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We develop a geometric mechanism to prove the existence of orbits that drift along a prescribed sequence of cylinders, under some general conditions on the dynamics. This mechanism can be used to prove the existence of Arnold diffusion for large families of perturbations of Tonelli Hamiltonians on A^3. Our approach can also be applied to more general Hamiltonians that are not necessarily convex. The main geometric objects in our framework are –dimensional invariant cylinders with boundary (not necessarily hyperbolic), which are assumed to admit center-stable and center-unstable manifolds. These enable us to define chains of cylinders, i.e., finite, ordered families of cylinders where each cylinder admits homoclinic connections, and any two consecutive cylinders in the chain admit heteroclinic connections. Our main result is on the existence of diffusing orbits which drift along such chains of cylinders, under precise conditions on the dynamics on the cylinders – i.e., the existence of Poincaré sections with the return maps satisfying a tilt condition – and on the geometric properties of the intersections of the center-stable and center-unstable manifolds of the cylinders – i.e., certain compatibility conditions between the tilt map and the homoclinic maps associated to its essential invariant circles. We give two proofs of our result, a very short and abstract one, and a more constructive one, aimed at possible applications to concrete systems.
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Effective reduced models from delay differential equations: Bifurcations, tipping solution paths, and ENSO variability
Conceptual delay models have played a key role in the analysis and understanding of El Niño-Southern Oscillation (ENSO) variability. Based on such delay models, we propose in this work a novel scenario for the fabric of ENSO variability resulting from the subtle interplay between stochastic disturbances and nonlinear invariant sets emerging from bifurcations of the unperturbed dynamics. To identify these invariant sets we adopt an approach combining Galerkin–Koornwinder (GK) approximations of delay differential equations and center-unstable manifold reduction techniques. In that respect, GK approximation formulas are reviewed and synthesized, as well as analytic approximation formulas of center-unstable manifolds. The reduced systems derived thereof enable us to conduct a thorough analysis of the bifurcations arising in a standard delay model of ENSO. We identify thereby a saddle-node bifurcation of periodic orbits co-existing with a subcritical Hopf bifurcation, and a homoclinic bifurcation for this model. We show furthermore that the computation of unstable periodic orbits (UPOs) unfolding through these bifurcations is considerably simplified from the reduced systems. These dynamical insights enable us in turn to design a stochastic model whose solutions---as the delay parameter drifts slowly through its critical values---produce a wealth of temporal patterns resembling ENSO events and exhibiting also decadal variability. Our analysis dissects the origin of this variability and shows how it is tied to certain transition paths between invariant sets of the unperturbed dynamics (for ENSO’s interannual variability) or simply due to the presence of UPOs close to the homoclinic orbit (for decadal variability). In short, this study points out the role of solution paths evolving through tipping ‘‘points’’ beyond equilibria, as possible mechanisms organizing the variability of certain climate phenomena.
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- Award ID(s):
- 2108856
- PAR ID:
- 10509776
- Publisher / Repository:
- Elsevier
- Date Published:
- Journal Name:
- Physica D: Nonlinear Phenomena
- Volume:
- 460
- ISSN:
- 0167-2789
- Page Range / eLocation ID:
- 134058
- Subject(s) / Keyword(s):
- Center manifold reduction Galerkin–Koornwinder approximations Stochastic modeling Transition paths
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1016/j.physd.2024.134058
