Critical Orbits Research Articles

Equivariant Conley index versus degree for equivariant gradient maps

In this article we study the relationship between the degree forinvariant strongly indefinite functionals and the equivariantConley index. We prove that, under certain assumptions, achange of the equivariant Conley indices is equivalent to thechange of the degrees for equivariant gradient maps. Moreover, weformulate easy to verify sufficient conditions for theexistence of a global bifurcation of critical orbits of invariantstrongly indefinite functionals.

Periodic solutions in the Sun–Jupiter–Trojan Asteroid–Spacecraft system

In this paper we investigate families of non-symmetric periodic orbits of the restricted four-body problem where the three primary bodies are set in the stable Lagrangian equilateral triangle configuration. More precisely, we consider the primary bodies Sun, Jupiter and an hypothetical Trojan Asteroid lie at the apices of an equilateral triangle and the fourth massless body (Spacecraft) is moving under the Newtonian gravitational attraction of the primaries. The problem admits eight non-collinear equilibrium points. Four of them are close to the Asteroid, two are stable and two are unstable. We focus our investigation on families of periodic orbits around Jupiter and/or the Asteroid. We also study analytically the solutions in the neighborhood of the stable equilibrium points. New families, namely short and long-period ones of non-symmetric periodic orbits, for each stable equilibrium point exist. The linear stability of each periodic solution is also studied. Special generating horizontal and vertical critical periodic orbits of each family are calculated.

Statistical stability and limit laws for Rovella maps

We consider a family of one-dimensional maps arising from the contracting Lorenz attractors studied by Rovella. Benedicks–Carleson techniques were used by Rovella to prove that there is a one-parameter family of maps whose derivatives along their critical orbits increase exponentially fast and the critical orbits have slow recurrence to the critical point. Metzeger proved that these maps have a unique absolutely continuous ergodic invariant probability measure (SRB measure).Here we use the technique developed by Freitas and show that the tail set (the set of points which at a given time have not achieved either the exponential growth of derivative or the slow recurrence) decays exponentially fast as time passes. As a consequence, we obtain the continuous variation (in the L1-norm) of the densities of the SRB measures and associated metric entropies with the parameter. Our main result also implies some statistical properties for these maps.

Primitive Prime Divisors in the Critical Orbit of zd+c

We prove the finiteness of the Zsigmondy set associated to the critical orbit of f(z) = z + c for rational values of c by uniformly bounding the size of the Zsigmondy set for all c ∈ Q and all d ≥ 2. We prove further that there exists an effectively computable bound M(c) on the largest element of the Zsigmondy set, and that under mild additional hypotheses on c, we have M(c) ≤ 3.

Fekete configuration, quantitative equidistribution and wandering critical orbits in non-archimedean dynamics

Let $f$ be a rational function of degree $d>1$ on the projective line over a possibly non-archimedean algebraically closed field. A well-known process initiated by Brolin considers the pullbacks of points under iterates of $f$, and produces an important equilibrium measure. We define the asymptotic Fekete property of pullbacks of points, which means that they mirror the equilibrium measure appropriately. As application, we obtain an error estimate of equidistribution of pullbacks of points for $C^1$-test functions in terms of the proximity of wandering critical orbits to the initial points, and show that the order is $O(\sqrt{kd^{-k}})$ upto a specific exceptional set of capacity 0 of initial points, which is contained in the set of superattracting periodic points and the omega-limit set of wandering critical points from the Julia set or the presingular domains of $f$. As an application in arithmetic dynamics, together with a dynamical Diophantine approximation, these estimates recover Favre and Rivera-Letelier's quantitative equidistribution in a purely local manner.

Elliptic orbits and satellite constellations for coverage of illuminated areas

Synthesis of systems of continuous or periodic cov� erage of the Earth (from global coverage to survey of separate regions) is a traditional approach to the design of satellite constellations (SC). Following this line of research, the methods of ballistic design have been developed for various types of orbits and forms of coverage (1-7). A brief historical overview of develop� ment and modern state of the art of the theory of SC for continuous coverage is presented in (8). The SC in elliptic orbits of the Molniya type (9-11) form a spe� cial class. These orbits are characterized by the use of critical orbit inclination (i = 63.4° or i = 116.6°) and by position of the orbit apogee in the hemisphere including a coverage region. In this class socalled oneroute (or onepath) systems were developed (10) whose paths on the ground surface coincide (i.e., parameters of the orbits and Greenwich longitudes of their ascending nodes are identical). For the systems designed for surveying the Earth's surface and connected with illumination of observed regions by the Sun one can use SC of continuous cov� erage in highly elliptical orbits (11). However, this approach can be nonoptimal as far as the number of required satellites is concerned. The use of SC of peri� odic coverage, as a rule, cannot solve this problem, since arising gaps in survey have cyclic character and are not connected with illumination by the Sun. The goal of this work is to develop the methods of choosing economical SC designed for observation of illumi� nated regions and based on a new type of highly elliptic orbits of Molniya type. The latter are synchronized with the solar day, as opposed to the classical Molniya orbits that are synchronized with the Earth's rotation (i.e., with the sidereal day). It is worthy of noting that such orbits can also be used for the inverse problem: to survey some areas during nighttime (for example, for their illumination). Another possible application is satellite communication systems in which the intervals of visibility, determining the possibility of communi� cation, are connected to the local time with maximum communication traffic. PseudoSunSynchronous Orbits of the Molniya Type. Let us consider a certain region of observation. Let the region center have geographic longitude λН and corresponding local time scale tm. We assume that there is a certain highly elliptical orbit with the orbit period multiple to the solar day duration

