We study the approximation of finite-dimensional rate-independent quasistatic systems, via a vanishing-inertia asymptotic analysis of dynamic evolutions. We prove the uniform convergence of dynamic solutions to a rate-independent one, employing the variational concept of energetic solution. Motivated by applications in soft locomotion, we allow time-dependence of the dissipation potential, and translation invariance of the potential energy.

VL - 60 SN - 1432-0835 UR - https://doi.org/10.1007/s00526-021-02067-6 IS - 5 JO - Calculus of Variations and Partial Differential Equations ER - TY - JOUR T1 - An avoiding cones condition for the Poincaré–Birkhoff Theorem JF - Journal of Differential Equations Y1 - 2017 A1 - Alessandro Fonda A1 - Paolo Gidoni KW - Avoiding cones condition KW - Hamiltonian systems KW - Periodic solutions KW - Poincaré–Birkhoff theorem AB -We provide a geometric assumption which unifies and generalizes the conditions proposed in [11], [12], so to obtain a higher dimensional version of the Poincaré–Birkhoff fixed point Theorem for Poincaré maps of Hamiltonian systems.

VL - 262 UR - http://www.sciencedirect.com/science/article/pii/S0022039616303278 ER - TY - JOUR T1 - On the genesis of directional friction through bristle-like mediating elements JF - ESAIM: COCV Y1 - 2017 A1 - Paolo Gidoni A1 - Antonio DeSimone AB -We propose an explanation of the genesis of directional dry friction, as emergent property of the oscillations produced in a bristle-like mediating element by the interaction with microscale fluctuations on the surface. Mathematically, we extend a convergence result by Mielke, for Prandtl–Tomlinson-like systems, considering also non-homothetic scalings of a wiggly potential. This allows us to apply the result to some simple mechanical models, that exemplify the interaction of a bristle with a surface having small fluctuations. We find that the resulting friction is the product of two factors: a geometric one, depending on the bristle angle and on the fluctuation profile, and a energetic one, proportional to the normal force exchanged between the bristle-like element and the surface. Finally, we apply our result to discuss the with the nap/against the nap asymmetry.

VL - 23 UR - https://doi.org/10.1051/cocv/2017030 ER - TY - JOUR T1 - Stasis domains and slip surfaces in the locomotion of a bio-inspired two-segment crawler JF - Meccanica Y1 - 2017 A1 - Paolo Gidoni A1 - Antonio DeSimone AB -We formulate and solve the locomotion problem for a bio-inspired crawler consisting of two active elastic segments (i.e., capable of changing their rest lengths), resting on three supports providing directional frictional interactions. The problem consists in finding the motion produced by a given, slow actuation history. By focusing on the tensions in the elastic segments, we show that the evolution laws for the system are entirely analogous to the flow rules of elasto-plasticity. In particular, sliding of the supports and hence motion cannot occur when the tensions are in the interior of certain convex regions (stasis domains), while support sliding (and hence motion) can only take place when the tensions are on the boundary of such regions (slip surfaces). We solve the locomotion problem explicitly in a few interesting examples. In particular, we show that, for a suitable range of the friction parameters, specific choices of the actuation strategy can lead to net displacements also in the direction of higher friction.

VL - 52 UR - https://doi.org/10.1007/s11012-016-0408-0 ER - TY - JOUR T1 - Generalizing the Poincaré–Miranda theorem: the avoiding cones condition JF - Annali di Matematica Pura ed Applicata (1923 -) Y1 - 2016 A1 - Alessandro Fonda A1 - Paolo Gidoni AB -After proposing a variant of the Poincaré–Bohl theorem, we extend the Poincaré–Miranda theorem in several directions, by introducing an avoiding cones condition. We are thus able to deal with functions defined on various types of convex domains, and situations where the topological degree may be different from \$\$\backslashpm \$\$±1. An illustrative application is provided for the study of functionals having degenerate multi-saddle points.

VL - 195 UR - https://doi.org/10.1007/s10231-015-0519-6 ER - TY - JOUR T1 - Periodic perturbations of Hamiltonian systems JF - Advances in Nonlinear Analysis Y1 - 2016 A1 - Alessandro Fonda A1 - Maurizio Garrione A1 - Paolo Gidoni AB -We prove existence and multiplicity results for periodic solutions of Hamiltonian systems, by the use of a higher dimensional version of the Poincaré–Birkhoff fixed point theorem. The first part of the paper deals with periodic perturbations of a completely integrable system, while in the second part we focus on some suitable global conditions, so to deal with weakly coupled systems.

