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DUCCIO PAPINI

Professore Associato
Dipartimento di Scienze e Metodi dell'Ingegneria

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Pubblicazioni

2024 - Infinitely many periodic solutions to a Lorentz force equation with singular electromagnetic potential [Articolo su rivista]
Boscaggin, A.; Dambrosio, W.; Papini, D.
abstract

We consider the Lorentz force equation in the physically relevant case of a singular electric field E. Assuming that E and B are T-periodic in time and satisfy suitable further conditions, we prove the existence of infinitely many T-periodic solutions. The proof is based on a min-max principle of Lusternik-Schnirelmann type, in the framework of non-smooth critical point theory. Applications are given to the problem of the motion of a charged particle under the action of a Liénard-Wiechert potential and to the relativistic forced Kepler problem.

2022 - A Low-Complexity Method to Address Process Variability in True Random Number Generators based on Digital Nonlinear Oscillators [Relazione in Atti di Convegno]
Addabbo, T.; Fort, A.; Mugnaini, M.; Moretti, R.; Vignoli, V.; Papini, D.
abstract

We discuss a monitoring system aiming to select, among a set of integrated entropy sources affected by process variability, the source guarantying the highest worst-case entropy. The approach is particularly suitable when considering True Random Number Generators based on Digital Nonlinear Oscillators, since multiple instances of the entropy sources can be implemented at a reduced hardware cost. In general, the approach can be applied for TRNGs based on parametric systems, thus offering entropy tuning capabilities. The original theoretical results have been validated with experiments.

2022 - A Stochastic Algorithm to Design Min-Entropy Tuning Controllers for True Random Number Generators [Articolo su rivista]
Addabbo, T.; Fort, A.; Moretti, R.; Mugnaini, M.; Papini, D.; Vignoli, V.
abstract

We discuss a stochastic algorithm to design tuning controllers for cryptographic True Random Number Generators, compliant to NIST recommendations, as an effective low-complexity solution to counteract entropy variability in integrated architectures implementing tunable entropy sources. Taking as a reference the min-entropy concept, we discussed the proposal from both the theoretical and hardware design points of view, validating claims with proofs and experiments. Depending on the target accuracy, the proposed architecture is scalable, and its profitable use in TRNG design strongly depends on the kind of core entropy sources taken into account. Furthermore, we show that the low-complexity entropy measurement techniques exploited in this proposal can be used to design a legitimate alternative to the Adaptive Proportion Health Test recommended in the NIST 800.90B publication.

2022 - Unbounded Solutions to Systems of Differential Equations at Resonance [Articolo su rivista]
Boscaggin, A.; Dambrosio, W.; Papini, D.
abstract

We deal with a weakly coupled system of ODEs of the type xj′′+nj2xj+hj(x1,…,xd)=pj(t),j=1,…,d,with hj locally Lipschitz continuous and bounded, pj continuous and 2 π-periodic, nj∈ N (so that the system is at resonance). By means of a Lyapunov function approach for discrete dynamical systems, we prove the existence of unbounded solutions, when either global or asymptotic conditions on the coupling terms h1, … , hd are assumed.

2022 - Unbounded Solutions to a System of Coupled Asymmetric Oscillators at Resonance [Articolo su rivista]
Boscaggin, A.; Dambrosio, W.; Papini, D.
abstract

We deal with the following system of coupled asymmetric oscillators {x¨1+a1x1+-b1x1-+ϕ1(x2)=p1(t)x¨2+a2x2+-b2x2-+ϕ2(x1)=p2(t),where ϕi: R→ R is locally Lipschitz continuous and bounded, pi: R→ R is continuous and 2 π-periodic and the positive real numbers ai, bi satisfy 1ai+1bi=2n,forsomen∈N.We define a suitable function L: T2→ R2, appearing as the higher-dimensional generalization of the well known resonance function used in the scalar setting, and we show how unbounded solutions to the system can be constructed whenever L has zeros with a special structure. The proof relies on a careful investigation of the dynamics of the associated (four-dimensional) Poincaré map, in action-angle coordinates.

