arXiv subjects

Paolo Gidoni

Publications and source records attributed to Paolo Gidoni.

17 recordsLinked to original sources

arXiv2026

Existence of a periodic solution for superquadratic Hamiltonian systems with possible finite-time blow-up

We prove a sufficient condition for the existence of a $T$-periodic solution for the planar system $\dot z=F(t,z)$, characterized by the growth to infinity of the rotations made in one period by solutions starting at increasingly large initial values. Our result applies in particular to superquadratic Hamiltonian systems satisfying the Ambrosetti--Rabinowitz condition. The key novelty of the paper is that we do not require any growth condition on the flow to ensure global existence of solutions, allowing finite-time blow-up. Our method is based on a fixed-point theorem which exploits the rotational properties of the dynamics. To conclude, we discuss a family of examples of Hamiltonian systems showing finite-time blow-up.

math.CA↗

arXiv2024

A Massera-type Theorem on relative-periodic solutions for a second-order model of rectilinear locomotion

We study the existence of a global periodic attractor for the reduced dynamics of a discrete toy model for rectilinear crawling locomotion, corresponding to a limit cycle in the shape and velocity variables. The body of the crawler consists of a chain of point masses, joined by active elastic links and subject to smooth friction forces, so that the dynamics is described by a system of second order differential equations. Our main result is of Massera-type, namely we show that the existence of a bounded solution implies the existence of the global periodic attractor for the reduced dynamics. In establishing this result, a contractive property of the dynamics of our model plays a central role. We then prove sufficient conditions on the friction forces for the existence of a bounded solution, and therefore of the attractor. We also provide an example showing that, if we consider more general friction forces, such as smooth approximations of dry friction, bounded solutions may not exist.

math.DS↗

arXiv2024

Gait controllability of length-changing slender microswimmers

Controllability results of four models of two-link microscale swimmers that are able to change the length of their links are obtained. The problems are formulated in the framework of Geometric Control Theory, within which the notions of fiber, total, and gait controllability are presented, together with sufficient conditions for the latter two. The dynamics of a general two-link swimmer is described by resorting to Resistive Force Theory and different mechanisms to produce a length-change in the links, namely, active deformation, a sliding hinge, growth at the tip, and telescopic links. Total controllability is proved via gait controllability in all four cases, and illustrated with the aid of numerical simulations.

math.OC↗

arXiv2023

A topological degree theory for rotating solutions of planar systems

We present a generalized notion of degree for rotating solutions of planar systems. We prove a formula for the relation of such degree with the classical use of Brouwer's degree and obtain a twist theorem for the existence of periodic solutions, which is complementary to the Poincaré-Birkhoff Theorem. Some applications to asymptotically linear and superlinear differential equations are discussed.

math.DS↗

arXiv2022

Limit cycles for dynamic crawling locomotors with periodic prescribed shape

We study the asymptotic behaviour of a family of dynamic models of crawling locomotion, with the aim of characterizing a gait as a limit property. The locomotors, which might have a discrete or continuous body, move on a line with a periodic prescribed shape change, and might possibly be subject to external forcing (e.g., crawling on a slope). We discuss how their behaviour is affected by different types of friction forces, including also set-valued ones such as dry friction. We show that, under mild natural assumptions, the dynamics always converge to a relative periodic solution. The asymptotic average velocity of the crawler yet might still depend on its initial state, so we provide additional assumption for its uniqueness. In particular, we show that the asymptotic average velocity is unique both for strictly monotone friction forces, and also for dry friction, provided in the latter case that the actuation is sufficiently smooth (for discrete models) or that the friction coefficients are always nonzero (for continuous models). We present several examples and counterexamples illustrating the necessity of our assumptions.

math-ph↗

arXiv2021

Existence of a periodic solution for superlinear second order ODEs

We prove a necessary and sufficient condition for the existence of a $T$-periodic solution for the time-periodic second order differential equation $\ddot{x}+f(t,x)+p(t,x,\dot x)=0$, where $f$ grows superlinearly in $x$ uniformly in time, while $p$ is bounded. Our method is based on a fixed-point theorem which uses the rotational properties of the dynamics.

math.CA↗

arXiv2021

Stabilization of periodic sweeping processes and asymptotic average velocity for soft locomotors with dry friction

We study the asymptotic stability of periodic solutions for sweeping processes defined by a polyhedron with translationally moving faces. Previous results are improved by obtaining a stronger $W^{1,2}$ convergence. Then we present an application to a model of crawling locomotion. Our stronger convergence allows us to prove the stabilization of the system to a running-periodic (or derivo-periodic, or relative-periodic) solution and the well-posedness of an average asymptotic velocity depending only on the gait adopted by the crawler. Finally, we discuss some examples of finite-time versus asymptotic-only convergence.

