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

Abstract

We deal with the periodic boundary value problem for a second-order nonlinear ODE which includes the case of the Nagumo-type equation vxx - gv + n (x) F (v) = 0, previously considered 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

Complex dynamics in a nerve fiber model with periodic coefficients

“…In some recent papers [20,21] we extended the above recalled Chen and Bell result toward different directions. In particular, in [20], we proved (via a variational approach) that the existence of at least two positive periodic solutions is guaranteed for an arbitrary weight function n(x), provided that β 0 n(x) dx is sufficiently large.…”

“…In particular, in [20], we proved (via a variational approach) that the existence of at least two positive periodic solutions is guaranteed for an arbitrary weight function n(x), provided that β 0 n(x) dx is sufficiently large. On the other hand, in [21], we obtained (via the Poincaré-Birkhoff fixed point theorem) the existence of many positive periodic solutions under more restrictive hypotheses on n(x) (but general enough to include the piecewise constant case) and removing the requirement that 1 0 F (s) ds > 0. In this work we come back to the original assumptions on n(x) of Chen and Bell's paper with the aim to prove that their model presents solutions with a very complicated behavior.…”

Computational methods for a mathematical model of propagation of nerve impulses in myelinated axons

“…Equations of this form were analysed, for example, in [22]. As shown in [2] and [7], the boundary value problem is solvable only for a certain value of d. In the particular case of (20), assuming that a < 1/2, this value can be determined analytically (see [13] ):…”

“…We shall now describe an alternative approach to obtain an estimate of τ and an approximate solution, which does not require the use of the formula (22). Taking into account the conditions (12) and (13) and the asymptotic behaviour of v for large values of t, described by (15) and (18), we search for an approximate solution of the considered problem in the form…”