PB - De Gruyter VL - 5 ER - TY - THES T1 - Two explorations in Dynamical Systems and Mechanics Y1 - 2016 A1 - Paolo Gidoni KW - Poincaré-Birkhoff Theorem AB - This thesis contains the work done by Paolo Gidoni during the doctorate programme in Matematical Analysis at SISSA, under the supervision of A. Fonda and A. DeSimone. The thesis is composed of two parts: "Avoiding cones conditions and higher dimensional twist" and "Directional friction in bio-inspired locomotion". PB - SISSA U1 - 35527 U2 - Mathematics U4 - 1 U5 - MAT/05 ER - TY - JOUR T1 - Liquid crystal elastomer strips as soft crawlers JF - Journal of the Mechanics and Physics of Solids Y1 - 2015 A1 - Antonio DeSimone A1 - Paolo Gidoni A1 - Giovanni Noselli KW - Crawling motility KW - Directional surfaces KW - Frictional interactions KW - Liquid crystal elastomers KW - Soft biomimetic robots AB -In this paper, we speculate on a possible application of Liquid Crystal Elastomers to the field of soft robotics. In particular, we study a concept for limbless locomotion that is amenable to miniaturisation. For this purpose, we formulate and solve the evolution equations for a strip of nematic elastomer, subject to directional frictional interactions with a flat solid substrate, and cyclically actuated by a spatially uniform, time-periodic stimulus (e.g., temperature change). The presence of frictional forces that are sensitive to the direction of sliding transforms reciprocal, ‘breathing-like’ deformations into directed forward motion. We derive formulas quantifying this motion in the case of distributed friction, by solving a differential inclusion for the displacement field. The simpler case of concentrated frictional interactions at the two ends of the strip is also solved, in order to provide a benchmark to compare the continuously distributed case with a finite-dimensional benchmark. We also provide explicit formulas for the axial force along the crawler body.

VL - 84 UR - http://www.sciencedirect.com/science/article/pii/S0022509615300430 ER - TY - JOUR T1 - A permanence theorem for local dynamical systems JF - Nonlinear Analysis: Theory, Methods & Applications Y1 - 2015 A1 - Alessandro Fonda A1 - Paolo Gidoni KW - Lotka–Volterra KW - permanence KW - Predator–prey KW - Uniform persistence AB -We provide a necessary and sufficient condition for permanence related to a local dynamical system on a suitable topological space. We then present an illustrative application to a Lotka–Volterra predator–prey model with intraspecific competition.

VL - 121 UR - http://www.sciencedirect.com/science/article/pii/S0362546X14003332 N1 - Nonlinear Partial Differential Equations, in honor of Enzo Mitidieri for his 60th birthday ER - TY - JOUR T1 - Crawling on directional surfaces JF - International Journal of Non-Linear Mechanics Y1 - 2014 A1 - Paolo Gidoni A1 - Giovanni Noselli A1 - Antonio DeSimone KW - Bio-mimetic micro-robots KW - Cell migration KW - Crawling motility KW - Directional surfaces KW - Self-propulsion AB -In this paper we study crawling locomotion based on directional frictional interactions, namely, frictional forces that are sensitive to the sign of the sliding velocity. Surface interactions of this type are common in biology, where they arise from the presence of inclined hairs or scales at the crawler/substrate interface, leading to low resistance when sliding ‘along the grain’, and high resistance when sliding ‘against the grain’. This asymmetry can be exploited for locomotion, in a way analogous to what is done in cross-country skiing (classic style, diagonal stride). We focus on a model system, namely, a continuous one-dimensional crawler and provide a detailed study of the motion resulting from several strategies of shape change. In particular, we provide explicit formulae for the displacements attainable with reciprocal extensions and contractions (breathing), or through the propagation of extension or contraction waves. We believe that our results will prove particularly helpful for the study of biological crawling motility and for the design of bio-mimetic crawling robots.

VL - 61 UR - http://www.sciencedirect.com/science/article/pii/S0020746214000213 ER -