Invariant Orbits Research Articles

Monodromies and poincaré series of quasihomogeneous complete intersections

We give a formula connecting the Saito duals of the reduced zeta functions of the monodromies of defining equations of a quasihomogeneous complete intersection, the Poincare series of its coordinate ring, and orbit invariants with respect to the natural ℂ*-action.

Hamiltonian modelling of relative motion.

This paper presents a Hamiltonian approach to modelling relative spacecraft motion based on derivation of canonical coordinates for the relative state-space dynamics. The Hamiltonian formulation facilitates the modelling of high-order terms and orbital perturbations while allowing us to obtain closed-form solutions to the relative motion problem. First, the Hamiltonian is partitioned into a linear term and a high-order term. The Hamilton-Jacobi equations are solved for the linear part by separation, and new constants for the relative motions are obtained, they are called epicyclic elements. The influence of higher order terms and perturbations, such as the oblateness of the Earth, are incorporated into the analysis by a variation of parameters procedure. Closed-form solutions for J(2-) and J(4-)invariant orbits and for periodic high-order unperturbed relative motion, in terms of the relative motion elements only, are obtained.

Free Energy as a Dynamical Invariant (or Can You Hear the Shape of a Potential?)

Classical lattice spin systems provide an important and illuminating family of models in statistical physics. An interaction Φ on a lattice L⊂ℤd determines a lattice spin system with potential AΦ. The pressure P(AΦ) and free energy FAΦ(β)=−(1/β)P(βAΦ) are fundamental characteristics of the system. However, even for the simplest lattice spin systems, the information about the potential that the free energy captures is subtle and poorly understood. We study whether, or to what extent, (microscopic) potentials are determined by their (macroscopic) free energy. In particular, we show that for a one-dimensional lattice spin system, the free energy of finite range interactions typically determines the potential, up to natural equivalence, and there is always at most a finite ambiguity; we exhibit exceptional potentials where uniqueness fails; and we establish deformation rigidity for the free energy. The proofs use a combination of thermodynamic formalism, algebraic geometry, and matrix algebra. In the language of dynamical systems, we study whether a Holder continuous potential for a subshift of finite type is naturally determined by its periodic orbit invariants: orbit spectra (Birkhoff sums over periodic orbits with various types of labeling), beta function (essentially the free energy), or zeta function. These rigidity problems have striking analogies to fascinating questions in spectral geometry that Kac adroitly summarized with the question ``Can you hear the shape of a drum?''.

DYNAMICAL CHARACTERISTICS OF DISCRETIZED CHAOTIC PERMUTATIONS

Chaos theory has been applied to various fields where appropriate random sequences are required. The randomness of chaotic sequences is characteristic of continuous-state systems. Accordingly, the discrepancy between the characteristics of spatially discretized chaotic dynamics and those of original analog dynamics must be bridged to justify applications of digital orbits generated, for example, from digital computers simulating continuous-state chaos. The present paper deals with the chaotic permutations appearing in a chaotic cryptosystem. By analysis of cycle statistics, the convergence of the invariant measure and periodic orbit skeletonization, we show that the orbits in chaotic permutations are ergodic and chaotic enough for applications. In the consequence, the systematic differences in the invariant measures and in the Lyapunov exponents of two infinitesimally L∞-close maps are also investigated.

Singular Calabi–Yau manifolds and ADE classification of CFTs

We study superstring propagations on the Calabi-Yau manifold which develops an isolated ADE singularity. This theory has been conjectured to have a holographic dual description in terms of N=2 Landau-Ginzburg theory and Liouville theory. If the Landau-Ginzburg description precisely reflects the information of ADE singularity, the Landau-Ginzburg model of $D_4,E_6,E_8$ and Gepner model of $A_2\otimes A_2, A_2\otimes A_3, A_2\otimes A_4$ should give the same result. We compute the elements of $D_4,E_6,E_8$ modular invariants for the singular Calabi-Yau compactification in terms of the spectral flow invariant orbits of the tensor product theories with the theta function which encodes the momentum mode of the Liouville theory. Furthermore we find the interesting identity among characters in minimal models at different levels. We give the complete proof for the identity.

On eigenvalues on the orbit space

A map P equivariant with respect to a compact Lie group induces a map Q on the orbit space, which is differentiable provided that P is sufficiently smooth. An equilibrium of Q corresponds to an invariant orbit of P. We compare the eigenvalues of the linearization at the equilibrium with the eigenvalues of a kind of linearization at the invariant orbit. It turns out that the former are products of the latter. Furthermore, the eigenvalues on the orbit space suffice to determine asymptotic stability or instability of the invariant orbit. This generalizes a similar result for vector fields by Kœnig, Math. Proc. Cambridge, Philos. Soc. 121 (1997) 401–424 and applies also to relative periodic points.

