Unpredictable solutions of linear differential and discrete equations

Abstract

In this study, the existence and uniqueness of the unpredictable solution for a non-homogeneous linear system of ordinary differential equations is considered. The hyperbolic case is under discussion. New properties of unpredictable functions are discovered. The presence of the solutions confirms the existence of Poincaré chaos. Simulations illustrating the chaos are provided. existence of unpredictable solutions simultaneously means the presence of Poincaré chaos, i.e., unpredictable solutions are "irregular". This makes the subject attractive for applications. Finally, we consider new properties of the functions. Sufficient conditions are provided such that a linear transformation of an unpredictable function is unpredictable, and it is proved that the sum of an unpredictable function and a periodic function is an unpredictable function.Ke α(t−s) g + (s + t n ) − g + (s) ds ≤ d t Ke α(t−s) ξds+

The Unpredictable Point and Poincaré Chaos

“…where n ∈ Z and λ is a parameter. Based on the result of the papers [12] and [18], it was demonstrated in paper [13] that for λ ∈ [3 + (2/3) 1/2 , 4] the map (4.6) possesses an unpredictable solution. For such values of the parameter, the unit interval [0, 1] is invariant under the iterations of the map [19].…”

“…The research as well as the origin of the chaos [8] gave us strong arguments for the development of motions in classical dynamical systems theory [9] by proceeding behind Poisson stable points to unpredictable points [10]. Then, the dynamics has been specified such that a function that is bounded on the real axis is an unpredictable point [11]- [13]. In the papers [11]- [13], we have demonstrated that unpredictable functions are easy to be analyzed as solutions of differential equations.…”

“…Then, the dynamics has been specified such that a function that is bounded on the real axis is an unpredictable point [11]- [13]. In the papers [11]- [13], we have demonstrated that unpredictable functions are easy to be analyzed as solutions of differential equations. This paradigm is not completed, if one does not consider discrete equations.…”

“…Definition 2.2 A uniformly continuous and bounded function ϕ : R → R p is unpredictable if there exist positive numbers ǫ 0 , δ, and sequences {t k } , {τ k }, both of which diverge to infinity, such thatThe last definition can be considered as a more restrictive version of the next two, which will be useful in the future for applications of functional analysis methods in the theory of differential equations. Definition 2.3 A continuous and bounded function ϕ : R → R p is unpredictable if there exist positive numbers ǫ 0 , δ, and sequences {t k } , {τ k }, both of which diverge to infinity, such that ϕ(t + t k ) − ϕ(t) → 0 as k → ∞ uniformly on compact subsets of R and ϕ(t k + τ k ) − ϕ(τ k ) ≥ ǫ 0 for each k ∈ N. Definition 2.4 A continuous and bounded function ϕ : R → R p is unpredictable if there exist positive numbers ǫ 0 , δ, and sequences {t k } , {τ k }, both of which diverge to infinity, such that ϕThe following definition of an unpredictable sequence was first mentioned in paper [13] as an analogue for Definition 2.2.Definition 2.5 A bounded sequence {ϕ n } , n ∈ Z, in R p is called unpredictable if there exist a positive number ǫ 0 and sequences {ζ k } , {η k }, k ∈ N, of positive integers, both of which diverge to infinity, such that ϕ n+ζ k − ϕ n → 0 uniformly as k → ∞ for each n in bounded intervals of integers and ϕ ζ k +η k − ϕ η k ≥ ǫ 0 for each k ∈ N.…”

“…The following definition of an unpredictable sequence was first mentioned in paper [13] as an analogue for Definition 2.2.…”