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 vx x - g v + n (x) F (v) = 0, previously considered by Grindrod and Sleeman [P. Grindrod, B.D. Sleeman, A model of a myelinated nerve axon: threshold behaviour and propagation, J. Math. Biol. 23 (1985) 119-135. [6]] and by Chen and Bell [P.-L. Chen, J. Bell, Spine-density dependence of the qualitative behavior of a model of a nerve fiber with excitable spines, J. Math. Anal. Appl. 187 (1994) 384-410.] 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 upon 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)