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 ′′ + c u ′ + (λ a + (x) - μ a - (x)) g (u) = 0, u^{primeprime}+cu^{prime}+igl{(}lambda a^{+}(x)-mu a^{-}(x)igr{)}g(u)% =0, where λ, μ > 0 {lambda,mu>0} are parameters, c {cinmathbb{R}}, a (x) {a(x)} is a locally integrable P-periodic sign-changing weight function, and g: [ 0, 1 ] → {gcolon{[0,1]} omathbb{R}} is a continuous function such that g (0) = g (1) = 0 {g(0)=g(1)=0}, g (u) > 0 {g(u)>0} for all u ] 0, 1 [ {uin{]0,1[}}, with superlinear growth at zero. A typical example for g (u) {g(u)}, that is of interest in population genetics, is the logistic-type nonlinearity g (u) = u 2 (1 - u) {g(u)=u^{2}(1-u)}. Using a topological degree approach, we provide high multiplicity results by exploiting the nodal behavior of a (x) {a(x)}. More precisely, when m is the number of intervals of positivity of a (x) {a(x)} in a P-periodicity interval, we prove the existence of 3 m - 1 {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.