Showing papers on "Phase space published in 2006"

A quantum Newton's cradle

Abstract: It is a fundamental assumption of statistical mechanics that a closed system with many degrees of freedom ergodically samples all equal energy points in phase space. To understand the limits of this assumption, it is important to find and study systems that are not ergodic, and thus do not reach thermal equilibrium. A few complex systems have been proposed that are expected not to thermalize because their dynamics are integrable. Some nearly integrable systems of many particles have been studied numerically, and shown not to ergodically sample phase space. However, there has been no experimental demonstration of such a system with many degrees of freedom that does not approach thermal equilibrium. Here we report the preparation of out-of-equilibrium arrays of trapped one-dimensional (1D) Bose gases, each containing from 40 to 250 (87)Rb atoms, which do not noticeably equilibrate even after thousands of collisions. Our results are probably explainable by the well-known fact that a homogeneous 1D Bose gas with point-like collisional interactions is integrable. Until now, however, the time evolution of out-of-equilibrium 1D Bose gases has been a theoretically unsettled issue, as practical factors such as harmonic trapping and imperfectly point-like interactions may compromise integrability. The absence of damping in 1D Bose gases may lead to potential applications in force sensing and atom interferometry.

Nonlinear dynamics of networks: the groupoid formalism

Abstract: A formal theory of symmetries of networks of coupled dynamical systems, stated in terms of the group of permutations of the nodes that preserve the network topology, has existed for some time. Global network symmetries impose strong constraints on the corresponding dynamical systems, which affect equilibria, periodic states, heteroclinic cycles, and even chaotic states. In particular, the symmetries of the network can lead to synchrony, phase relations, resonances, and synchronous or cycling chaos. Symmetry is a rather restrictive assumption, and a general theory of networks should be more flexible. A recent generalization of the group-theoretic notion of symmetry replaces global symmetries by bijections between certain subsets of the directed edges of the network, the ‘input sets’. Now the symmetry group becomes a groupoid, which is an algebraic structure that resembles a group, except that the product of two elements may not be defined. The groupoid formalism makes it possible to extend group-theoretic methods to more general networks, and in particular it leads to a complete classification of ‘robust’ patterns of synchrony in terms of the combinatorial structure of the network. Many phenomena that would be nongeneric in an arbitrary dynamical system can become generic when constrained by a particular network topology. A network of dynamical systems is not just a dynamical system with a high-dimensional phase space. It is also equipped with a canonical set of observables—the states of the individual nodes of the network. Moreover, the form of the underlying ODE is constrained by the network topology—which variables occur in which component equations, and how those equations relate to each other. The result is a rich and new range of phenomena, only a few of which are yet properly understood.

Simulating rare events in equilibrium or nonequilibrium stochastic systems.

Abstract: We present three algorithms for calculating rate constants and sampling transition paths for rare events in simulations with stochastic dynamics. The methods do not require a priori knowledge of the phase-space density and are suitable for equilibrium or nonequilibrium systems in stationary state. All the methods use a series of interfaces in phase space, between the initial and final states, to generate transition paths as chains of connected partial paths, in a ratchetlike manner. No assumptions are made about the distribution of paths at the interfaces. The three methods differ in the way that the transition path ensemble is generated. We apply the algorithms to kinetic Monte Carlo simulations of a genetic switch and to Langevin dynamics simulations of intermittently driven polymer translocation through a pore. We find that the three methods are all of comparable efficiency, and that all the methods are much more efficient than brute-force simulation.

Retrieval of the Green’s Function from Cross Correlation: The Canonical Elastic Problem

Abstract: In realistic materials, multiple scattering takes place and average field intensities or energy densities follow diffusive processes. Multiple P to S energy conversions by the random inhomogeneities result in equipartition of elastic waves, which means that in the phase space the available elastic energy is distributed among all the possible states of P and S waves, with equal amounts in average. In such diffusive regimes, the P to S energy ratio equilibrates in a universal way independent of the particular details of the scattering. We study the canonical problem of isotropic plane waves in an elastic medium and show that the Fourier transform of azimuthal average of the cross correlation of motion between two points within an elastic medium is proportional to the imaginary part of the exact Green’s tensor function between these points, provided the energy ratio ES / EP is the one predicted by equipartition in two and three dimensions, respectively. These results clearly show that equipartition is a necessary condition to retrieve the exact Green’s function from correlations of the elastic field. However, even if there is not an equipartitioned regime and correlations do not allow to retrieve precisely the exact Green’s function, the correlations may provide valuable results of physical significance by reconstructing specific arrivals.

