We consider the localized nonlinear breathing modes that emerge inheterogeneous granular configurations of two materials with a periodicity ofthree and four beads. We examine as characteristic examples chains with 1 steeland 2 alumnium beads, as well as ones with 1 steel and three aluminum beads. Weanalyze the higher order gaps that emerge in such settings and explore theintrinsic localized modes that bifurcate from the edge of the upper bands. Ageneric surprising feature of such states is that they appear to be more robustthan their counterparts bifurcating from the edges of the lower bands. Directnumerical simulations, using driving of the system at suitable frequenciesthrough an actuator or taking advantage of the modulational instabilities ofextended band edge states in the system illustrate the spontaneous formation oflocalized modes within the corresponding nearest gaps.
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