math.DS↗

arXiv2020

A vanishing inertia analysis for finite dimensional rate-independent systems with nonautonomous dissipation, and an application to soft crawlers

We study the approximation of quasistatic evolutions, formulated as abstract finite-dimensional rate-independent systems, via a vanishing-inertia asymptotic analysis of dynamic evolutions. We prove the uniform convergence of dynamical solutions to the quasistatic one, employing the 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.

math-ph↗

arXiv2020

On the optimal control of rate-independent soft crawlers

Existence of optimal solutions and necessary optimality conditions for a controlled version of Moreau's sweeping process are derived. The control is a measurable ingredient of the dynamics and the constraint set is a polyhedron. The novelty consists in considering time periodic trajectories, adding the requirement that the control have zero average, and considering an integral functional that lacks weak semicontinuity. A model coming from the locomotion of a soft-robotic crawler, that motivated our setting, is analysed in detail. In obtaining necessary conditions, an improvement of the method of discrete approximations is used.

math.OC↗

arXiv2019

Multiplicity of clines for systems of indefinite differential equations arising from a multilocus population genetics model

We investigate sufficient conditions for the presence of coexistence states for different genotypes in a diploid diallelic population with dominance distributed on a heterogeneous habitat, considering also the interaction between genes at multiple loci. In mathematical terms, this corresponds to the study of the Neumann boundary value problem \begin{equation*} \begin{cases} \, p_{1}''+λ_{1} w_{1}(x,p_{2}) f_{1}(p_{1}) = 0, &\text{in $Ω$,} \, p_{2}''+λ_{2} w_{2}(x,p_{1}) f_{2}(p_{2}) = 0, &\text{in $Ω$,} \, p_{1}'=p_{2}'=0, &\text{on $\partialΩ$,} \end{cases} \end{equation*} where the coupling-weights $w_{i}$ are sign-changing in the first variable, and the nonlinearities $f_{i}\colon\mathopen{[}0,1\mathclose{]}\to\mathopen{[}0,+\infty\mathclose{[}$ satisfy $f_{i}(0)=f_{i}(1)=0$, $f_{i}(s)>0$ for all $s\in\mathopen{]}0,1\mathclose{[}$, and a superlinear growth condition at zero. Using a topological degree approach, we prove existence of $2^{N}$ positive fully nontrivial solutions when the real positive parameters $λ_{1}$ and $λ_{2}$ are sufficiently large.

math.CA↗

arXiv2018

Lower bound on the number of periodic solutions for asymptotically linear planar Hamiltonian systems

In this work we prove the lower bound for the number of $T$-periodic solutions of an asymptotically linear planar Hamiltonian system. Precisely, we show that such a system, $T$-periodic in time, with $T$-Maslov indices $i_0,i_\infty$ at the origin and at infinity, has at least $|i_\infty-i_0|$ periodic solutions, and an additional one if $i_0$ is even. Our argument combines the Poincaré--Birkhoff Theorem with an application of topological degree. We illustrate the sharpness of our result, and extend it to the case of second orders ODEs with linear-like behaviour at zero and infinity.

math.DS↗

arXiv2018

Existence and regularity of solutions for an evolution model of perfectly plastic plates

We continue the study of a dynamic evolution model for perfectly plastic plates, recently derived from three-dimensional Prandtl-Reuss plasticity. We extend the previous existence result by introducing non-zero external forces in the model, and we discuss the regularity of the solutions thus obtained. In particular, we show that the first derivatives with respect to space of the stress tensor are locally square integrable.

math.AP↗

arXiv2017

Rate-independent soft crawlers

This paper applies the theory of rate-independent systems to model the locomotion of bio-mimetic soft crawlers. We prove the well-posedness of the approach and illustrate how the various strategies adopted by crawlers to achieve locomotion, such as friction anisotropy, complex shape changes and control on the friction coefficients, can be effectively described in terms of stasis domains. Compared to other rate-independent systems, locomotion models do not present any Dirichlet boundary condition, so that all rigid translations are admissible displacements, resulting in a non-coercivity of the energy term. We prove that existence and uniqueness of solution are guaranteed under suitable assumptions on the dissipation potential. Such results are then extended to the case of time-dependent dissipation.

math.FA↗

arXiv2016

Stasis domains and slip surfaces in the locomotion of a bio-inspired two-segment crawler

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.

physics.bio-ph↗

arXiv2016

On the genesis of directional friction through bristle-like mediating elements

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.

math-ph↗

arXiv2016

Liquid Crystal Elastomer Strips as Soft Crawlers

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.

cond-mat.soft↗

arXiv2014

Crawling on directional surfaces

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.

math-ph↗