By applying the Horn’s fixed point theorem, we prove the existence of T0-periodic PC-mild solution of impulsive periodic systems when PC-mild solutions are ultimate bounded.

In this paper, we consider an n-species Lotka-Volterra periodic competition system. Using a comparison method and the Brouwer fixed point theorem, we obtain some sufficient conditions for the ultimate boundedness of solutions and the existence and global attractivity of a positive periodic solution. We also point out that these results constitute a generalization of K. Gopalsamy and J. M. Cushi...

We present a chaotic frequency-modulated microwave source that is governed by a simple, first-order nonlinear delay differential equation. When a sinusoidal nonlinearity is incorporated, the dynamical behaviors range from fixed-point to periodic to chaotic, depending on the feedback strength. When the sinusoidal nonlinearity is replaced by a binary nonlinearity, the system exhibits a complex pe...

By using the critical point theory, the existence of periodic solutions for 2nth-order nonlinear pLaplacian difference equations is obtained. The main approaches used in our paper are variational techniques and the Saddle Point theorem. The problem is to solve the existence of periodic solutions for 2nth-order p-Laplacian difference equations. The results obtained successfully generalize and co...

On the other hand, let S be a subset of the real numbers and let / be a function from S into itself. For every positive integer n, we let f denote the n iterate of f : f =f and f=f<> f~ for n > 2. For every #0 G s > w e call the set {f(xQ)\k > 0} the orbit of xQ under /. If XQ satisfies f{x$) = XQ for some positive integer 77?, then we call XQ a periodic point of / and call the smallest such po...

For the family of exponential maps z 7→ exp(z) + κ, we show the following analog of a theorem of Douady and Hubbard concerning polynomials. Suppose that g is a periodic external ray of an exponential map with nonescaping singular value. Then g lands at a repelling or parabolic periodic point. We also show that there are periodic external rays landing at all periodic points of such an exponentia...

Isochrons are foliations of phase space that extend the notion of phase of a stable periodic orbit to the basin of attraction of this periodic orbit. Each point in the basin of attraction lies on only one isochron and two points on the same isochron converge to the periodic orbit with the same phase. These properties allow one to define so-called phase models that reduce the dimension of an osc...

‎In this work‎, ‎a regular polygon with $n$ sides is described by a periodic (circular) sequence with period $n$‎. ‎Each element of the sequence represents a vertex of the polygon‎. ‎Each symmetry of the polygon is the rotation of the polygon around the center-point and/or flipping around a symmetry axis‎. ‎Here each symmetry is considered as a system that takes an input circular sequence and g...

By counting the numbers of periodic points of all periods for some interval maps, we obtain infinitely many new congruence identities in number theory. Let S be a nonempty set and let f be a map from S into itself. For every positive integer n, we define the n iterate of f by letting f 1 = f and f = f ◦ f for n ≥ 2. For y ∈ S, we call the set { f(y) : k ≥ 0 } the orbit of y under f . If f(y) = ...

Let T be a tree with n vertices. Let f : T → T be continuous and suppose that the n vertices form a periodic orbit under f . We show: (1) (a) If n is not a divisor of 2k then f has a periodic point with period 2k. (b) If n = 2pq, where q > 1 is odd and p ≥ 0, then f has a periodic point with period 2pr for any r ≥ q. (c) The map f also has periodic orbits of any period m where m can be obtained...