Quantitative perturbative study of convergence to equilibrium for collisional kinetic models in the torus

Abstract: For a general class of linear collisional kinetic models in the torus, including in particular the linearized Boltzmann equation for hard spheres, the linearized Landau equation with hard and moderately soft potentials and the semi-classical linearized fermionic and bosonic relaxation models, we prove explicit coercivity estimates on the associated integro-differential operator for some modified Sobolev norms. We deduce the existence of classical solutions near equilibrium for the full nonlinear models associated with explicit regularity bounds, and we obtain explicit estimates on the rate of exponential convergence towards equilibrium in this perturbative setting. The proof is based on a linear energy method which combines the coercivity property of the collision operator in the velocity space with transport effects, in order to deduce coercivity estimates in the whole phase space.

Radial diffusion and MHD particle simulations of relativistic electron transport by ULF waves in the september 1998 storm

Abstract: [1] In an MHD particle simulation of the September 1998 magnetic storm the evolution of the radiation belt electron radial flux profile appears to be diffusive, and diffusion caused by ULF waves has been invoked as the probable mechanism. In order to separate adiabatic and nonadiabatic effects and to investigate the radial diffusion mechanism during this storm, in this work we solve a radial diffusion equation with ULF wave diffusion coefficients and a time-dependent outer boundary condition, and the results are compared with the phase space density of the MHD particle simulation. The diffusion coefficients include contributions from both symmetric resonance modes (ω ≈ mωd, where ω is the wave frequency, m is the azimuthal wave number, and ωd is the bounce-averaged drift frequency) and asymmetric resonance modes (ω ≈ (m ± 1)ωd). ULF wave power spectral densities are obtained from a Fourier analysis of the electric and magnetic fields of the MHD simulation and are used in calculating the radial diffusion coefficients. The asymmetric diffusion coefficients are proportional to the magnetic field asymmetry, which is also calculated from the MHD field. The resulting diffusion coefficients vary with the radial coordinate L (the Roederer L-value) and with time during different phases of the storm. The last closed drift shell defines the location of the outer boundary. Both the location of the outer boundary and the value of the phase space density at the outer boundary are time-varying. The diffusion calculation simulates a 42-hour period during the 24–26 September 1998 magnetic storm, starting just before the storm sudden commencement and ending in the late recovery phase. The differential flux calculated in the MHD particle simulation is converted to phase space density. Phase space densities in both simulations (diffusion and MHD particle) are functions of Roederer L-value for fixed first and second adiabatic invariants. The Roederer L-value is calculated using drift shell tracing in the MHD magnetic field, and particles have zero second invariant. The radial diffusion calculation reproduces the main features of the MHD particle simulation quite well. The symmetric resonance modes dominate the radial diffusion, especially in the inner and middle L region, while the asymmetric resonances are more important in the outer region. Using both symmetric and asymmetric terms gives a better result than using only one or the other and is better than using a simple power law diffusion coefficient. We find that it is important to specify the value of the phase space density on the outer boundary dynamically in order to get better agreement between the radial diffusion simulation and the MHD particle simulation.

Port-Hamiltonian systems: an introductory survey

Abstract: The theory of port-Hamiltonian systems provides a framework for the geometric description of network models of physical systems. It turns out that port-based network models of physical systems immediately lend themselves to a Hamiltonian description. While the usual geometric approach to Hamiltonian systems is based on the canonical symplectic structure of the phase space or on a Poisson structure that is obtained by (symmetry) reduction of the phase space, in the case of a port-Hamiltonian system the geometric structure derives from the interconnection of its sub-systems. This motivates to consider Dirac structures instead of Poisson structures, since this notion enables one to define Hamiltonian systems with algebraic constraints. As a result, any power-conserving interconnection of port-Hamiltonian systems again defines a port-Hamiltonian system. The port-Hamiltonian description offers a systematic framework for analysis, control and simulation of complex physical systems, for lumped-parameter as well as for distributed-parameter models.

