Related papers: Periodic orbits 1-5 of quadratic polynomials on a …

Iteration of the quadratic map produces sequences of polynomials whose degrees {\sl explode} as the orbital period grows more and more. The polynomial mixing all 335 period-12 orbits has degree $4020$, while for the $52,377$ period-20…

In this note, we consider the following two families of quadratic polynomials $S_{a,c}(x) = a - cx^2$ and $T_{a,c}(x) = a - c(1 + x^2)$ and show that their respective period-3 orbits live very different lives.

Explicit formulas for {\sl orbital carriers} of periods $4$, $5$, and $6$ are reported for discrete-time quadratic dynamics. A systematic investigation of {\sl orbital inheritance} for periods as high as $k\leq 12$ is also reported.…

The planetary dynamics of $4/3$, $3/2$, $5/2$, $3/1$ and $4/1$ mean motion resonances is studied by using the model of the general three body problem in a rotating frame and by determining families of periodic orbits for each resonance.…

We study the origin and bifurcations of typical classes of unstable periodic orbits in a jet flow that was introduced before as a kinematic model of chaotic advection, transport and mixing of passive scalars in meandering oceanic and…

The aim of this paper is to numerically investigate the orbital dynamics of the circular planar restricted problem of five bodies. By numerically integrating several large sets of initial conditions of orbits we classify them into three…

Given a quadratic polynomial with rational coefficients, we investigate the existence of consecutive squares in the orbit of a rational point under the iteration of the polynomial. We display three different constructions of $1$-parameter…

We present a novel numerical method to calculate periodic orbits for dynamical systems by an iterative process which is based directly on the action integral in classical mechanics. New solutions are obtained for the planar motion of three…

We introduce a restricted four body problem in a 2+2 configuration extending the classical Sitnikov problem to the Double Sitnikov problem. The secondary bodies are moving on the same perpendicular line to the planewhere the primaries…

Let $q$ be an odd prime power. Let $f\in \mathbb{F}_q[x]$ be a polynomial having degree at least $2$, $a\in \mathbb{F}_q$, and denote by $f^n$ the $n$-th iteration of $f$. Let $\chi$ be the quadratic character of $\mathbb{F}_q$, and…

Periodic solutions of the three body problem are very important for understanding its dynamics either in a theoretical framework or in various applications in celestial mechanics. In this paper we discuss the computation and continuation of…

Odd degree Chebyshev polynomials over a ring of modulo $2^w$ have two kinds of period. One is an "orbital period". Odd degree Chebyshev polynomials are bijection over the ring. Therefore, when an odd degree Chebyshev polynomial iterate…

By folding an autonomous system of rational equations in the plane to a scalar difference equation, we show that the rational system has coexisting periodic orbits of all possible periods as well as stable aperiodic orbits for certain…

We construct a highly-symmetric periodic orbit of six bodies in three dimensions. In this orbit, binary collisions occur at the origin in a regular periodic fashion, rotating between pairs of bodies located on the coordinate axes.…

Classes of polynomial differential equations of degree n are considered. An explicit upper bound on the size of the coefficients are given which implies that each equation in the class has exactly n complex periodic solutions. In most of…

In this work the existence of periodic solutions is studied for the Hamiltonian functions (Formula presented.) where the first term consist of a harmonic oscillator and the second term are homogeneous polynomials of degree 5 defined by two…

In this paper, we study the critical orbit of a post-critically finite polynomial of the form $f_{c,d}(x) = x^d+c \in \mathbb{C}[x]$. We discover that in many cases the orbit elements satisfy some strong arithmetic properties. It is well…

This paper shows that orbital equations generated by iteration of polynomial maps do not have necessarily a unique representation. Remarkably, they may be represented in an infinity of ways, all interconnected by certain nonlinear…

It is proved that for every complex quadratic polynomial $f$ with Cremer's fixed point $z_0$ (or periodic orbit) for every $\delta>0$, there is at most one periodic orbit of minimal period $n$ for all $n$ large enough, entirely in the disc…

For a class of polynomial non-autonomous differential equations of degree n, we use phase plane analysis to show that each equation in this class has n periodic solutions. The result implies that certain rigid two-dimensional systems have…