Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Effective Reduced Models from Delay Differential Equations: Bifurcations, Tipping Solution Paths, and ENSO variability

Mickaël D. Chekroun, Honghu Liu

Abstract

Conceptual delay models have played a key role in the understanding of El Niño-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.

Effective Reduced Models from Delay Differential Equations: Bifurcations, Tipping Solution Paths, and ENSO variability

Abstract

Conceptual delay models have played a key role in the understanding of El Niño-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.
Paper Structure (19 sections, 3 theorems, 93 equations, 10 figures, 2 tables)

This paper contains 19 sections, 3 theorems, 93 equations, 10 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Galerkin-Koornwinder (GK) approximations of DDEs
  3. Koornwinder polynomials
  4. Space-time representation and GK approximations
  5. The formulas of GK approximations
  6. Center-Unstable Manifold Approximations from GK systems
  7. The leading-order approximation theorem
  8. Analytic formulas for higher-order approximations
  9. Bifurcations Analysis of a Delay El Niño-Southern Oscillation (ENSO) Model
  10. GK approximations of the Suarez and Schopf ENSO model
  11. Subcritical Hopf Bifurcation
  12. Saddle-Node bifurcation of periodic Orbits (SNO) and homoclinic orbit
  13. Approximation results of the stable and unstable DDE limit cycles
  14. Model error estimates
  15. Tipping Solution Paths and ENSO variability
  16. ...and 4 more sections

Key Result

Theorem 3.1

Assume that $G$ and $A(\tau)$ satisfy the assumptions recalled above, and that the PES condition PES is satisfied. Then for each $\tau$ in a neighborhood $\mathcal{U}$ of $\tau_c$, Eq. Eq_GK_tau admits a local invariant manifold, $\mathfrak{M}_\tau=\hbox{graph}(h_\tau)$, with $h_\tau$ that maps $H_\ where $G_k$ denotes the leading-order term in the Taylor expansion of $G$. Then, the Lyapunov-Perro

Figures (10)

  • Figure 1: Panel A shows a solution $u(t, \theta)$ to a transport problem of the form \ref{['lin_PDE']}--\ref{['PDE_BC']} obtained as a reformulation of the DDE $\dot{x}=a x(t-\tau)-b x(t-\tau)^3$ from CGLW16. Panel B shows the time series $u(t,0)$ at the right endpoint $\theta=0$. It coincides with the time series $x(t)$ obtained by solving the original DDE. The solution is shown for $a=0.5$, $b=20$ and $\tau=0.4$.
  • Figure 2: Eigenvalues dependence as $\tau$ crosses its critical value $\tau_c$. Are shown here, the first 10 pairs $(\lambda_j(\tau),\overline{\lambda_j(\tau)})$ as $\tau$ is increased from $\tau=1.3$ to $\tau=2.5$, for $\alpha=0.75$. These pairs move all from left to right. The pair of eigenvalues that crosses the imaginary axis for the critical delay parameter $\tau=\tau_c$ is shown in red, while the stable pairs are shown in black. These eigenvalues are computed from the GK linear part $A(\tau)$ given in \ref{['eq:A_SS']} in dimension $N=50$. It is noteworthy that these 10 pairs of GK eigenvalues (for $N=50$) satisfy the actual characteristic equation associated with the linear part of the DDE \ref{['Eq_SS_perturb']}, $\lambda = (1 - 3 T_{+}^2) - \alpha e^{-\lambda \tau}$, up to a maximal error of $10^{-4}$, as $\tau$ varies from $\tau=1.3$ to $\tau=2.5$.
  • Figure 3: Lyapunov coefficient $\ell^N_1(\tau_c)$ given in Eq. \ref{['Eq_l1_GK']}: $\alpha$-dependence. Here, the critical value $\tau_c$ and the entries in Eq. \ref{['Eq_l1_GK']} are computed from a high-dimensional GK approximation ($N=20$) of the (perturbed) Suarez and Schopf model \ref{['Eq_SS_perturb']}.
  • Figure 4: Bifurcation Diagram from the 2D Reduced GK System \ref{['Eq_EffectiveReduced_ENSO']}. This diagram describes precisely the local and global bifurcations occurring for the (perturbed) Suarez and Schopf model \ref{['Eq_SS_perturb']}, given the approximation skills of the reduced system \ref{['Eq_EffectiveReduced_ENSO']} in approximating the DDE's periodic orbits; see Sec. \ref{['Sec_approx_orbits']} below. As for Fig. \ref{['Fig_Hopf_SS']} below, it turned out to be sufficient to use $N=6$ in the construction of the parameterization $\Psi_{\tau}$ involved in Eq. \ref{['Eq_EffectiveReduced_ENSO']}, to reach such skills. In each inset, the stable steady states $T_+$ and $T_-$ given by Eq. \ref{['Eq_steady_states']} are marked by red dots with $T_+$ corresponding to $(0,0)$ since this diagram is computed for the perturbed DDE \ref{['Eq_EffectiveReduced_ENSO']}. This diagram reads as follows. The steady state $T_+$ (resp. $T_-$) is locally stable for all $\tau < \tau_c$, and loses its stability through a subcritical Hopf bifurcation at $\tau_c \approx 1.7408$; see Table \ref{['Table_Lyap_coeff']} and Eq. \ref{['Eq_tauc_true']}. As $\tau$ approaches $\tau^{\sharp} \approx 1.5906$ from above, the bifurcating UPOs (black dashed curves) merge into a homoclinic orbit (cyan curve) connecting the stable and unstable directions of the saddle steady state $T_0$, marked by an empty blue circle in inset A. This merging of UPOs terminates the branch of UPOs shown as dashed black line in the diagram. The diminishing of $\tau$ below $\tau^{\sharp}$ leads to new UPOs that encompass the homoclinic orbit, with amplitude that grows as $\tau$ approaches $\tau^{\sharp}$; see blue dashed curves in inset B. This latter branch of UPOs terminates at $\tau^* \approx 1.5592$ through a SNO bifurcation that gives rise to a branch of stable periodic orbits shown by the red curves in inset C.
  • Figure 5: Approximation results: DDE vs Effective Reduced GK system. The solutions are shown in lagged coordinates for $\alpha=0.75$. For different values of $\tau$ as indicated, the solutions to the DDE \ref{['Eq_SS_perturb']} are compared to those obtained from the formula of $T^\ast$ given by \ref{['Eq_SS_reconstruct2']}, built from the solutions of the 2D reduced GK system Eq. \ref{['Eq_EffectiveReduced_ENSO']} and the parameterization $\Psi_\tau$ given by \ref{['Eq_Phi']}. The stable (resp. unstable) DDE limit cycles are shown in black (resp. blue). The stable (resp. unstable) limit cycles, obtained from the 2D reduced system after lifting through \ref{['Eq_SS_reconstruct2']}, are shown by the orange (resp. cyan) dashed curves. The stable (resp. unstable) equilibria are shown as filled (resp. empty) circles. Here, it is sufficient to use a GK system of dimension $N=6$ to build the parameterization $\Psi_\tau$ and the modes and eigenvalues used in \ref{['Eq_SS_reconstruct2']}, in order to achieve such approximation skills.
  • ...and 5 more figures

Theorems & Definitions (5)

  • Theorem 3.1
  • Remark 3.1
  • Proposition 4.1
  • Proposition 1
  • Remark 1