ON STABLE QUADRATIC POLYNOMIALS

AbstractWe recall that a polynomial f(X) ∈ K[X] over a field K is called stable if all its iterates are irreducible over K. We show that almost all monic quadratic polynomials f(X) ∈ ℤ[X] are stable over ℚ. We also show that the presence of squares in so-called critical orbits of a quadratic polynomial f(X) ∈ ℤ[X] can be detected by a finite algorithm; this property is closely related to the stability of f(X). We also prove there are no stable quadratic polynomials over finite fields of characteristic 2 but they exist over some infinite fields of characteristic 2.

DYNAMICS AND BIFURCATIONS OF A FAMILY OF RATIONAL MAPS WITH PARABOLIC FIXED POINTS

We study a family of rational maps of the Riemann sphere with the property that each map has two fixed points with multiplier -1; moreover, each map has no period 2 orbits. The family we analyze is Ra(z) = (z3 - z)/(-z2 + az + 1), where a varies over all nonzero complex numbers. We discuss many dynamical properties of Ra including bifurcations of critical orbit behavior as a varies, connectivity of the Julia set J(Ra), and we give estimates on the Hausdorff dimension of J(Ra).

Settled polynomials over finite fields

We study the factorization into irreducibles of iterates of a quadratic polynomial f f over a finite field. We call f f settled when the factorization of its n n th iterate for large n n is dominated by “stable” polynomials, namely those that are irreducible under post-composition by any iterate of f f . We prove that stable polynomials may be detected by their action on the critical orbit of f f and that the critical orbit also gives information about the splitting of non-stable polynomials under post-composition by iterates of f f . We then define a Markov process based on the critical orbit of f f and conjecture that its limiting distribution describes the full factorization of large iterates of f f . This conjecture implies that almost all quadratic f f defined over a finite field are settled. We give several types of evidence for our conjecture.

On the Dynamics of the Rational Family $f_{t}(z)=- t/4{(z^{2}- 2)^{2}/(z^{2}- 1)}$

In this paper we discuss the dynamics as well as the structure of the parameter space of the one-parameter family of rational maps $${f_{t}(z)}= - {t\over 4} {(z^{2}- 2)^{2}\over {z^{2}- 1}}$$ with free critical orbit $$\pm\sqrt{2}\mathop \rightarrow \limits^{(2)}0 \mathop \rightarrow \limits^{(4)}t \mathop\rightarrow \limits^{(1)}\cdots.$$ In particular we show that for any escape parameter t, the boundary of the basin at infinity A t is either a Cantor set, a curve with infinitely many complementary components, or else a Jordan curve. In the latter case the Julia set is a Sierpinski curve.

Families of periodic orbits in the restricted four-body problem

In this paper, families of simple symmetric and non-symmetric periodic orbits in the restricted four-body problem are presented. Three bodies of masses m1, m2 and m3 (primaries) lie always at the apices of an equilateral triangle, while each moves in circle about the center of mass of the system fixed at the origin of the coordinate system. A massless fourth body is moving under the Newtonian gravitational attraction of the primaries. The fourth body does not affect the motion of the three bodies. We investigate the evolution of these families and we study their linear stability in three cases, i.e. when the three primary bodies are equal, when two primaries are equal and finally when we have three unequal masses. Series, with respect to the mass m3, of critical periodic orbits as well as horizontal and vertical-critical periodic orbits of each family and in any case of the mass parameters are also calculated.

Critical homoclinic orbits lead to snap-back repellers

When nondegenerate homoclinic orbits to an expanding fixed point of a map f : X → X , X ⊆ R n , exist, the point is called a snap-back repeller. It is known that the relevance of a snap-back repeller (in its original definition) is due to the fact that it implies the existence of an invariant set on which the map is chaotic. However, when does the first homoclinic orbit appear? When can other homoclinic explosions, i.e., appearance of infinitely many new homoclinic orbits, occur? As noticed by many authors, these problems are still open. In this work we characterize these bifurcations, for any kind of map, smooth or piecewise smooth, continuous or discontinuous, defined in a bounded or unbounded closed set. We define a noncritical homoclinic orbit and a homoclinic orbit of an expanding fixed point is structurally stable iff it is noncritical. That is, only critical homoclinic orbits are responsible for the homoclinic explosions. The possible kinds of critical homoclinic orbits will be also investigated, as well as their dynamic role.