2020 - Periodic solutions to a forced kepler problem in the plane [Articolo su rivista]
Boscaggin, A.; Dambrosio, W.; Papini, D.
abstract

Given a smooth function U(t, x), T-periodic in the first variable and satisfying U(t, x) = O(vertical bar x vertical bar(alpha)) for some alpha is an element of (0, 2) as vertical bar x vertical bar -> infinity, we prove that the forced Kepler problem(sic) = -x/vertical bar x vertical bar(3) + del U-x(t, x), x is an element of R-2,has a generalized T-periodic solution, according to the definition given in the paper by A. Boscaggin, R. Ortega, and L. Zhao [Trans. Amer. Math. Soc. 372 (2019), 677-703]. The proof relies on variational arguments.

2019 - Chaotic dynamics in a periodically perturbed Liénard system [Articolo su rivista]
Papini, D.; Villari, G.; Zanolin, F.
abstract

We prove the existence of infinitely many periodic solutions, as well as the presence of chaotic dynamics, for a periodically perturbed planar Liénard system of the form x' = y−F(x)+p(ωt), y' = −g(x). We consider the case in which the perturbing term is not necessarily small. Such a result is achieved by a topological method, that is by proving the presence of a horseshoe structure.

2019 - Coupling the Yoccoz-Birkeland population model with price dynamics: Chaotic livestock commodities market cycles [Articolo su rivista]
Arlot, S.; Marmi, Stefano; Papini, D.
abstract

We propose a new model for the time evolution of livestock commodities prices which exhibits endogenous deterministic stochastic behaviour. The model is based on the Yoccoz–Birkeland integral equation, a model first developed for studying the time-evolution of single species with high average fertility, a relatively short mating season and density-dependent reproduction rates. This equation is then coupled with a differential equation describing the price of a livestock commodity driven by the unbalance between its demand and supply. At its birth the cattle population is split into two parts: reproducing females and cattle for butchery. The relative amount of the two is determined by the spot price of the meat. We prove the existence of an attractor (theorem A) and of a non-trivial periodic solution (theorem B) and we investigate numerically the properties of the attractor: the strange attractor existing for the original Yoccoz–Birkeland model is persistent but its chaotic behaviour depends also on the time evolution of the price in an essential way.

2018 - Parabolic solutions for the planar N-centre problem: multiplicity and scattering [Articolo su rivista]
Boscaggin, Alberto; Dambrosio, Walter; Papini, Duccio; Boscaggin, Alberto; Dambrosio, Walter; Papini, Duccio
abstract

For the planar N-centre problem we prove the existence of entire parabolic trajectories, having prescribed asymptotic directions for t→±∞ and prescribed topological characterization with respect to the set of the centres.

2017 - Complex Dynamics in a ODE Model Related to Phase Transition [Articolo su rivista]
Papini, Duccio; Zanolin, Fabio
abstract

Motivated by some recent studies on the Allen–Cahn phase transition model with a periodic nonautonomous term, we prove the existence of complex dynamics for the second order equation −x¨+(1+ε−1A(t))G′(x)=0, where A(t) is a nonnegative T-periodic function and ε>0 is sufficiently small. More precisely, we find a full symbolic dynamics made by solutions which oscillate between any two different strict local minima x0 and x1 of G(x). Such solutions stay close to x0 or x1 in some fixed intervals, according to any prescribed coin tossing sequence. For convenience in the exposition we consider (without loss of generality) the case x0=0 and x1=1.

2017 - Multiple positive solutions to elliptic boundary blow-up problems [Articolo su rivista]
Boscaggin, Alberto; Dambrosio, Walter; Papini, Duccio
abstract

We prove the existence of multiple positive radial solutions to a sign-indefinite elliptic boundary blow-up problem where the nonlinearity is a function superlinear at zero and at infinity and is multiplied by a sign changing weight function. In particular, we show how the number of solutions is affected by the nodal behavior of the weight function. The proof is based on a careful shooting-type argument for the equivalent singular ODE problem. As a further application of this technique, the existence of multiple positive radial homoclinic solutions is also considered.

2016 - Multiple homoclinic solutions for a one-dimensional Schrödinger equation [Articolo su rivista]
Dambrosio, Walter; Papini, Duccio
abstract

In this paper we study the problem of the existence of homoclinic solutions to a Schrodinger equation of the form x''-V(t)x+x(3)=0, where V is a stepwise potential. The technique of proof is based on a topological method, relying on the properties of the transformation of continuous planar paths (the S.A.P. method), together with the application of the classical Conley-Wazewski method.

