  Conceptual delay models have played a key role in the understanding of El
Ni\~no-Southern Oscillation (ENSO) variability. Based on such delay models, we
propose 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.
