## 1 Introduction

[2] Prediction is the ultimate goal of meteorology and climatology, and the issue of predictability is crucial. Prediction in these disciplines is mostly probabilistic, but there may be different rationales for a probabilistic description. These rationales are intimately linked to the various meanings of the concepts of determinism, deterministic chaos, and stochasticity. The fundamental laws of classical physics are deterministic, since the future and past of the state of such a system is uniquely given by the state at a given time. The majority of models of weather and climate are deterministic in this sense. Predictability in chaotic systems is limited by the largest positive Lyapunov exponent. This may require a probabilistic description, not only in systems with a large number of degrees of freedom, but also in simple, low-dimensional, nonlinear systems. When dealing with data in climatology, either from observation or from large-scale simulations, the climate dynamicist will have to ask the question of whether prediction for the system/phenomenon of interest is better served by a low-dimensional chaotic model of climate variability, or by a high-dimensional (stochastic) model. Hence, there is a demand for methods by which it is possible to decide from the observation data whether the system dynamics can be uniquely projected onto an attractor in a low-dimensional phase space. If the system is autonomous and the attractor of the trajectory has dimension *d*, the Takens' time-delay method [*Takens*, 1981] can be used to construct an *m* > 2*d*–dimensional embedding space on which the attractor can be mapped continuously and one to one. In practice this method works only if the attractor dimension *d* is reasonably low. Dynamic systems with a large number of independent or weakly dependent degrees of freedom can only be described either by large-scale numerical simulation or by stochastic methods. For such systems, the phase-space attractor is also high dimensional and cannot be mapped one to one onto a low-dimensional time-delay embedding space. The computation of attractor dimension then typically fails to converge when embedding dimension *m* is increased, but such convergence can be difficult to detect if the time series is short. *Kaplan and Glass* [1992, 1993] devised a direct test for the existence of low-dimensional deterministic dynamics which is useful for short time series. This is the kind of test that will be employed in this paper to the instrumental time series for the El Niño Southern Oscillation (ENSO).

[3] The problem of determinism versus stochasticity in ENSO has been a subject of research for more than two decades, and conflicting results have been reported. *Bauer and Brown* [1992] analyzed Eastern tropical Pacific sea surface temperature (SST) anomalies via the reconstruction of the attractor by means of singular spectrum analysis and found indications of low-dimensional dynamics. Elsner and Tsonis [1993] found that the Southern Oscillation index (SOI) exhibits signatures of nonlinear dynamics by employing a nonlinear prediction algorithm to the SOI time series and to surrogate data exhibiting the same correlation structure but devoid of the nonlinearities. On the other hand, *Schreiber and Schmitz* [2000], using a more accurate scheme, came to the conclusion that the linear null hypothesis cannot be rejected on the basis of the SOI time series. The same conclusion was drawn by *Binder and Wilches* [2002] from the application of tests to the SOI which are similar to those we employ in the present paper. The data analyzed in all these papers are anomalies, i.e., the seasonal cycle is eliminated by considering the departure from the mean value for the season. There is a general consensus that ENSO is strongly and nonlinearly coupled and phase synchronized with the annual cycle [*Stein et al*., 2011], so this is not an issue of controversy. The response of the tropical Pacific to the seasonal forcing is nonlinear, and the seasonal timing of ENSO episodes (its phase) is nonlinearly connected to the seasonal forcing. But that does not imply that the trigging of a Niño episode one year, and not another year, is a result of low-dimensional, nonlinear dynamics alone. In this paper, we explore the significance of this distinction by analyzing the Niño 3 index, which is the average of the SST in the region 150°W–90°W and 5°N–5°S, both with the seasonal cycle included and as an anomaly with the cycle subtracted. For the former, we find low-dimensional determinism and nonlinearity, and for the latter, we do not.

[4] The remainder of the paper is structured as follows. In section 2 we briefly describe a time-delay equation for equatorial wave dynamics and ENSO, and in section 3 we review the tests for low-dimensional determinism and nonlinearity. In section 4 these techniques are first applied to the SOI and fail to reveal low-dimensional deterministic dynamics, in agreement with *Binder and Wilches* [2002]. We then employ them to the Niño 3 data and to numerical solutions of the time-delay equation with and without seasonality subtracted and with and without stochastic forcing. We also apply a superposed-epoch analysis to these data to highlight the characteristic waveforms of ENSO episodes as manifested in the Niño 3 signal. The implications of our findings are discussed in this section and summarized in section 5.