On critical orbits and sectional hyperbolicity of the nonwandering set for flows

In this note, we introduce the notion of nonuniformly sectional hyperbolic set and use it to prove that any C1-open set which contains a residual subset of vector fields with nonuniformly sectional hyperbolic critical set also contains a residual subset of vector fields with sectional hyperbolic nonwandering set. This not only extends Theorem A of Castro [11], but using suspensions we recover it.

PARAMETERIZED DYNAMICS FOR THE WEIERSTRASS ELLIPTIC FUNCTION OVER SQUARE PERIOD LATTICES

We iterate the Weierstrass elliptic ℘ function in order to understand the dependence of the dynamics on the underlying period lattice L. We focus on square lattices and use the holomorphic dependence on the classical invariants (g2, g3) = (g2, 0) to show that in parameter space (g2-space) one sees both quadratic-like attracting orbit behavior and prepole dynamics. In the case of prepole parameters all critical orbits terminate at poles and the Julia set of ℘L is the entire sphere. We show that both the Mandelbrot-like dynamics and the prepole parameters accumulate on prepole parameters of lower order providing results on the dynamics occurring in parameter space "between Mandelbrot sets".

Global bifurcations of critical orbits of [formula omitted]-invariant strongly indefinite functionals

Let H be a separable Hilbert space which is an orthogonal representation of a compact Lie group G and let Φ : H → R be a G -invariant strongly indefinite functional of the class C 1 . To study critical orbits of Φ we have defined the degree for G -invariant strongly indefinite functionals, which is an element of the Euler ring U ( G ) . Using this degree we have formulated the Rabinowitz alternative for G -invariant strongly indefinite functionals. The abstract results are applied to the study of global bifurcations of weak solutions of non-cooperative systems of elliptic differential equations.

Effect of 3rd-degree gravity harmonics and Earth perturbations on lunar artificial satellite orbits

In a previous work we studied the effects of (I) the J 2 and C 22 terms of the lunar potential and (II) the rotation of the primary on the critical inclination orbits of artificial satellites. Here, we show that, when 3rd-degree gravity harmonics are taken into account, the long-term orbital behavior and stability are strongly affected, especially for a non-rotating central body, where chaotic or collision orbits dominate the phase space. In the rotating case these phenomena are strongly weakened and the motion is mostly regular. When the averaged effect of the Earth’s perturbation is added, chaotic regions appear again for some inclination ranges. These are more important for higher values of semi-major axes. We compute the main families of periodic orbits, which are shown to emanate from the ‘frozen eccentricity’ and ‘critical inclination’ solutions of the axisymmetric problem (‘J 2 + J 3’). Although the geometrical properties of the orbits are not preserved, we find that the variations in e, I and g can be quite small, so that they can be of practical importance to mission planning.

A characterization of hyperbolic rational maps

In the early 1980's Thurston gave a topological characterization of rational maps whose critical points have finite iterated orbits (\cite{Th,DH1}): given a topological branched covering $F$ of the two sphere with finite critical orbits, if $F$ has no Thurston obstructions then $F$ possesses an invariant complex structure (up to isotopy), and is combinatorially equivalent to a rational map. We extend this theory to the setting of rational maps with infinite critical orbits, assuming a certain kind of hyperbolicity. Our study includes also holomorphic dynamical systems that arise as coverings over disconnected Riemann surfaces of finite type. The obstructions we encounter are similar to those of Thurston. We give concrete criteria for verifying whether or not such obstructions exist. Among many possible applications, these results can be used for example to construct holomorphic maps with prescribed dynamical properties; or to give a parameter description, both local and global, of bifurcations of complex dynamical systems.

Elliptic functions with critical orbits approaching infinity

We construct examples of elliptic functions, viewed as iterated meromorphic functions from the complex plane to the sphere, with the property that there exist one or more critical points which approach the essential singularity at ∞ under iteration but are not prepoles. We obtain many nonequivalent elliptic functions satisfying this property, including examples with Julia set the whole sphere as well as examples with nonempty Fatou set. These are the first examples of this type known to exist and provide examples to illustrate unusual chaotic measure theoretic behaviour studied by the third author and others.

Connectedness of the Arnold tongues for double standard maps

We show that Arnold tongues for the family of double standard maps \[ f a , b ( x ) = 2 x + a − ( b / π ) s i n ( 2 π x ) f_{a,b}(x)=2x+a-(b/\pi )sin(2 \pi x) \] are connected. This proof is accomplished in the complex domain by means of quasiconformal techniques and depends partly upon the fact that the complexification of f a , b f_{a,b} , has only one critical orbit taking symmetry into account.

On the length of critical orbits of stable quadratic polynomials

We use the Weil bound of multiplicative character sums together with some recent results of N. Boston and R. Jones, to show that the critical orbit of quadratic polynomials over a finite field of q elements is of length O q 3/4 � , improving upon the trivial bound q.