2015 - Asymptotic and chaotic solutions of a singularly perturbed Nagumo-type equation [Articolo su rivista]
Boscaggin, Alberto; Dambrosio, Walter; Papini, Duccio
abstract

We deal with the singularly perturbed Nagumo-type equation where ε > 0 is a real parameter and a : R -> R is a piecewise constant function satisfying 0  <  a(s)  <  1 for all s. For small ε, we prove the existence of chaotic, homoclinic and heteroclinic solutions. We use a dynamical systems approach, based on the Stretching Along Paths technique and on the Conley–Wazewski's method.

2015 - Linear and nonlinear eigenvalue problems for Dirac systems in unbounded domains [Articolo su rivista]
Capietto, Anna; Dambrosio, Walter; Papini, Duccio
abstract

We first study the linear eigenvalue problem for a planar Dirac system in the open half-line and describe the nodal properties of its solutions by means of the rotation number. We then give a global bifurcation result for a planar nonlinear Dirac system in the open half-line. As an application, we provide a global continuum of solutions of the nonlinear Dirac equation which have a special form.

2015 - Nontrivial, nonnegative periodic solutions of a system of singular-degenerate parabolic equations with nonlocal terms [Articolo su rivista]
Fragnelli, Genni; Mugnai, Dimitri; Nistri, Paolo; Papini, Duccio
abstract

We study the existence of nontrivial, nonnegative periodic solutions for systems of singular-degenerate parabolic equations with nonlocal terms and satisfying Dirichlet boundary conditions. The method employed in this paper is based on the Leray–Schauder topological degree theory. However, verifying the conditions under which such a theory applies is more involved due to the presence of the singularity. The system can be regarded as a possible model of the interactions of two biological species sharing the same isolated territory, and our results give conditions that ensure the coexistence of the two species.

2013 - Corrigendum: Non-trivial non-negative periodic solutions of a system of doubly degenerate parabolic equations with nonlocal terms [Articolo su rivista]
Genni, Fragnelli; Nistri, Paolo; Papini, Duccio
abstract

2012 - A global bifurcation result for a second order singular equation [Articolo su rivista]
Capietto, A.; Dambrosio, W.; Papini, D.
abstract

Dedicated, with gratefulness and friendship, to Professor Fabio Zanolin on the occasion of his 60th birthday Abstract. We deal with a boundary value problem associated to a second order singular equation in the open interval (0, 1]. We first study the eigenvalue problem in the linear case and discuss the nodal properties of the eigenfunctions. We then give a global bifurcation result for nonlinear problems.

2012 - Periodic solutions to nonlinear equations with oblique boundary conditions [Articolo su rivista]
Allegretto, W; Papini, Duccio
abstract

We study the existence of positive periodic solutions to nonlinear elliptic and parabolic equations with oblique and dynamical boundary conditions and non-local terms. The results are obtained through fixed point theory, topological degree methods and properties of related linear elliptic problems with natural boundary conditions and possibly nonsymmetric principal part. As immediate consequences, we also obtain estimates on the principal eigenvalue for non-symmetric elliptic eigenvalue problems. © 2012 Juliusz Schauder University Centre for Nonlinear Studies.

2011 - Coexistence and optimal control problems for a degenerate predator-prey model [Articolo su rivista]
Allegretto, W.; Fragnelli, G.; Nistri, P.; Papini, Duccio
abstract

In this paper we present a predator-prey mathematical model for two biological populations which dislike crowding. The model consists of a system of two degenerate parabolic equations with nonlocal terms and drifts. We provide conditions on the system ensuring the periodic coexistence, namely the existence of two non-trivial non-negative periodic solutions representing the densities of the two populations. We assume that the predator species is harvested if its density exceeds a given threshold. A minimization problem for a cost functional associated with this process and with some other significant parameters of the model is also considered. © 2010 Elsevier Inc.

