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We prove that if a certain entry in the map of the Hadamard-Perron theorem is $T$-periodic in one of the variables, then the stable manifold guaranteed by the Hadamard-Perron theorem is a graph of a $T$-periodic function. As an application, we extend the classical Levinson's result about the occurrence of an attracting closed invariant curve near a stable cycle of a system of autonomous equations under periodic perturbations to hybrid differential equations.

A recent result by Gudoshnikov et al [SIAM J. Control Optim. 2022] ensures finite-time convergence of the stress-vector of an elastoplastic lattice spring model under assumption that the vector $g'(t)$ of the applied displacement controlled-loading lies strictly inside the normal cone to the associated polyhedral set (that depends on mechanical parameters of the springs). Determination of the terminal distribution of stresses has been thereby linked to a problem of spotting a face on the boundary of the polyhedral set where the normal cone contains vector $g'(t)$. In this paper the above-mentioned problem of spotting an eligible face is converted into a search for an eligible set of springs that have a certain topological property with respect to the entire graph (of springs). Specifically, we prove that eligible springs are those that keep non-zero lengths after their nodes are collapsed with the nodes of the displacement-controlled loading after a finite number of eligible displacements of the nodes of the graph. The proposed result allows to judge about possible distribution of plastic deformations in elastoplastic lattice spring models directly from the topology of the associated graph of springs. A benchmark example is provided.

Disordered network materials abound in both nature and synthetic situations while rigorous analysis of their nonlinear mechanical behaviors still is very challenging. The purpose of this paper is to connect the mathematical framework of sweeping process originally proposed by Moreau to a generic class of Lattice Spring Models with plasticity phenomenon. We explicitly construct a sweeping process and provide numerical schemes to find the evolution of stresses in a Lattice Spring Model with infinitesimal strains and perfect plasticity. In particular, we develop a highly efficient "leapfrog" computational framework that allow ones to rigorously track the progression of plastic events in the system based on the sweeping process theory. The utility of our framework is demonstrated by analyzing the elastoplastic stresses in a novel class of disordered network materials exhibiting the property of hyperuniformity, in which the infinite wave-length density fluctuations associated with the distribution of network nodes are completely suppressed. We find enhanced mechanical properties such as increasing stiffness, yield strength and tensile strength as the degree of hyperuniformity of the material system increases. Our results have implications for optimal network material design and our event-based framework can be readily generalized for nonlinear stress analysis of other heterogeneous material systems. We also include some insights to the model from the viewpoint of the rigidity theory.