Period-doubling cascades are among the most prominent features of many smooth one-parameter families of maps, F : R×M → M, where M is a locally compact manifold without boundary, typically R . In particular, we investigate F (μ, ·) for μ ∈ J = [μ1, μ2], when F (μ1, ·) has only finitely many periodic orbits while F (μ2, ·) has exponential growth of the number of periodic orbits as a function of ...

Let (X, ‖ · ‖) be a Banach space, f : X → X continous Freche’t differentiable map.Denote the set of almost periodic point by A(f).In this paper,we prove that there exists an uncountable set Λ such that f |Λ is distributionally chaotic,and Λ ⊂ A(f).

The Peano kernel method is used in order to calculate the best possible constants c 2n+1 , independent of M, f 2 Lip M 1 and x, in the estimate jf(x) ? intpollf](x)j M c 2n+1 sin n + 1 2 x : Here, Lip M 1 denotes the class of all 2-periodic functions with Lipschitz-constant M, and intpolf] is the trigonometric interpolation polynomial of f with respect to the 2n + 1 nodes x = 2 2n+1. Let f be i...

We study transcendental singularities of a Schröder map arising from a rational function f , using results from complex dynamics and Nevanlinna theory. These maps are transcendental meromorphic functions of finite order in the complex plane. We show that their transcendental singularities lie over the set where f is not semihyperbolic (unhyperbolic). In addition, if they are direct, then they l...

We show that if the Julia set J(f) of a rational function f is invariant under translation by one and infinity is a periodic or preperiodic point for f , then J(f) must either be a line or the Riemann sphere.

We consider the equation x00(t) = x0(t) + F (x(t); t) signx(t h); = const > 0; h = const > 0; which is a model for a scalar system with a discontinuous negative delayed feedback, and study the dynamics of oscillations with emphasis on the existence, frequency and stability of periodic oscillations. Our main conclusion is that, in the autonomous case F (x; t) F (x), for jF (x)j < 1, there are pe...

This paper studies the boundedness, global asymptotic stability, and periodicity of positive solutions of the equation xn f xn−2 /g xn−1 , n ∈ N0, where f, g ∈ C 0,∞ , 0,∞ . It is shown that if f and g are nondecreasing, then for every solution of the equation the subsequences {x2n} and {x2n−1} are eventually monotone. For the case when f x α βx and g satisfies the conditions g 0 1, g is nondec...

We present a proof of the Sturm-Hurwitz theorem, using basic calculus. Let f : R → R be a continuous 2π-periodic function, and let us assume that its Fourier series is given by (1) f(x) ∼ ∞

Existence of nonnegative solutions are established for the periodic problem y = f(t, y) a.e. on [0, T ], y(0) = y(T ). Here the nonlinearity f satisfies a Landesman Lazer type condition.

The Li\'{e}nard system $\frac{dx}{dt}=y,\ \frac{dy}{dt}=-f(x)y-g(x)$ is considered. Under some assumptions on functions $f(x)$ and $g(x)$, we prove the existence of a periodic solution this system.