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High Multiplicity and Chaos for an Indefinite Problem Arising from Genetic Models
Advanced Nonlinear Studies ( IF 2.1 ) Pub Date : 2020-08-01 , DOI: 10.1515/ans-2020-2094
Alberto Boscaggin 1 , Guglielmo Feltrin 2 , Elisa Sovrano 3
Affiliation  

Abstract 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^{\prime\prime}+cu^{\prime}+\bigl{(}\lambda a^{+}(x)-\mu a^{-}(x)\bigr{)}g(u)% =0, where λ , μ > 0 {\lambda,\mu>0} are parameters, c ∈ ℝ {c\in\mathbb{R}} , a ⁢ ( x ) {a(x)} is a locally integrable P-periodic sign-changing weight function, and g : [ 0 , 1 ] → ℝ {g\colon{[0,1]}\to\mathbb{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 [ {u\in{]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.

中文翻译:

遗传模型引起的不确定问题的高度多样性和混沌

摘要 我们处理与参数相关的二阶非线性微分方程 u ′′ + c ⁢ u ′ + ( λ ⁢ a + ⁢ ( x ) - μ ⁢ a - ⁢ ( x ) ) ⁢ 相关的周期性边值问题g ⁢ ( u ) = 0 , u^{\prime\prime}+cu^{\prime}+\bigl{(}\lambda a^{+}(x)-\mu a^{-}(x) \bigr{)}g(u)% =0, 其中 λ , μ > 0 {\lambda,\mu>0} 是参数, c ∈ ℝ {c\in\mathbb{R}} , a ⁢ ( x ) {a(x)} 是一个局部可积的 P 周期符号变化权重函数,g : [ 0 , 1 ] → ℝ {g\colon{[0,1]}\to\mathbb{R}} 是一个连续函数使得 g ⁢ ( 0 ) = g ⁢ ( 1 ) = 0 {g(0)=g(1)=0} , g ⁢ ( u ) > 0 {g(u)>0} 对所有 u ∈ ] 0 , 1 [ {u\in{]0,1[}} ,超线性增长为零。g ⁢ ( u ) {g(u)} 的一个典型例子是群体遗传学中感兴趣的逻辑型非线性 g ⁢ ( u ) = u 2 ⁢ ( 1 - u ) {g(u)= u^{2}(1-u)}。使用拓扑度方法,我们通过利用 a ⁢ ( x ) {a(x)} 的节点行为来提供高多重性结果。更准确地说,当 m 是 P 周期区间中 a ⁢ ( x ) {a(x)} 的正区间数时,我们证明 3 m - 1 {3^{m}-1} 非-常数正 P 周期解,只要参数 λ 和 μ 为正且足够大。这样的结果扩展到次谐波解决方案的情况。此外,通过近似论证,我们展示了一系列具有复杂行为的全局定义解的存在,由三个符号的(可能是非周期性的)双无限序列编码。当 m 是 P 周期区间中 a ⁢ ( x ) {a(x)} 的正区间数时,我们证明存在 3 m - 1 {3^{m}-1} 非常数正P 周期解,只要参数 λ 和 μ 为正且足够大。这样的结果扩展到次谐波解决方案的情况。此外,通过近似论证,我们展示了一系列具有复杂行为的全局定义解的存在,由三个符号的(可能是非周期性的)双无限序列编码。当 m 是 P 周期区间中 a ⁢ ( x ) {a(x)} 的正区间数时,我们证明存在 3 m - 1 {3^{m}-1} 非常数正P 周期解,只要参数 λ 和 μ 为正且足够大。这样的结果扩展到次谐波解决方案的情况。此外,通过近似论证,我们展示了一系列具有复杂行为的全局定义解的存在,由三个符号的(可能是非周期性的)双无限序列编码。
更新日期:2020-08-01
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