We report phase diagrams detailing the intransitivity observed in the climate scenarios supported by a prototype atmospheric general circulation model, namely, the Lorenz-84 low-order model. So far, this model was known to have a pair of coexisting climates described originally by Lorenz. Bifurcation analysis allows the identification of a remarkably wide parameter region where up to four climates coexist simultaneously. In this region the dynamical behavior depends crucially on subtle and minute tuning of the model parameters. This strong parameter sensitivity makes the Lorenz-84 model a promising candidate of testing ground to validate techniques of assessing the sensitivity of low-order models to perturbations of parameters.

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