Hidden and unstable periodic orbits as a result of homoclinic bifurcations in the Suarez-Schopf delayed oscillator and the irregularity of ENSO

Abstract

We revisit the classical Suarez-Schopf delayed oscillator. Special attention is paid to the region of linear stability in the space of parameters. By means of the theory of inertial manifolds developed in our adjacent papers, we provide analytical-numerical evidence for the existence of two-dimensional inertial manifolds in the model. This allows to suggest a complete qualitative description of the dynamics in the region of linear stability. We show that there are two subregions corresponding to the existence of hidden or self-excited periodic orbits. These subregions must be separated by a curve on which homoclinic "figure eights", bifurcating into a single one or a pair of unstable periodic orbits, should exist. We relate the observed hidden oscillations and homoclinics to the irregularity theories of ENSO and provide numerical evidence that chaotic behavior may appear if a small periodic forcing is applied to the model. We also use parameters from the Suarez-Schopf model to discover hidden and self-excited asynchronous periodic regimes in a ring array of coupled lossless transmission lines studied by J. Wu and H. Xia.

Figures (12)

• Figure 1: The lower hidden curve (green), the upper hidden curve (orange) and the neutral curve (blue). In the region $\Omega_{se}$ between the orange and the blue curves, only phase portraits with self-excited periodic orbits corresponding to Fig. \ref{['fig: SSPPunstableTwo']} are observed. In the region $\Omega_{hid}$ between the green and the orange curves, only hidden periodic orbits with phase portraits corresponding to Fig. \ref{['fig: SSPPunstable']} are observed. The region $\Omega_{grad}$ under the green curve is expected to be nonoscillatory with a gradient-like behavior. At the points from the upper hidden curve phase portraits with homoclinic "figure eights" corresponding to Fig. \ref{['fig: SSPPHomo']} are expected.
• Figure 1: A self-excited symmetric periodic orbit with period $\sigma \approx 12.3$ ($\sigma_{years} \approx 8.16$) of \ref{['EQ: ElNinoSSmodel']}, where $\alpha=0.75$ and $\tau=1.65$, which can be localized from a small neighborhood of $\phi^{0}$. The blue and orange trajectories are attracted by the symmetric equilibria. All the trajectories are projected onto the $(c_{1},c_{2})$-plane by the projector $\Pi$.
• Figure 1: (Left): Some of the limiting regimes for iterations of the Poincaré map for the solutions starting from $\phi_{1} \equiv 5$ (red) and $\phi_{2} \equiv 5 + 0.00001$ (blue) of \ref{['EQ: SSmodelForcedAbstract']} with $W(t) = A\sin(t)$, $\tau=1.596$, $\alpha=0.75$ and different values of the amplitude $A$. All the iterations are projected onto the $(c_{1},c_{2})$-plane by the projector $\Pi$. (Right): The difference between the corresponding solutions.
• Figure 1: A numerically obtained region (blue) in the space of parameters $(\tau,\alpha)$, where $0 \leq \tau \leq 2$, of system \ref{['EQ: ElNinoSSmodel']}, for which there is a squeezing of two-dimensional volumes at the zero stationary state. The green curve is the lower hidden curve from Fig. \ref{['FIG: SSHiddenCurves']}.
• Figure 1: Results of the continuation by parameter procedure. At the initial step $\varepsilon=0$ we see the self-excited periodic orbit, the existence of which is guaranteed by Proposition \ref{['PROP: SSmodelHidden']}. At $\varepsilon=1$ this orbit evolves into a hidden periodic orbit of \ref{['EQ: ElNinoSSmodel']}. All the trajectories are projected onto the $(\phi(-\tau),\phi(0))$ plane.

Theorems & Definitions (12)

• Theorem 2.1
• Remark 2.2
• Remark 2.3
• Remark 2.4
• Remark 3.1
• Remark 3.2
• Remark 3.3
• Remark 4.1
• Proposition A.1
• Proof 1