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.

10aAvoiding cones condition10aHamiltonian systems10aPeriodic solutions10aPoincaré–Birkhoff theorem1 aFonda, Alessandro1 aGidoni, Paolo uhttp://www.sciencedirect.com/science/article/pii/S002203961630327801320nas a2200133 4500008004100000245008300041210006900124300001400193490000700207520089100214100001801105700002201123856004101145 2017 eng d00aOn the genesis of directional friction through bristle-like mediating elements0 agenesis of directional friction through bristlelike mediating el a1023-10460 v233 aWe 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.

1 aGidoni, Paolo1 aDeSimone, Antonio uhttps://doi.org/10.1051/cocv/201703001538nas a2200157 4500008004100000022001400041245009300055210006900148260000800217300001400225490000700239520104800246100001801294700002201312856004601334 2017 eng d a1572-964800aStasis domains and slip surfaces in the locomotion of a bio-inspired two-segment crawler0 aStasis domains and slip surfaces in the locomotion of a bioinspi cFeb a587–6010 v523 aWe 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.

1 aGidoni, Paolo1 aDeSimone, Antonio uhttps://doi.org/10.1007/s11012-016-0408-000945nas a2200157 4500008004100000022001400041245007900055210007200134260000800206300001600214490000800230520046300238100002200701700001800723856004600741 2016 eng d a1618-189100aGeneralizing the Poincaré–Miranda theorem: the avoiding cones condition0 aGeneralizing the Poincaré–Miranda theorem the avoiding cones con cAug a1347–13710 v1953 aAfter 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.

1 aFonda, Alessandro1 aGidoni, Paolo uhttps://doi.org/10.1007/s10231-015-0519-600883nas a2200157 4500008004100000245005000041210005000091260001500141300001400156490000600170520040100176100002200577700002300599700001800622856008500640 2016 eng d00aPeriodic perturbations of Hamiltonian systems0 aPeriodic perturbations of Hamiltonian systems bDe Gruyter a367–3820 v53 aWe 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.

1 aFonda, Alessandro1 aGarrione, Maurizio1 aGidoni, Paolo uhttps://www.math.sissa.it/publication/periodic-perturbations-hamiltonian-systems00745nas a2200121 4500008004100000245005600041210005600097260001000153520032000163653003100483100001800514856009100532 2016 en d00aTwo explorations in Dynamical Systems and Mechanics0 aTwo explorations in Dynamical Systems and Mechanics bSISSA3 aThis 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".10aPoincaré-Birkhoff Theorem1 aGidoni, Paolo uhttps://www.math.sissa.it/publication/two-explorations-dynamical-systems-and-mechanics01765nas a2200217 4500008004100000022001400041245005300055210005300108300001400161490000700175520110000182653002201282653002501304653002801329653003001357653002701387100002201414700001801436700002201454856007101476 2015 eng d a0022-509600aLiquid crystal elastomer strips as soft crawlers0 aLiquid crystal elastomer strips as soft crawlers a254 - 2720 v843 aIn 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.

10aCrawling motility10aDirectional surfaces10aFrictional interactions10aLiquid crystal elastomers10aSoft biomimetic robots1 aDeSimone, Antonio1 aGidoni, Paolo1 aNoselli, Giovanni uhttp://www.sciencedirect.com/science/article/pii/S002250961530043000827nas a2200193 4500008004100000022001400041245005300055210005100108300001200159490000800171520026300179653002100442653001500463653002000478653002400498100002200522700001800544856007100562 2015 eng d a0362-546X00aA permanence theorem for local dynamical systems0 apermanence theorem for local dynamical systems a73 - 810 v1213 aWe 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.

10aLotka–Volterra10apermanence10aPredator–prey10aUniform persistence1 aFonda, Alessandro1 aGidoni, Paolo uhttp://www.sciencedirect.com/science/article/pii/S0362546X1400333201733nas a2200217 4500008004100000022001400041245003700055210003700092300001200129490000700141520111900148653002901267653001901296653002201315653002501337653002001362100001801382700002201400700002201422856007101444 2014 eng d a0020-746200aCrawling on directional surfaces0 aCrawling on directional surfaces a65 - 730 v613 aIn 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.

10aBio-mimetic micro-robots10aCell migration10aCrawling motility10aDirectional surfaces10aSelf-propulsion1 aGidoni, Paolo1 aNoselli, Giovanni1 aDeSimone, Antonio uhttp://www.sciencedirect.com/science/article/pii/S0020746214000213