Non-standard particle orbits in a thermonuclear tokamak plasma

The problem of determining the orbits of energetic ions in the confining magnetic field of a tokamak plasma and their phase space characteristics is considered analytically. First, starting from the orbit invariants in the simplified magnetic field of a tokamak plasma, a quartic equation describing the trajectories in configuration space is derived. Then from an examination of the coefficients of the orbit equation, the orbit types and their classification are deduced. Analytic solutions of the orbit equation are given and the partition of phase space into regions of trapped (passing) particles and loss cones is determined. Finally, as a specific application, the trajectories and phase space characteristics of 3.5 MeV α particles in the magnetic field of the JET tokamak are calculated.

Spacecraft Formation Flying Control Using Mean Orbit Elements

Two nonlinear feedback control laws are presented for reestablishing a desired J2 invariant relative orbit. Since it is convenient to describe the relative orbit of a deputy with respect to a chief satellite in terms of mean orbit element differences, and because the conditions for a relative orbit being J2 invariant are expressed in terms of mean orbit elements, the first control law feeds back errors in terms of mean orbit elements. Dealing with mean orbit elements has the advantage that short period oscillations are not perceived as tracking errors; rather, only the long term tracking errors are compensated for. The second control law feeds back traditional Cartesian position and velocity tracking errors. For both of the control laws, the desired orbit is computed using mean orbit elements. A numerical study compares and contrasts the two feedback laws.

Apparently Chaotic Orbits Embedded in Closed Curves

In [S. Bahar, Chaos Solitons Fractals, 5 (1995), pp. 1001--1006; 7 (1996), pp. 41--47], pictures were presented of the forward orbit {zn}n in the plane, where zn satisfies zn+1 = znAn+1 for certain affine transformations An . These orbits (a) appear to lie on closed curves, and (b) assume forms reminiscent of strange attractors. We answer the questions raised in these references and prove that indeed in the aperiodic case the invariant orbits of this "iterated function system" lie on closed curves and that in some sense the maps act "chaotically" on the set of closed curves. A key role is being played by almost periodic functions. Furthermore, we provide an expression for the top Lyapunov exponent, that is, a sum in the periodic case and an integral in the aperiodic case. Our results have important implications in applications of computer visualization and imaging and shed some light on nonlinear dynamical systems.

Orbital stability index for stochastic systems

The Known concepts of Lyapunov exponent, moment Lyapunov exponents, and stability index for stationary points of stochastic systems are carried over for invariant orbits with nonvanishing diffusion. The obtained geneal results are applied to investiating stochastic stability and stabilization of orbits on the plane. These questions are considered under small diffusion as well.

Invariant theory, orbits and non-decomposable quadruples of subspaces having non-zero defect

Let V be a vector space having dimension n ⩾ 3 over an algebraically closed field k of characteristic 0. Quadruples of subspaces of V were classified by Gelf and Ponomarev; more recently, Howe and Huang explicitly described the homogeneous generatorss F i of the corresponding algebra of invariants. The purpose of this paper is to connect these two theories in the case where there is a non-decomposable quadruple, having non-zero defect, in the dimension class of the quadruples. We show that the non-decomposables form an open orbit characterized by the non-vanishing of each F i ; furthermore, the SL(V)-orbit of a non-decomposable is completely determined by the values of the F i . The orbit structure for decomposable quadruples where some F i does not vanish is more complicated but is completely described in three tables.

Parasitic absorption by fusion born -particles in ICRF heated plasmas

A method has been developed to calculate the dielectric tensor in a toroidal plasma from a distribution function represented numerically by a discrete set of orbit invariants. Assuming a locally homogeneous plasma, parasitic -particle absorption of the fast magnetosonic wave is studied for different orbit widths in JET-like scenarios. In addition, the beneficial effect observed in JET of introducing a small concentration in second-harmonic tritium heating scenarios is studied.

Bifurcation analysis for spherically symmetric systems using invariant theory

We study a degenerate steady state bifurcation problem with spherical symmetry. This singularity, with the five dimensional irreducible action ofO(3), has been studied by several authors for codimensions up to 2. We look at the case where the topological codimension is 3, theC∞-codimension is 5. We find a tertiary Hopf bifurcation and a heteroclinic orbit. Our analysis does not use any specific properties of the five dimensional representation and can in principle be used for higher representations as well. The computations are based on invariant theory and orbit space reduction.

