arXiv subjects

Fabio Zanolin

Publications and source records attributed to Fabio Zanolin.

23 records · Page 2Linked to original sources

arXiv2015

Pairs of positive periodic solutions of nonlinear ODEs with indefinite weight: a topological degree approach for the super-sublinear case

We study the periodic and the Neumann boundary value problems associated with the second order nonlinear differential equation \begin{equation*} u'' + c u' + λa(t) g(u) = 0, \end{equation*} where $g \colon \mathopen{[}0,+\infty\mathclose{[}\to \mathopen{[}0,+\infty\mathclose{[}$ is a sublinear function at infinity having superlinear growth at zero. We prove the existence of two positive solutions when $\int_{0}^{T} a(t) \!dt < 0$ and $λ> 0$ is sufficiently large. Our approach is based on Mawhin's coincidence degree theory and index computations.

math.CA↗

arXiv2008

Chaotic dynamics in the Volterra predator-prey model via linked twist maps

We prove the existence of infinitely many periodic solutions and complicated dynamics, due to the presence of a topological horseshoe, for the classical Volterra predator--prey model with a periodic harvesting. The proof relies on some recent results about chaotic planar maps combined with the study of geometric features which are typical of linked twist maps.

math.DS↗

arXiv2007

Complex dynamics in a nerve fiber model with periodic coefficients

We deal with the periodic boundary value problem for a second-order nonlinear ODE which includes the case of the Nagumo type equation $v_{xx} - g v + n(x) F(v) = 0,$ previously considered by Grindrod and Sleeman and by Chen and Bell in the study of nerve fiber models. In some recent works we discussed the case of nonexistence of nontrivial solutions as well as the case in which many positive periodic solutions may arise, the different situations depending by threshold parameters related to the weight function $n(x).$ Here we show that for a step function $n(x)$ (or for small perturbations of it) it is possible to obtain infinitely many periodic solutions and chaotic dynamics, due to the presence of a topological horseshoe (according to Kennedy and Yorke).

math.DS↗

arXiv2007

Cutting surfaces and applications to periodic points and chaotic-like dynamics

In this paper we propose an elementary topological approach which unifies and extends various different results concerning fixed points and periodic points for maps defined on sets homeomorphic to rectangles embedded in euclidean spaces. We also investigate the associated discrete semidynamical systems in view of detecting the presence of chaotic-like dynamics.

math.DS↗

arXiv2006

Multiplicity of periodic solutions for differential equations arising in the study of a nerve fiber model

We deal with the periodic boundary value problem for a second-order nonlinear ODE which includes the case of the Nagumo type equation $v_{xx} - g v + n(x) F(v) = 0,$ previously considered by Grindrod and Sleeman and by Chen and Bell in the study of the model of a nerve fiber with excitable spines. In a recent work we proved a result of nonexistence of nontrivial solutions as well as a result of existence of two positive solutions, the different situations depending by a threshold parameter related to the integral of the weight function $n(x).$ Here we show that the number of positive periodic solutions may be very large for some special choices of a (large) weight $n.$ We also obtain the existence of subharmonic solutions of any order. The proofs are based on the Poincaré - Bikhoff fixed point theorem.

math.CA↗