Here's another suggestions to add to the dummy-variable approach suggested by @Aksakal.
From a signal-processing point of view, we can consider the different components of your series to be produced by several different waves of varying frequency. A classic example would be to express your time series $X_t$ as $$X_t = f(t) + g(t) + \epsilon_t$$
where $f(t)$ is some trend component, $g(t)$ is a seasonal/cyclical component, and $\epsilon_t$ is some perhaps serially-correlated high-frequency term that
can be modeled via ARIMA or your favorite time series regression. Since you only observe $X_t$, you need a way to discern these many different components from each other. One way to achieve this would be to consider decomposing the signal into its constituent component frequencies. For example, consider the Wavelet transform discussed in section 5.9 of the Hastie Tibshirani Friedman text. Here's a photographic example from that chapter:
As you can see, decomposing the wave exposes the underlying frequencies that drive what you observe in $X_t$. From here, you may consider checking the ACF/PACF of each component frequency and perhaps even thresholding some of the noise before inverting back to the original signal. If you find that some of the lower frequencies exhibit serial dependence, then you may be able to argue that there is an underlying cyclical component.
Certainly, the graphical nature of this approach is impractical and complicated in higher-dimensions, yet there are certainly tools available to automate or deal with this problem accordingly.