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This paper introduces a small-gain sufficient condition for $2$-contraction
of feedback interconnected systems, on the basis of individual gains of
suitable subsystems arising from a modular decomposition of the second additive
compound equation. The condition applies even to cases when individual
subsystems might fail to be contractive (due to the extra margin of contraction
afforded by the second additive compound matrix). Examples of application are
provided to illustrate the theory and show its degree of conservatism and scope
of applicability.
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