Reduced phase space quantization and Dirac observables

Abstract: In her recent work, Dittrich generalized Rovelli's idea of partial observables to construct Dirac observables for constrained systems to the general case of an arbitrary first class constraint algebra with structure functions rather than structure constants. Here we use this framework and propose how to implement explicitly a reduced phase space quantization of a given system, at least in principle, without the need to compute the gauge equivalence classes. The degree of practicality of this programme depends on the choice of the partial observables involved. The (multi-fingered) time evolution was shown to correspond to an automorphism on the set of Dirac observables, so generated and interesting representations of the latter will be those for which a suitable preferred subgroup is realized unitarily. We sketch how such a programme might look for general relativity. We also observe that the ideas by Dittrich can be used in order to generate constraints equivalent to those of the Hamiltonian constraints for general relativity such that they are spatially diffeomorphism invariant. This has the important consequence that one can now quantize the new Hamiltonian constraints on the partially reduced Hilbert space of spatially diffeomorphism invariant states, just as for the recently proposed master constraint programme.

Phase space density analysis of the outer radiation belt energetic electron dynamics

Abstract: [1] We present an analysis of the electron phase space density in the Earth's outer radiation belt during three magnetically disturbed periods to determine the likely roles of inward radial diffusion and local acceleration in the energization of electrons to relativistic energies. During the recovery phase of the 9 October 1990 storm and the period of prolonged substorms between 11 and 16 September 1990, the relativistic electron phase space density increases substantially and peaks in the phase space density occur in the region 4.0 ∼0.8 MeV. The peaks in the phase space density are associated with prolonged substorm activity, enhanced chorus amplitudes, and predominantly low values of the ratio between the electron plasma frequency, fpe, and the electron gyrofrequency, fce (fpe/fce < ∼4). The data provide further evidence for a local wave acceleration process in addition to radial diffusion operating in the heart of the outer radiation belt. During the recovery phase of the 9 October 1990 storm the peaks are more pronounced at large M (550 MeV/G) and large Kaufmann K (0.11 RE) than large M (700 MeV/G) and small K (0.025 RE), which suggests that radial diffusion is more effective below about 0.7 MeV for 5.0 < L* < 5.5 during this period. At low M (M ≤ 250 MeV/G), corresponding to energies, E < ∼0.8 MeV, there is no evidence for a peak in phase space density and the data are more consistent with inward radial diffusion and losses to the atmosphere by pitch angle scattering. During the 26 August 1990 storm there is a net loss in the relativistic electron phase space density for 3.3 < L* < 6.0. At low M (M ≤ 250 MeV/G) the phase space density decreases by almost a constant factor and the gradient remains positive for all L*, but at high M (M ≥ 550 MeV/G) the decrease in phase space density is greater at larger L* and the gradient changes from positive to negative. The data show that it is possible to have inward radial diffusion at low energies and outward radial diffusion at higher energies, which would fill the outer radiation belt.

Sub-Planck phase-space structures and Heisenberg-limited measurements

Abstract: We show how sub-Planck phase-space structures in the Wigner function [W. H. Zurek, Nature (London) 412, 712 (2001)] can be used to achieve Heisenberg-limited sensitivity in weak-force measurements. Nonclassical states of harmonic oscillators, consisting of superpositions of coherent states, are shown to be useful for the measurement of weak forces that cause translations or rotations in phase space, which is done by entangling the quantum oscillator with a two-level system. Implementations of this strategy in cavity QED and ion traps are described.

Wigner-Weyl correspondence in quantum mechanics for continuous and discrete systems-a Dirac-inspired view

Abstract: Drawing inspiration from Dirac's work on functions of non-commuting observables, we develop an approach to phase-space descriptions of operators and the Wigner-Weyl correspondence in quantum mechanics, complementary to standard formulations. This involves a two-step process: introducing phase-space descriptions based on placing position dependences to the left of momentum dependences (or the other way around); then carrying out a natural transformation to eliminate a kernel which appears in the expression for the trace of the product of two operators. The method works uniformly for both continuous Cartesian degrees of freedom and for systems with finite-dimensional state spaces. It is interesting that the kernel encountered is naturally expressible in terms of geometric phases, and its removal involves extracting its square root in a suitable manner.

