  We show that the continuous wavelet transform can provide a unique
decomposition of a timeseries in to 'signal-like' and 'noise-like' components:
  From the overall wavelet spectrum two mutually independent skeleton spectra
can be extracted, allowing the separate detection and monitoring in even
non-stationary timeseries of the evolution of (a) both stable but also
transient, evolving periodicities, such as the output of low dimensional
dynamical systems and (b) scale-invariant structures, such as discontinuities,
self-similar structures or noise. An indicative application to the
monthly-averaged sunspot index reveals, apart from the well-known 11-year
periodicity, 3 of its harmonics, the 2-year periodicity (quasi-biennial
oscillation, QBO) and several more (some of which detected previously in
various solar, earth-solar connection and climate indices), here proposed being
just harmonics of the QBO, in all supporting the double-cycle solar magnetic
dynamo model (Benevolenskaya, 1998, 2000). The scale maximal spectrum reveals
the presence of 1/f fluctuations with timescales up to 1 year in the sunspot
number, indicating that the solar magnetic configurations involved in the
transient solar activity phenomena with those characteristic timescales are in
a self-organized-critical state (SOC), as previously proposed for the solar
flare occurence (Lu and Hamilton, 1991).
