Butterfly Effects of the First and Second Kinds in Lorenz Models

Over the span of 50 years, the pioneering study with the three-dimensional Lorenz model (3DLM; Lorenz,1963) and follow-up studies in 1969 and 1972 have changed our view on the predictability of weather and climate by revealing the so-called butterfly effect. Although the Lorenz ‘63 and ‘72 studies emphasize nonlinear dynamics (i.e., chaotic dynamics), people often apply a “simple” conceptual model that contains monotonic positive feedback but no negative nonlinear feedback to understand the characteristics of nonlinear solutions in the 3DLM. In this study, we (1) present examples to discuss common misunderstandings regarding the butterfly effect, (2) define the butterfly effect of the first and second kinds as the sensitive dependence of the solution on initial conditions and the enabling role of a tiny perturbation in producing an organized large-scale system (e.g., a tornado), respectively; (3) illustrate important but overlooked features, i.e., the boundedness and recurrence of the butterfly solutions in the 3DLM and high-dimensional LMs; (4) discuss the “diverged” relationship between the butterfly effect of the first and second kinds by analyzing the Lorenz models developed in 1963 and 1969/1972.

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