The Aharonov-Bohm effect in noncommutative quantum mechanics

Abstract: The Aharonov–Bohm effect in noncommutative (NC) quantum mechanics is studied. First, by introducing a shift for the magnetic vector potential we give the Schrodinger equations in the presence of a magnetic field on NC space and NC phase space, respectively. Then, by solving the Schrodinger equations, we obtain the Aharonov–Bohm phase on NC space and NC phase space, respectively.

Stickiness in Hamiltonian systems: from sharply divided to hierarchical phase space.

Abstract: We investigate the dynamics of chaotic trajectories in simple yet physically important Hamiltonian systems with nonhierarchical borders between regular and chaotic regions with positive measures. We show that the stickiness to the border of the regular regions in systems with such a sharply divided phase space occurs through one-parameter families of marginally unstable periodic orbits and is characterized by an exponent $\ensuremath{\gamma}=2$ for the asymptotic power-law decay of the distribution of recurrence times. Generic perturbations lead to systems with hierarchical phase space, where the stickiness is apparently enhanced due to the presence of infinitely many regular islands and Cantori. In this case, we show that the distribution of recurrence times can be composed of a sum of exponentials or a sum of power laws, depending on the relative contribution of the primary and secondary structures of the hierarchy. Numerical verification of our main results are provided for area-preserving maps, mushroom billiards, and the newly defined magnetic mushroom billiards.

Periodic modulation effect on self-trapping of two weakly coupled Bose-Einstein condensates

Abstract: With phase space analysis approach, we investigate the self-trapping phenomenon for two weakly coupled Bose-Einstein condensates (BECs) in a symmetric double well potential. We identify two kinds of self-trapping by their different relative phase behavior. By applying a periodic modulation on the energy bias of the system we find that the self-trapping can be controlled, in other words, the transition parameters can be adjusted effectively by the periodic modulation. Analytic expressions for the dependence of the transition parameters on the modulation parameters are derived for high- and low-frequency modulations. For an intermediate-frequency modulation, we find the resonance between the periodic modulation and nonlinear Rabi oscillation dramatically affect the tunneling dynamics and demonstrate many phenomena. Finally, we study the effects of many-body quantum fluctuation on the self-trapping and discuss the possible experimental realization.

q -breathers in Fermi-Pasta-Ulam chains: Existence, localization, and stability

Abstract: The Fermi-Pasta-Ulam (FPU) problem consists of the nonequipartition of energy among normal modes of a weakly anharmonic atomic chain model. In the harmonic limit each normal mode corresponds to a periodic orbit in phase space and is characterized by its wave number q. We continue normal modes from the harmonic limit into the FPU parameter regime and obtain persistence of these periodic orbits, termed here q-Breathers (QB). They are characterized by time periodicity, exponential localization in the q-space of normal modes and linear stability up to a size-dependent threshold amplitude. Trajectories computed in the original FPU setting are perturbations around these exact QB solutions. The QB concept is applicable to other nonlinear lattices as well.

Semiclassical structure of chaotic resonance eigenfunctions.

Abstract: We study the resonance (or Gamow) eigenstates of open chaotic systems in the semiclassical limit, distinguishing between left and right eigenstates of the nonunitary quantum propagator and also between short-lived and long-lived states. The long-lived left (right) eigenstates are shown to concentrate as @ ! 0 on the forward (backward) trapped set of the classical dynamics. The limit of a sequence of eigenstates f � @�g @!0 is found to exhibit a remarkably rich structure in phase space that depends on the corresponding limiting decay rate. These results are illustrated for the open baker’s map, for which the probability density in position space is observed to have self-similarity properties. In closed systems, the two most fundamental semiclassical properties are that the mean density of states is given by the Weyl law [1], which associates with each quantum state a Planck cell in the available region of phase space, and that in classically chaotic systems the stationary states have Wigner functions which are semiclassically uniform over the energy shell [2], in agreement with the quantum ergodicity theorem [3]. It is remarkable that it is still not known, in general, how these fundamental properties extend to open (scattering) systems. In open systems, the lack of unitarity of the quantum propagator gives rise to nonorthogonal decaying eigenstates with complex energies (resonances), the imaginary parts of which are interpreted as decay rates. In the case of open chaotic systems, the classical mechanics is structured in phase space around fractal sets associated with trajectories that remain trapped for infinite times, either in the future (forward-trapped set K� ) or in the past (backwardtrapped set K� ). The mean density of resonances is believed (but not, in general, proved) to be determined by the fractal dimension of the invariant set K0 � K� \K� , the classical repeller. This is the fractal Weyl law [4 –6]. (Note that this is different to the resonance statistics in weakly open systems, for which the size of the opening vanishes in the semiclassical limit [7].) Much less is known about the resonance (or Gamow) eigenstates. These are important in many areas of physics [8] and chemistry [9], because they have a marked influence on observable quantities such as scattering cross sections and reaction rates (they are a component of the Siegert pseudostates basis in terms of which the scattering wave functions and S matrix, for example, can conveniently be expanded [8]). Following the well established idea that in the semiclassical limit time-independent quantum properties should be related to time-independent classical sets, it is natural to expect that long-lived eigenstates of open systems should be determined by the structure of ~

