Let M be a smooth compact manifold of dimension at least 2 and Diff (M) be the space of C smooth diffeomorphisms of M . Associate to each diffeomorphism f ∈ Diff (M) the sequence Pn(f ) of the number of isolated periodic points for f of period n. In this paper we exhibit an open set N in the space of diffeomorphisms Diff (M) such for a Baire generic diffeomorphism f ∈ N the number of periodic p...

The study of periodic orbits for dynamical systems dates back to the very origins of the subject. Given a diieomorphism f : M ! M of a compact manifold it is a classical problem to count the number of periodic orbits fx; fx; : : : ; f n?1 xg, with f n x = x. For each n 1, we denote by N(n) the number of periodic points of period n. General Problem. What can we say about the numbers N(n)? What d...

and Applied Analysis 3 The previous work mainly focuses on the autonomous differential-difference equation 1.2 . However, some papers 13, 24 contain some interesting nonautonomous differential difference equations arising in economics and population biology where the delay r of 1.2 depends on time t instead of a positive constant. Motivated by the lack of more results on periodic solutions for ...

We study the ordinary differential equation εẍ+ ẋ+ ε g(x) = εf(ωt), with f and g analytic and f quasi-periodic in t with frequency vector ω ∈ R. We show that if there exists c0 ∈ R such that g(c0) equals the average of f and the first non-zero derivative of g at c0 is of odd order n, then, for ε small enough and under very mild Diophantine conditions on ω, there exists a quasi-periodic solution...

We consider abstract differential equations of the form u′(t) = Au(t)+f(t) or u′′(t)=Au(t)+f(t) in Banach spaces X, where f(·), R→X is almost-periodic, while A is a linear operator, (A)⊂X →X. If the solution u(·) is likewise almost-periodic, R→X, we establish connections between their Bohr-transforms, û(λ) and f̂ (λ). 2000 Mathematics Subject Classification. 34C27, 34G10.

Let D be a dendrite and f : D → D a continuous map. Denote by E(D) and B(D) the sets of endpoints and branch points of D respectively. We show that if E(D) is countable (resp. B(D) is discrete) then f is pointwise-recurrent if and only if f is pointwise periodic homeomorphism (resp. every point in D\E(D) is periodic).

We consider the semilinear Schrödinger equation { −4 u+ V (x)u = f(x, u), x ∈ RN , u ∈ H1(RN ), where V (x) is asymptotically periodic and sign-changing, f(x, u) is a superlinear, subcritical nonlinearity. Under asymptotically periodic V (x) and a super-quadratic condition about f(x, u). We prove that the above problem has a ground state solution which minimizes the corresponding energy among a...

Let f be a subhyperbolic entire transcendental function of finite order and let z0 be a repelling periodic point of f . We show that there exists at least one dynamic ray (injective curve to ∞ consisting of escaping points) that lands at z0. In fact, our result holds more generally, namely for any subhyperbolic entire function f for which each periodic address is realised by some dynamic ray in...

Abstract: We study a dynamic system described by the Kuramoto-Sivashinsky equation with an external excitation f posed on a finite domain. Firstly it shows that under the given boundary feedback conditions it admits a unique solution and the solution is stable. Secondly it proves that if the external excitation f is a time periodic function, then the system under the boundary conditions admits ...

Let (X, d) be a nonempty metric space, and let (2X , Hd) be the hyperspace of all nonempty compact subsets of X with the Hausdorff metric. Let F :X → 2X be an ε-contractive map. A general condition is given that guarantees the existence of a periodic point of F (the theorem extends a result of Edelstein to multi-valued maps). The condition holds when X is compact; hence, F has a periodic point ...