The knowledge of periodic orbits of the model helps to explain some of these features like clear bimodality of PDFs of principal parameters. These PDFs have been reconstructed as weighted averages of periodic orbits with weights proportional to the period of the orbit and inversely proportional to its sum of positive Lyapunov exponents.
The fraction of time spent in the vicinity of each periodic orbit has been compared with its instability characteristics. The relationship between these values shows the 93\% correlation. The attractor dimension of the model has also been approximated as a weighted average of local attractor dimensions in vicinities of periodic orbits.
Three UPOs are presented below: one always skew-symmetric orbit wit respect to the middle line with T=38 days; one orbit with skew-symmetric average with T=41 days; and one orbit with no symmetry with T=216 days.
|Click images to see the streamfunction evolution (animated gif files, about 500K)|
|Energy vs Enstrophy for UPO T=38 days||Energy vs Enstrophy for UPO T=41 days|
Energy vs Enstrophy for UPO T=216 days
Click the image to see the streamfunction evolution (animated gif file, about 2300K)
and E. Kazantsev. Unstable periodic orbits and Attractors of Non Linear Dynamical Models. XXIII General Assembly of the European Geophysical Society. Nice, April 1998. Slides ( .ps.gz file).