Weyl-Wigner Formulation of Noncommutative Quantum Mechanics

Abstract: We address the phase space formulation of a noncommutative extension of quantum mechanics in arbitrary dimension, displaying both spatial and momentum noncommutativity. By resorting to a covariant generalization of the Weyl-Wigner transform and to the Seiberg-Witten map we construct an isomorphism between the operator and the phase space representations of the extended Heisenberg algebra. This map provides a systematic approach to derive the entire structure of noncommutative quantum mechanics in phase space. We construct the extended starproduct, Moyal bracket and propose a general definition of noncommutative states. We study the dynamical and eigenvalue equations of the theory and prove that the entire formalism is independent of the particular choice of Seiberg-Witten map. Our approach unifies and generalizes all the previous proposals for the phase space formulation of noncommutative quantum mechanics. For concreteness we rederive these proposals by restricting our formalism to some 2-dimensional spaces.

State space for two qutrits has a phase space structure in its core

Abstract: We investigate the state space of bipartite qutrits. For states which are locally maximally mixed we obtain an analog of the ``magic'' tetrahedron for bipartite qubits---a magic simplex $\mathcal{W}$. This is obtained via the Weyl group which is a kind of ``quantization'' of classical phase space. We analyze how this simplex $\mathcal{W}$ is embedded in the whole state space of two qutrits and discuss symmetries and equivalences inside the simplex $\mathcal{W}$. Because we are explicitly able to construct optimal entanglement witnesses we obtain the border between separable and entangled states. With our method we find also the total area of bound entangled states of the parameter subspace under intervestigation. Our considerations can also be applied to higher dimensions.

Two methods for the computation of nonlinear modes of vibrating systems at large amplitudes

Abstract: The aim of this paper is to present two methods for the calculation of the nonlinear normal modes of vibration for undamped nonlinear mechanical systems: the time integration periodic orbit method and the modal representation method. In the periodic orbit method, the nonlinear normal mode is obtained by making the continuation of branches of periodic orbits of the equation of motion. The terms “periodic orbits” means a closed trajectory in the phase space, which is obtained by time integration. In the modal representation method, the nonlinear normal mode is constructed in terms of amplitude, phase, mode shape, and frequency, with the distinctive feature that the last two quantities are amplitude and total phase dependent. The methods are compared on two DOF strongly nonlinear systems.

Definability of no-return transition states in the high-energy regime above the reaction threshold.

Abstract: No-return transition states (TSs) defined in multidimensional phase space, where recrossing trajectories through the commonly used "configuration" TS pass only once, robustly exist up to a moderately high-energy regime above the reaction threshold, even when nonlinear resonances among the bath degrees of freedom perpendicular to the reaction coordinate result in local chaos. However, at much higher energy when global chaos appears in the bath space, the separability of the reaction coordinate from the bath degrees of freedom starts to lose locally. In the phase space near the saddles, it is found that the slower the system passes the TS, the more recrossing trajectories reappear. Their implications and mechanisms are discussed concerning to what extent one can define no-return TSs in the high-energy regime above the reaction threshold.

Hamiltonian Perspective on Generalized Complex Structure

Abstract: In this note we clarify the relation between extended world-sheet super-symmetry and generalized complex structure. The analysis is based on the phase space description of a wide class of sigma models. We point out the natural isomorphism between the group of orthogonal automorphisms of the Courant bracket and the group of local canonical transformations of the cotangent bundle of the loop space. Indeed this fact explains the natural relation between the world-sheet and the geometry of T⊕T*. We discuss D-branes in this perspective.

