Superlinear Indefinite Equations on the Real Line and Chaotic Dynamics

Abstract

In this paper we are concerned with a differential equation of the formwhere À14a5b4 þ 1; q has infinitely many zeros in ða; bÞ; and g is superlinear.We prove the existence of solutions with prescribed nodal properties in the intervals of negativity and positivity of q: When c ¼ 0 and q is periodic we show that the equation under consideration exhibits chaotic-like dynamics. # 2002 Elsevier Science (USA)

Chaotic dynamics in periodically forced asymmetric ordinary differential equations

“…Definition 1.2 agrees with other ones considered in the literature about chaotic dynamics for ODEs with periodic coefficients (see [7,31,35], where ψ is the Poincaré map associated to a differential system). We notice that if the map ψ fulfills Definition 1.2 and is also continuous and injective on…”

Example of a suspension bridge ODE model exhibiting chaotic dynamics: A topological approach

“…, m} Z , there exists a corresponding sequence (w i ) i∈Z ∈ D Z with w i ∈ K s i and w i+1 = ψ(w i ), ∀i ∈ Z (1) and, whenever (s i ) i∈Z is a k-periodic sequence (that is, s i+k = s i , ∀i ∈ Z) for some k 1, there exists a k-periodic sequence (w i ) i∈Z ∈ D Z satisfying (1). 1 This definition is taken from [11] (with minor variants) and agrees with other ones considered in the literature about chaotic dynamics for ODEs with periodic coefficients (see [2,21]), where ψ is the Poincaré map associated to a differential system. We notice that if the map ψ fulfils Definition 1.1 and is continuous on…”

“…Our main result is the following: (2) induces chaotic dynamics on m symbols in the plane. The precise behavior of the solutions can be described as follows:…”

Periodic Solutions and Chaotic Dynamics in Forced Impact Oscillators

“…In an informal way, we say that a map induces chaotic dynamics on two symbols if there exists an invariant set Λ being semiconjugate to the Bernoulli shift, topologically transitive, and having infinitely many periodic points (see Definition 5.1 and Theorem 5.1 in the appendix). This definition of chaos has been used before by several authors in [1,7,8,27,42].…”

“…(see [5,7,8,17,21,32,37,42,45]). In this paper we study the notion of chaos under the perspective of topological horseshoes; see [21] and [47].…”