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Understanding instabilities in dynamical systems drives to the heart ofmodern chaos theory, whether forecasting or attempting to control futureoutcomes. Instabilities in the sense of locally maximal stretching in maps iswell understood, and is connected to the concepts of Lyapunovexponents/vectors, Oseledec spaces and the Cauchy--Green tensor. In this paper,we extend the concept to global optimization of stretching, as this forms askeleton organizing the general instabilities. The `map' is general butincorporates the inevitability of finite-time as in any realistic application:it can be defined via a finite sequence of discrete maps, or a finite-time flowassociated with a continuous dynamical system. Limiting attention totwo-dimensions, we formulate the global optimization problem as one over arestricted class of foliations, and establish the foliations which bothmaximize and minimize global stretching. A classification of nondegeneratesingularities of the foliations is obtained. Numerical issues in computingoptimal foliations are examined, in particular insights into special curvesalong which foliations appear to veer and/or do not cross, and foliationbehavior near singularities. Illustrations and validations of the results tothe H\'{e}non map, the double-gyre flow and the standard map are provided.

Globally optimal stretching foliations of dynamical systems reveal the organizing skeleton of intensive instabilities