Complex Order from Disorder and from Simple Order in Coarse-Graining Invariant Orbits of Certain Two-Dimensional Linear Cellular Automata

This paper considers three-dimensional coarse-graining invariant orbits for two-dimensional linear cellular automata over a finite field, as a nontrivial extension of the two-dimensional coarse-graining invariant orbits for one-dimensional CA that were studied in an earlier paper. These orbits can be found by solving a particular kind of recursive equations (renormalizing equations with rescaling term). The solution starts from some seed that has to be determined first. In contrast with the one-dimensional case, the seed has infinite support in most cases. The way for solving these equations is discussed by means of some examples. Three categories of problems (and solutions) can be distinguished (as opposed to only one in the one-dimensional case). Finally, the morphology of a few coarse-graining invariant orbits is discussed: Complex order (of quasiperiodic type) seems to emerge from random seeds as well as from seeds of simple order (for example, constant or periodic seeds).

Linearization of vector fields on the orbit space of the action of a compact Lie group

In this paper we present results which relate the linearization of a G-equivariant vector field at an equilibrium to the linearization of the vector field obtained by projection on the orbit space of the group action. This allows us to propose a notion of hyperbolicity for an equilibrium of the projected vector field which is compatible with the orbit space structure. It provides an alternative to the notion of normal hyperbolicity for relative equilibria (flow invariant group orbits) developed by Field [9, 10].

COARSE-GRAINING INVARIANT PATTERNS OF ONE-DIMENSIONAL TWO-STATE LINEAR CELLULAR AUTOMATA

We consider one-dimensional two state cellular automata and their orbital patterns. We are particulary interested in initial states and their orbital pattern which are invariant under a certain coarse-graining operation. We show that invariant initial states are automatic. Moreover, we numerically study the complex nature of the initial states and their invariant orbits.

ON ORBIT CONNECTEDNESS, ORBIT CONVEXITY, AND ENVELOPES OF HOLOMORPHY

The univalence and description of the envelope of holomorphy is considered for a domain having a compact Lie group action. The main result is the following: Let be a holomorphic Stein -manifold, and a -invariant orbit connected domain. Then is schlicht and orbit convex if and only if is schlicht. Moreover, in this case, .

Extensions and applications of Fuller's theorem on periodic points

If $f X\arrow X$ is a continuous function, $x\in X$ is called a periodic point if some iterate $f^{k}$ of $f$ leaves $x$ fixed. In 1953 F. B. Fuller proved a basic result for the existence of periodic points. He showed that if $X$ is a compact absolute neighborhood retract with non-zero Euler characteristic and if $N(X)$ is the larger of the sums of the odd and even real Betti numbers of $X$, then for any homeomorphism $f$ on $X$, $f^{k}$ has a fixed point for some $k, $1\leq k\leq N(X)$. Actually Fuller’s result holds for any space$X$ for which the Lefschetz Fixed Point Theorem is valid.In this paper the authors first establish a number of extensions of the Fuller Theorem. They next consider G-spaces and extend results of Mann on periodic points and invariant orbits.In the last section they turn to the Riemannian category using the volume function on the orbit space to obtain additional results and examples on periodic points and invariant orbits for groups of isometries. They conclude the paper by using the Fuller Theorem, together with a result of Kobayashi relating the Lefschetz number of an isometry and the Euler characteristic of its fixed point set, to establish a new result on the existence of ${\it common}$ periodic points for ${\it commuting}$ isometries.

Invariant measures and orbit closures for unipotent actions on homogeneous spaces

Invariant measures and orbit closures for unipotent actions on homogeneous spaces

Type-I intermittency for the Hénon-map family

Type-I intermittency was defined as the precursor quasiregular behavior to every tangent bifurcation for the one-dimensional (1D) quadratic-map family. We have extended the definition to the two-dimensional H\'enon-map family with the following results. The gradual increase of bistability and multistability that accompanies the reduction of dissipation from the 1D limit carries with it the eventual decrease and disappearance of the intermittency associated with the occurrence of each stable periodic orbit. Numerically we observe qualitative changes in the invariant orbit density and the Lyapunov exponent in the transition region, the latter showing a discontinuous, or first-order, rather than the usual continuous phase transition. A hysteretic response of the dynamics to slow parameter change, which usually accompanies a first-order transition, is also noted. The changes associated with various periodic orbits, which are tabulated, show both similarities and differences. A considerable understanding of these phenomena is achieved by an in-depth study of the topology of, and dynamics in, the phase plane of the H\'enon system. Heuristic pictures are developed for some surprising bifurcation structures.