High multiplicity and chaos for an indefinite problem arising from genetic models

We deal with the periodic boundary value problem associated with the parameter-dependent second-order nonlinear differential equation u'' + cu' + (λ a+(x) - μ a-(x)) g(u) = 0, where λ,μ>0 are parameters, c∈R, a(x) is a locally integrable P-periodic sign-changing weight function, and g: [0,1]→R is a continuous function such that g(0)=g(1)=0, g(u)>0 for all u∈]0,1[, with superlinear growth at zero. A typical example for g(u), that is of interest in population genetics, is the logistic-type nonlinearity g(u)=u^2(1-u). Using a topological degree approach, we provide high multiplicity results by exploiting the nodal behavior of a(x). More precisely, when m is the number of intervals of positivity of a(x) in a P-periodicity interval, we prove the existence of 3^m-1 non-constant positive P-periodic solutions, whenever the parameters λ and μ are positive and large enough. Such a result extends to the case of subharmonic solutions. Moreover, by an approximation argument, we show the existence of a family of globally defined solutions with a complex behavior, coded by (possibly non-periodic) bi-infinite sequences of three symbols.