The odd symmetry of the wave sloshing in the previous comment may be classified as a tripole of sorts.

The period doubling seems to be an emergent feature of nonlinear waves, as modeled by the Mathieu equation

> ![mathieu](http://imageshack.com/a/img923/6401/tORmHR.png)

> from Rajchenbach, Jean, and Didier Clamond. "Faraday waves: their dispersion relation, nature of bifurcation and wavenumber selection revisited." Journal of Fluid Mechanics 777 (2015): R2.

The nonlinearity comes from mutiplicative mixing of waves of different periods. Period doubling is somewhat reinforcing, as a harmonic that has a period of one year when mixed with a period of two years will recover the two year period.

$ sin(2\pi t)sin(\pi t) = \frac{1}{2} (cos(\pi t) - cos(3\pi t) )$

But the "somewhat" is a caveat, as what decides the alignment of the doubling on an odd versus even year is inherently arbitrary and therefore metastable.

One can almost see the potential tri-fold symmetry of the wave sloshing in this paper discussing a Pacific Ocean tripole.

> ![tripole](http://imageshack.com/a/img923/1705/QrHU54.png)

> from Henley, Benjamin J., et al. "A tripole index for the interdecadal Pacific oscillation." Climate Dynamics 45.11-12 (2015): 3077-3090.

My own theory is that climate shifts such as occur with the IPO (circa late 1970's) are associated with a period doubling phase inversion between even and odd years.

The period doubling seems to be an emergent feature of nonlinear waves, as modeled by the Mathieu equation

> ![mathieu](http://imageshack.com/a/img923/6401/tORmHR.png)

> from Rajchenbach, Jean, and Didier Clamond. "Faraday waves: their dispersion relation, nature of bifurcation and wavenumber selection revisited." Journal of Fluid Mechanics 777 (2015): R2.

The nonlinearity comes from mutiplicative mixing of waves of different periods. Period doubling is somewhat reinforcing, as a harmonic that has a period of one year when mixed with a period of two years will recover the two year period.

$ sin(2\pi t)sin(\pi t) = \frac{1}{2} (cos(\pi t) - cos(3\pi t) )$

But the "somewhat" is a caveat, as what decides the alignment of the doubling on an odd versus even year is inherently arbitrary and therefore metastable.

One can almost see the potential tri-fold symmetry of the wave sloshing in this paper discussing a Pacific Ocean tripole.

> ![tripole](http://imageshack.com/a/img923/1705/QrHU54.png)

> from Henley, Benjamin J., et al. "A tripole index for the interdecadal Pacific oscillation." Climate Dynamics 45.11-12 (2015): 3077-3090.

My own theory is that climate shifts such as occur with the IPO (circa late 1970's) are associated with a period doubling phase inversion between even and odd years.