2011 - Non-trivial non-negative periodic solutions of a system of doubly degenerate parabolic equations with nonlocal terms [Articolo su rivista]
Fragnelli, G.; Nistri, P.; Papini, Duccio
abstract

The aim of the paper is to provide conditions ensuring the ex- istence of non-trivial non-negative periodic solutions to a system of doubly degenerate parabolic equations containing delayed nonlocal terms and satis- fying Dirichlet boundary conditions. The employed approach is based on the theory of the Leray-Schauder topological degree theory, thus a crucial purpose of the paper is to obtain a priori bounds in a convenient functional space, here L 2(QT ), on the solutions of certain homotopies. This is achieved under different assumptions on the sign of the kernels of the nonlocal terms. The considered system is a possible model of the interactions between two biologi- cal species sharing the same territory where such interactions are modeled by the kernels of the nonlocal terms. To this regard the obtained results can be viewed as coexistence results of the two biological populations under different intra and inter specific interferences on their natural growth rates.

2011 - Positive periodic solutions and optimal control for a distributed biological model of two interacting species [Articolo su rivista]
Fragnelli, G.; Nistri, P; Papini, Duccio
abstract

The paper deals with the existence of positive periodic solutions to a system of degenerate parabolic equations with delayed nonlocal terms and Dirichlet boundary conditions. Taking in each equation a meaningful function as a control parameter, we show that for a suitable choice of a class of such controls we have, for each of them, a time-periodic response of the system under different assumptions on the kernels of the nonlocal terms. Finally, we consider the problem of the minimization of a cost functional on the set of pairs: control-periodic response. The considered system may be regarded as a possible model for the coexistence problem of two biological populations, which dislike crowding and live in a common territory, under different kind of intra- and inter-specific interferences. © 2010 Elsevier Ltd. All rights reserved.

2010 - Common asymptotic behavior of solutions and almost periodicity for discontinuous, delayed, and impulsive neural networks [Articolo su rivista]
Allegretto, W; Forti, Mauro; Papini, Duccio
abstract

The paper considers a general neural network model with impulses at a given sequence of instants, discontinuous neuron activations, delays, and time-varying data and inputs. It is shown that when the neuron interconnections satisfy an M-matrix condition, or a dominance condition, then the state solutions and the output solutions display a common asymptotic behavior as time t → +∞. It is also shown, via a new technique based on prolonging the solutions of the delayed neural network to −∞, that it is possible to select a unique special solution that is globally exponentially stable and can be considered as the unique global attractor for the network. Finally, this paper shows that for almost periodic data and inputs the selected solution is almost periodic; moreover, it is robust with respect to a large class of perturbations of the data. Analogous results also hold for periodic data and inputs. A by-product of the analysis is that a sequence of almost periodic impulses is able to induce in the generic case (nonstationary) almost periodic solutions in an otherwise globally convergent nonimpulsive neural network. To the authors’ knowledge the results in this paper are the only available results on global exponential stability of the unique periodic or almost periodic solution for a general neural network model combining three main features, i.e., impulses, discontinuous neuron activations and delays. The results in this paper are compared with several results in the literature dealing with periodicity or almost periodicity of some subclasses of the neural network model here considered and some hints for future work are given.

2009 - An efficient and accurate method for the estimation of entropy and other dynamical invariants for piecewise affine chaotic maps [Articolo su rivista]
Addabbo, T.; Fort, A.; Papini, D.; Rocchi, S.; Vignoli, V.
abstract

In this paper, we discuss an efficient iterative method for the estimation of the chief dynamical invariants of chaotic systems based on stochastically stable piecewise affine maps (e.g. the invariant measure, the Lyapunov exponent as well as the KolmogorovSinai entropy). The proposed method represents an alternative to the Monte-Carlo methods and to other methods based on the discretization of the FrobeniusPerron operator, such as the well known Ulam's method. The proposed estimation method converges not slower than exponentially and it requires a computation complexity that grows linearly with the iterations. Referring to the theory developed by C. Liverani, we discuss a theoretical tool for calculating a conservative estimation of the convergence rate of the proposed method. The proposed approach can be used to efficiently estimate any order statistics of a symbolic source based on a piecewise affine mixing map. © 2009 World Scientific Publishing Company.

