The intersections of orbits by a surface section either form closed invariant curves or are scattered in some zones instability." We follow the evolution potential V = 21 (A x2+By2) - exy2 as perturbation increases while total energy is constant. instability increase with increasing e and them combine; then we see "heteroclinic" doubly asymptotic new periodic belonging to "irregular" families. There also open very complicated form. points intersection most zone seem ergodically distributed....

Abstract We study the approximate (formal) integrals of motion in Hamiltonian <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi>H</mml:mi> <mml:mo>=</mml:mo> <mml:mfrac> <mml:mrow> <mml:mn>1</mml:mn> </mml:mrow> <mml:mn>2</mml:mn> </mml:mfrac> <mml:mfenced close=")" open="("> <mml:msup> <mml:mover accent="true"> <mml:mi>x</mml:mi> <mml:mo>̇</mml:mo> </mml:mover> </mml:msup> <mml:mo>+</mml:mo> <mml:mi>y</mml:mi> </mml:mfenced> <mml:mi>ϵ</mml:mi> <mml:mspace...

In a Hamiltonian system of three degrees freedom we have found large stochastic region (the "big sea"), some other regions, apparently separated from the above ("small seas"), and ordered regions. regions maximal Lyapunov characteristic number vanishes, while it has finite values in However, these are different big sea small seas. Three formal integrals were constructed they truncated at orders 2,3,...,11. The numerical along several orbits calculated. variations all decrease with order...

We present a generic criterion which can be used in gravitational-wave data analysis to distinguish an extreme-mass-ratio inspiral into Kerr background spacetime from one non-Kerr spacetime. exploit the fact that when integrable system, such as system describes geodesic orbits spacetime, is perturbed, tori phase space initially corresponded resonances disintegrate so form Birkhoff chains on surface of section. The KAM curves islands chain share same ratio frequencies, even though frequencies...

Periodic orbits are the main "landmarks" in study of perturbed dynamical systems. We find characteristic curves many families periodic a "galactic-type" potential with increasing perturbation. There two types families: (a) regular which generated (directly or through intermediate families) from unperturbed system, and (b) irregular ones, independent above. The appearance seems to be connected "dissolution" invariant nearby nonperiodic orbits. number crossing x axis given times increases...

By detecting gravitational wave signals from extreme mass ratio inspiraling sources (EMRIs) we will be given the opportunity to check our theoretical expectations regarding nature of supermassive bodies that inhabit central regions galaxies. We have explored some qualitatively new features a perturbed Kerr metric induces in its geodesic orbits. Since generic does not possess all special symmetries metric, equations former case are described by slightly nonintegrable Hamiltonian system....

The aim of this paper is to unify the work done until now from different points view on a third integral motion besides energy and angular momentum integrals, present number applica- tions generalizations. In Introduction, methods for finding integrals are dis- cussed. explicitly calculated by means von Zeipel's method. next section definitions integrable systems given distinction between useful nonuseful made. Poincare 5 nonexistence theorem mentioned its many exceptions pointed out. Then...

We show that in deterministic dynamical systems any orbit is associated with an invariant spectrum of stretching numbers, i.e. numbers expressing the logarithmic divergences neighbouring orbits within one period. The first moment this maximal Lyapunov characteristic number (LCN). In case a chaotic domain, single characterizes whole domain. invariance allows estimation LCN by calculating, for short times, many initial conditions same region instead calculating extremely long times. However,...

We calculate orbits of photons and particles in the relativistic problem two extreme Reissner-Nordström black holes (fixed). In case there are three types non-periodic orbits, namely falling into M 1 2 (types I II), escaping to infinity (III). The various separated by asymptotic main (unstable) periodic orbits: ( a ) around , b c both . Between different third type. initial conditions form Cantor sets, this fact is manifestation chaos. role higher-order explained. situation similar parabolic...

In this paper a general discussion of the resonance cases in an axially-symmetric potential field is pre- sented, when unperturbed frequencies radial and z direction have rational ratio. The form third integral not valid these because appearance divisors (m2P-n2Q), which become zero cases. However, new isolating case available, can be used to construct power series eliminate all secular terms. Three are distinguished, (a) m+n>4, (p) m+n=4, ( ) m+n <4. first orbits rather similar those...

view Abstract Citations (67) References (9) Co-Reads Similar Papers Volume Content Graphics Metrics Export Citation NASA/ADS Inner Lindblad resonance in galaxies. Nonlinear theory. I. Contopoulos, G. In the reported investigation use is made of action-angle variables employed by Kalnajs (1971) study galactic resonant problems. The Hamiltonian corresponding to a spiral galaxy expressed with aid variables. A third-order epicyclic theory used and perturbation relations are developed. Attention...

It is shown that the line element, which forms basis of post-Newtonian theory Einstein, Infeld and Hoffmann for motion mass points under their mutual gravita­tional attractions, invariant in form to a certain post-Galilean transformation. necessary transformation, expressed as an expansion inverse powers c 2 ( velocity light), include terms O -2 ) transformation spatial coordinate C -4 time coordinate. Comparison with Lorentz (between two frames uniform relative V), expanded similarly ,...

This paper contains a complete description of the resonance case A =B, i.e., when unperturbed fre- quencies in two perpendicular directions are equal. The form third integral is different from that non case. secular terms eliminated, step by step, and higher-order calculated means computer. better conserved actual orbits more included it. invariant curves give main characteristics orbits. theoretical in- variant represent sufficiently well empirically found (by orbital cal- culations) up to...

view Abstract Citations (32) References (8) Co-Reads Similar Papers Volume Content Graphics Metrics Export Citation NASA/ADS A Classification of the Integrals Motion. Contopoulos, George distinction is made between isolating, quasi-isolating, and ergodic integrals motion a Hamiltonian system In quasi-isolating case there eventually set isolating integral hypersurfaces. new proof given theorem that, if H series in variables, another coinciding with one up to terms any degree has as many are...

This paper presents a number of numerical investigations orbits in the de Broglie–Bohm version quantum mechanics. We first clarify how notion chaos should be implemented case Bohmian orbits. Then, we investigate three different characteristic systems: (a) superposition stationary states Hamiltonian two uncoupled harmonic oscillators with incommensurable frequencies, (b) wave packets Hénon–Heiles-type and (c) modified two-slit experiment. In these examples, identify regular or chaotic also...

We study the orbits of particles (time-like geodesics) around two fixed black holes when energy is elliptic, i. e. it does not allow motion to extend infinity. Most are chaotic, but in many cases there also ordered motions stable periodic orbits. The that fall into first or second hole separated by unstable These satellite they exist. But for certain intervals parameters no hole. Then limiting like arcs hyperbolae, reaching curve zero velocity.

We distinguish two types of stickiness in systems degrees freedom: (a) around an island stability, and (b) chaos, along the unstable asymptotic curves periodic orbits. In fact, there are orbits near outer boundary that remain close to for some time, then extend large distances into surrounding chaotic sea. But later return contribute overall produces dark regions islands lines extending far from islands. have studied these effects standard map with a rather nonlinearity K = 5, we emphasized...