Consider a1, . . . , an ∈ R arbitrary elements. We characterize those functions f : R → R that decompose into the sum of aj-periodic functions, i.e., f = f1+· · ·+fn with ∆aj f(x) := f(x+aj)−f(x) = 0. We show that f has such a decomposition if and only if for all partitions B1∪B2∪· · ·∪BN = {a1, . . . , an} with Bj consisting of commensurable elements with least common multiples bj one has ∆b1 ...

In a recent work, we developed three new compact numerical quadrature formulas for finite-range periodic supersingular integrals $$I[f]={\mathop\int{\!\!\!\!\!\!=}}^{\,\,b}_{\!\!a} f(x)\,dx$$ , where $$f(x)=g(x)/(x-t)^3,$$ assuming that $$g\in C^\infty [a,b]$$ and f(x) is T-periodic, $$T=b-a$$ . With $$h=T/n$$ these read $$\begin{aligned} {\widehat{T}}{}^{(0)}_n[f]&=h\sum ^{n-1}_{j=1}f(t+jh) -\...

In this paper, we discuss the existence of positive periodic solutions to the nonlinear differential equation u′′(t)+ a(t)u(t) = f (t, u(t)), t ∈ R, where a : R → [0,+∞) is an ω-periodic continuous function with a(t) ≡ 0, f : R × [0,+∞) → [0,+∞) is continuous and f (·, u) : R → [0,+∞) is also an ω-periodic function for each u ∈ [0,+∞). Using the fixed point index theory in a cone, we get an ess...

We show there is a residual subset S(M) of Diff1(M) such that, for every f ∈ S(M), any homoclinic class of f containing periodic saddles p and q of indices α and β, respectively, where α < β, has superexponential growth of the number of periodic points inside the homoclinic class. Furthermore, it is shown the super-exponential growth occurs for hyperbolic periodic points of index γ inside the h...

For nonlinear difference equations of the form xn F n, xn−1, . . . , xn−m , it is usually difficult to find periodic solutions. In this paper, we consider a class of difference equations of the form xn anxn−1 bnf xn−k , where {an}, {bn} are periodic sequences and f is a nonlinear filtering function, and show how periodic solutions can be constructed. Several examples are also included to illust...

Two examples of dynamical systems with spatial almost periodicity are considered and the complexity growth during the time evolution is quantified. The first example deals with almost periodic cellular automata (CA). The growth rate of the information which is needed to store a configuration can quantitatively be given by a real number which is bounded above by the dimension of the CA. In the s...

We study the discrete dynamics of standard (or left) polynomials $f(x)$ over division rings $D$. define their fixed points to be $\lambda \in D$ for which $f^{\circ n}(\lambda)=\lambda$ any $n \mathbb{N}$, where n}(x)$ is defined recursively by n}(x)=f(f^{\circ (n-1)}(x))$ and 1}(x)=f(x)$. Periodic are similarly defined. prove that $\lambda$ a point if only $f(\lambda)=\lambda$, enables use kno...

the purpose of this paper is to study the higher order asymptotic distributions of the eigenvalues associated with a class of sturm-liouville problem with equation of the form w??=(?2f(x)?r(x)) (1), on [a,b, where ? is a real parameter and f(x) is a real valued function in c2(a,b which has a single zero (so called turning point) at point 0x=x and r(x) is a continuously differentiable function. ...

Let ∆ be a ball in the complex vector space C centered at the origin, let f : ∆ → C be a holomorphic mapping, with f(0) = 0, and let M be a positive integer. If the origin 0 is an isolated fixed point of the M th iteration f of f, then one can define the number OM (f, 0) of periodic orbits of f with period M hidden at the fixed point 0, which has the meaning: any holomorphic mapping g : ∆ → C s...

In this note, we show a polynomial bound for the growth of the lap number of a piecewise monotone and piecewise continuous interval map with nitely many periodic points. We use Milnor and Thurston's kneading theory with the coordinates of Baladi and Ruelle, which are useful for extending the theory to the non continuous case. We say that f : 0; 1] ! 0; 1] is piecewise strictly monotone piecewis...