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) = ...

By studying various rational integrable maps on Ĉ with p invariants, we show that periodic points form an invariant variety of dimension p for each period if p ≥ d/2, in contrast to the case of nonintegrable maps in which they are isolated. We prove the theorem: ‘If there is an invariant variety of periodic points of some period, there is no set of isolated periodic points in the map.’

the problem of pole assignment, also known as an eigenvalue assignment, in linear discrete-time periodic systems in discs was solved by a novel method which employs elementary similarity operations. the former methods tried to assign the points inside the unit circle while preserving the stability of the discrete time periodic system. nevertheless, now we can obtain the location of eigenvalues ...

In this paper we study symplectic maps with a continuous symmetry group arising by periodic forcing of symmetric Hamiltonian systems. By Noether’s Theorem, for each continuous symmetry the symplectic map has a conserved momentum. We study the persistence of relative periodic points of the symplectic map when momentum is varied and also treat subharmonic persistence and relative subharmonic bifu...

We show that S. Saito’s fixed point formula serves as a powerful tool for counting the number of isolated periodic points of an area-preserving surface map admitting periodic curves. His notion of periodic curves of types I and II plays a central role in our discussion. We establish a Shub-Sullivan type result on the stability of local indices under iterations of the map, the finiteness of the ...

In this paper, we study bifurcations of equilibrium points and periodic solutions observed in a resistively coupled oscillator with voltage ports. We classify equilibrium points and periodic solutions into four and eight di erent types, respectively, according to their symmetrical properties. By calculating D-type of branching sets (symmetry-breaking bifurcations) of equilibrium points and peri...

The Kowalevski workshop on mathematical methods of regular dynamics was organized by Professor Vadim Kuznetsov in April 2000 at the University of Leeds [1]. In his introductory talk about the Kowalevski top, Professor Kuznetzov [2] had shown his strong interest on the subject and motivated the authors to work on classical tops. In our recent paper [3] we have studied the behaviour of periodic p...

We prove the Conley conjecture for a closed symplectically aspherical symplectic manifold: a Hamiltonian diffeomorphism of a such a manifold has infinitely many periodic points. More precisely, we show that a Hamiltonian diffeomorphism with finitely many fixed points has simple periodic points of arbitrarily large period. This theorem generalizes, for instance, a recent result of Hingston estab...