2009 - Invariant measures of tunable chaotic sources: Robustness analysis and efficient estimation [Articolo su rivista]
Addabbo, T.; Fort, A.; Papini, D.; Rocchi, S.; Vignoli, V.
abstract

In this paper, a theoretical approach for studying the robustness of the chaotic statistics of piecewise affine maps with respect to parameter perturbations is discussed. The approach is oriented toward the study of the effects that the nonidealities derived from the circuit implementation of these chaotic systems have on their dynamics. The ergodic behavior of these systems is discussed in detail, adopting the approach developed by Boyarsky and Góra, with particular reference to the family of sawtooth maps, and the robustness of their invariant measures is studied. Although this paper is particularly focused on this specific family of maps, the proposed approach can be generalized to other piecewise affine maps considered in the literature for information and communications technology applications. Moreover, in this paper, an efficient method for estimating the unique invariant density for stochastically stable piecewise affine maps is proposed. The method is an alternative to Monte Carlo methods and to other methods based on the discretization of the Frobenius-Perron operator. © 2009 IEEE.

2008 - Analysis of a lagoon ecological model with anoxic crises and impulsive harvesting [Articolo su rivista]
Allegretto, W.; Papini, Duccio
abstract

We analyze mathematically a system of impulsive nonlinear parabolic equations that model a shallow lagoon subject to anoxic crises and two types of impulsive harvesting. The main focus is on the existence and properties of periodic solutions. In particular we give conditions that ensure the existence of such solutions and examine the effect of harvesting on the occurrence of anoxic crises. Our approach is based on estimates of the principal eigenvalue of associated linear problems, and on results from Nonlinear Functional Analysis. In particular, we obtain explicit criteria that involve the integrals of coefficients rather than maxima and minima. This is significant due to the large seasonal variations in the coefficient values.

2008 - Synchronization problems for unidirectional feedback coupled nonlinear systems [Articolo su rivista]
Makarenkov, O.; Nistri, Paolo; Papini, Duccio
abstract

In this paper we consider three di®erent synchronization problems consisting in designing a nonlinear feedback unidirectional coupling term for two (possibly chaotic) dynamical systems in order to drive the trajectories of one of them, the slave system, to a reference trajectory or to a prescribed neighborhood of the reference trajectory of the second dynamical system: the master system. If the slave system is chaotic then synchronization can be viewed as the control of chaos; namely the coupling term allows to suppress the chaotic motion by driving the chaotic system to a prescribed reference trajectory. Assuming that the entire vector field representing the velocity of the state can be modified, three different methods to define the nonlinear feedback synchronizing controller are proposed: one for each of the treated problems. These methods are based on results from the small parameter perturbation theory of autonomous systems having a limit cycle, from nonsmooth analysis and from the singular perturbation theory respectively. Simulations to illustrate the effectiveness of the obtained results are also presented.

2007 - Some results on periodic points and chaotic dynamics arising from the study of the nonlinear Hill equations [Articolo su rivista]
Papini, Duccio; Zanolin, Fabio
abstract

2007 - Stability for delayed reaction-diffusion neural networks [Articolo su rivista]
Allegretto, W.; Papini, D.
abstract

We consider a Hopfield neural network model with diffusive terms, non-decreasing and discontinuous neural activation functions, time-dependent delays and time-periodic coefficients. We provide conditions on interconnection matrices and delays which guarantee that for each periodic input the model has a unique periodic solution that is globally exponentially stable. Even in the case without diffusion, such conditions improve recent results on classical delayed Hopfield neural networks with discontinuous activation functions. Numerical examples illustrate the results.

2005 - Detecting multiplicity for systems of second-order equations: An alternative approach [Articolo su rivista]
Capietto, A.; Dambrosio, W.; Papini, D.
abstract

In this paper we are concerned with a system of second-order differential equations of the form x&Ti + A(t, x)x = 0, t ∈ [0, π], x ∈ RN, where A(t, x) is a symmetric N × N matrix. We concentrate on an asymptotically linear situation and we prove the existence of multiple solutions to the Dirichlet problem associated to the system. Multiplicity is obtained by a comparison between the number of moments of verticality of the matrices A0(t) and A∞(t), which are the uniform limits of A(t, x) for |x| → 0 and |x| → +∞, respectively. For the proof, which is based on a generalized shooting approach, we provide a theorem on the existence of zeros of a class of N-dimensional vector fields.

