tau)]}+c\cos{(2\pi t)}, \] modelling the dynamics of the thermocline depth, \(h\), at the Eastern boundary of the Pacific Ocean. The delayed feedback term can cause oscillations in \(h\) which may then interact with the periodic forcing to give quasiperiodic solutions, each lying on a torus. As the parameters \(c\) and \(\tau\) are varied, tori may be destroyed in what appear to be saddle-node bifurcations. However, two smooth tori cannot meet and be destroyed in a simple way: they must break up and be destroyed in complicated bifurcation scenarios. These bifurcations occur within a Chenciner bubble in parameter space. This paper presents the first analysis of the bifurcations within a Chenciner bubble for a flow (as opposed to a map) and for a delay differential equation. The techniques used are the numerical continuation of periodic orbits and the calculation of their stable and unstable manifolds, and the direct numerical integration, of the above delay differential equation. Implications of the bifurcations in this climate model are also discussed.

MSC:

37M20 Computational methods for bifurcation problems in dynamical systems
37N10 Dynamical systems in fluid mechanics, oceanography and meteorology
37L05 General theory of infinite-dimensional dissipative dynamical systems, nonlinear semigroups, evolution equations
37G35 Dynamical aspects of attractors and their bifurcations
65P30 Numerical bifurcation problems
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