2005 - Global exponential stability and global convergence in finite time of delayed neural networks with infinite gain [Articolo su rivista]
Forti, Mauro; Nistri, Paolo; Papini, Duccio
abstract

This paper introduces a general class of neural networks with arbitrary constant delays in the neuron interconnections, and neuron activations belonging to the set of discontinuous monotone increasing and (possibly) unbounded functions. The discontinuities in the activations are an ideal model of the situation where the gain of the neuron amplifiers is very high and tends to infinity, while the delay accounts for the finite switching speed of the neuron amplifiers, or the finite signal propagation speed. It is known that the delay in combination with high-gain nonlinearities is a particularly harmful source of potential instability. The goal of this paper is to single out a subclass of the considered discontinuous neural networks for which stability is instead insensitive to the presence of a delay. More precisely, conditions are given under which there is a unique equilibrium point of the neural network, which is globally exponentially stable for the states, with a known convergence rate. The conditions are easily testable and independent of the delay. Moreover, global convergence in finite time of the state and output is investigated. In doing so, new interesting dynamical phenomena are highlighted with respect to the case without delay, which make the study of convergence in finite time significantly more difficult. The obtained results extend previous work on global stability of delayed neural networks with Lipschitz continuous neuron activations, and neural networks with discontinuous neuron activations but without delays.

2005 - Global exponential stability of the periodic solution of a delayed neural network with discontinuous activations [Articolo su rivista]
Papini, D; Taddei, Valentina
abstract

Delayed neural networks with periodic coefficients and discontinuous and/or unbounded activation functions are investigated by means of Lyapunov theory and fixed point theorems. We obtain conditions, independent from the delay, assuring the existence of an unique limit cycle, which is globally exponential stable.

2004 - Fixed points, periodic points, and coin-tossing sequences for mappings defined on two-dimensional cells [Articolo su rivista]
Papini, Duccio; Zanolin, F.
abstract

We propose, in the general setting of topological spaces, a definition of two-dimensional oriented cell and consider maps which possess a property of stretching along the paths with respect to oriented cells. For these maps, we prove some theorems on the existence of fixed points, periodic points, and sequences of iterates which are chaotic in a suitable manner. Our results, motivated by the study of the Poincaré map associated to some nonlinear Hill's equations, extend and improve some recent work. The proofs are elementary in the sense that only well known properties of planar sets and maps and a two-dimensional equivalent version of the Brouwer fixed point theorem are used.

2004 - On the periodic boundary value problem and chaotic-like dynamics for nonlinear Hill's equations [Articolo su rivista]
Papini, Duccio; Zanolin, Fabio
abstract

We present some results which show the rich and complicated structure of the solutions of the second order differential equation x''+ w(t)g(x) = 0 when the weight w(t) changes sign and g is sufficiently far from the linear case. New applications, motivated by recent studies on the superlinear Hill’s equation, are then proposed for some asymptotically linear equations and for some sublinear equations with a sign-indefinite weight. Our results are based on a fixed point theorem for maps which satisfy a stretching condition along the paths on two-dimensional cells.

2004 - Periodic Solutions of a Certain Generalized Liénard Equation [Articolo su rivista]
Papini, D.; Villari, G.
abstract

We are concerned with the existence of at least one periodic solution of a generalized nonlinear Lienard equation with a periodic forcing term. The main tool is a continuation theorem by Capietto, Mawhin and Zanolin. A priori bounds for the periodic solutions are obtained either by studying the behavior of the trajectories of a new equivalent system or by determining the nature of singular points at infinity of suitable autonomous systems in the usual phase plane. © 2004, Division of Functional Equations, The Mathematical Society of Japan. All rights reserved.

2004 - Periodic solutions of asymptotically linear second order equations with changing sign weight [Articolo su rivista]
Dambrosio, W.; Papini, Duccio
abstract

In this paper we study the ordinary differential equation x'' + q(t)g(x) = 0, where g is a locally Lipschitz continuous function that satisfies g(x)x > 0 for all non zero x and is asymptotically linear, while q is a continuous, π-periodic and changing sign weight. By the application of a recent result on the existence and multiplicity of fixed points of planar maps, we give conditions on q and on the behavior of the ratio g(x)/x near zero and near infinity in order to obtain multiple periodic solutions with the prescribed number of zeros in the intervals of positivity and negativity of q, as well as multiple subharmonics of any order and uncountably many bounded solutions.

2003 - Boundary blow-up for differential equations with indefinite weight [Articolo su rivista]
Mawhin, J.; Papini, Duccio; Zanolin, Fabio
abstract

We obtain results of existence and multiplicity of solutions for the second-order equation x'' + q(t)g(x) = 0 with x(t) defined for all t in ]0,1[ and such that x(t) goes to infinity as tends to 0+ and to 1-. We assume g having superlinear growth at infinity and q(t) possibly changing sign on [0, 1].

2003 - Prescribing the nodal behaviour of periodic solutions of a superlinear equation with indefinite weight [Articolo su rivista]
Papini, Duccio
abstract

This paper deals with the existence of periodic solutions to the differential equation x'' + q(t)g(x) = 0. Here g is Lipschitz, xg(x) > 0 for all non vanishing x, g has superlinear growth at infinity and q is continuous and is allowed to change sign finitely many times. We prove that there are two periodic solutions with a precise number of zeros in each interval of positivity of q and that, moreover, for each interval of negativity, one can fix a priori whether the solution will have exactly one zero and be strictly monotone or will have no zeros and exactly one zero of the derivative. The techniques are based on the study of the Poincaré map and a careful phase plane analysis. Generalizations are discussed in order to treat more gereal Floquet-type boundary conditions.

2002 - Differential equations with indefinite weight: boundary value problems and qualitative properties of the solutions [Relazione in Atti di Convegno]
Papini, Duccio; Zanolin, F.
abstract

We describe the qualitative properties of the solutions of the second order scalar equation x'' + q(t)g(x) = 0, where q is a changing sign function, and consider the problem of existence and multiplicity of solutions which satisfy various different boundary conditions. In particular we outline some difficulties which arise in the use of the shooting approach.

2002 - Periodic points and chaotic-like dynamics of planar maps associated to nonlinear Hill's equations with indefinite weight [Articolo su rivista]
Papini, Duccio; Zanolin, F.
abstract

We prove some results about the existence of fixed points, periodic points and chaotic-like dynamics for a class of planar maps which satisfy a suitable property of “arc expansion” type. We also outline some applications to the nonlinear Hill’s equations with indefinite weight.

2002 - Superlinear indefinite equations on the real line and chaotic dynamics [Articolo su rivista]
Capietto, A.; Dambrosio, W.; Papini, Duccio
abstract

In this paper we are concerned with a damped Hill equation on a time interval (a,b), where −∞⩽a

2001 - Boundary blow-up for some quasi-linear differential equation with indefinite weight [Relazione in Atti di Convegno]
Mawhin, J.; Papini, Duccio; Zanolin, F.
abstract

2000 - A topological approach to superlinear indefinite boundary value problems [Articolo su rivista]
Papini, Duccio; Zanolin, Fabio
abstract

We obtain the existence of infinitely many solutions with prescribed nodal properties for some boundary value problems associated to the second order scalar equation x''+q(t)g(x)=0, where g(x) has superlinear growth at infinity and q(t) changes sign.

2000 - Infinitely many solutions for a Floquet-type BVP with superlinearity indefinite in sign [Articolo su rivista]
Papini, Duccio
abstract

By the application of a technique developed by G. J. Butler we find infinitely many solutions of a Floquet-type BVP for the equation x''+q(t)g(x)=0, where q is a weight function that is allowed to change sign, g is is superlinear and such that g(x)x>0 for all non-zero x. The boundary condition is (x(b),x'(b))=L(x(a),x'(a)), where L is a continuous, positively homogeneous, and nondegenerate map. At first we apply the main result to obtain solutions with a prescribed large number of zeros when L is the rotation of a fixed angle l; second, we find infinitely many subharmonic solutions of any order and, again, solutions with a prescribed large number of zeros for the periodic problem associated to the equation x''+cx'+q(t)g(x)=0, with q and g as above and a constant c.

2000 - Periodic solutions for a class of Liénard equations [Articolo su rivista]
Papini, Duccio
abstract