Pair Tunneling through Single Molecules.

Abstract: By a polaronic energy shift, the effective charging energy of molecules can become negative, favoring ground states with even numbers of electrons. Here we show that charge transport through such molecules near ground-state degeneracies is dominated by tunneling of electron pairs which coexists with (featureless) single-electron cotunneling. Because of the restricted phase space for pair tunneling, the current-voltage characteristics exhibit striking differences from the conventional Coulomb blockade. In asymmetric junctions, pair tunneling can be used for gate-controlled current rectification and switching.

Gaussian phase-space representations for fermions

Abstract: We introduce a positive phase-space representation for fermions, using the most general possible multimode Gaussian operator basis. The representation generalizes previous bosonic quantum phase-space methods to Fermi systems. We derive equivalences between quantum and stochastic moments, as well as operator correspondences that map quantum operator evolution onto stochastic processes in phase space. The representation thus enables first-principles quantum dynamical or equilibrium calculations in many-body Fermi systems. Potential applications are to strongly interacting and correlated Fermi gases, including coherent behavior in open systems and nanostructures described by master equations. Examples of an ideal gas and the Hubbard model are given, as well as a generic open system, in order to illustrate these ideas.

Exploring the Phase Space of the Quantum δ -Kicked Accelerator

Abstract: We experimentally explore the underlying pseudoclassical phase space structure of the quantum $\ensuremath{\delta}$-kicked accelerator. This was achieved by exposing a Bose-Einstein condensate to the spatially corrugated potential created by pulses of an off-resonant standing light wave. For the first time quantum accelerator modes were realized in such a system. By utilizing the narrow momentum distribution of the condensate we were able to observe the discrete momentum state structure of a quantum accelerator mode and also to directly measure the size of the structures in the phase space.

The topological AC effect on noncommutative phase space

Abstract: The Aharonov-Casher (AC) effect in non-commutative(NC) quantum mechanics is studied. Instead of using the star product method, we use a generalization of Bopp's shift method. After solving the Dirac equations both on noncommutative space and noncommutative phase space by the new method, we obtain the corrections to AC phase on NC space and NC phase space respectively.

Spherical 2+p spin-glass model : An analytically solvable model with a glass-to-glass transition

Abstract: We present the detailed analysis of the spherical $s+p$ spin-glass model with two competing interactions: among $p$ spins and among $s$ spins. The most interesting case is the $2+p$ model with $p\ensuremath{\geqslant}4$ for which a very rich phase diagram occurs, including, next to the paramagnetic and the glassy phase represented by the one step replica symmetry breaking ansatz typical of the spherical $p$-spin model, another two amorphous phases. Transitions between two contiguous phases can also be of a different kind. The model can thus serve as a mean-field representation of amorphous-amorphous transitions (or transitions between undercooled liquids of different structure). The model is analytically solvable everywhere in the phase space, even in the limit where the infinite replica symmetry breaking ansatz is required to yield a thermodynamically stable phase.

Complementarity in Classical Dynamical Systems

Abstract: The concept of complementarity, originally defined for non-commuting observables of quantum systems with states of non-vanishing dispersion, is extended to classical dynamical systems with a partitioned phase space. Interpreting partitions in terms of ensembles of epistemic states (symbols) with corresponding classical observables, it is shown that such observables are complementary to each other with respect to particular partitions unless those partitions are generating. This explains why symbolic descriptions based on an ad hoc partition of an underlying phase space description should generally be expected to be incompatible. Related approaches with different background and different objectives are discussed.

Symmetric and asymmetric librations in extrasolar planetary systems: a global view

Abstract: We present a global view of the resonant structure of the phase space of a planetary system with two planets, moving in the same plane, as obtained from the set of the families of periodic orbits. An important tool to understand the topology of the phase space is to determine the position and the stability character of the families of periodic orbits. The region of the phase space close to a stable periodic orbit corresponds to stable, quasi periodic librations. In these regions it is possible for an extrasolar planetary system to exist, or to be trapped following a migration process due to dissipative forces. The mean motion resonances are associated with periodic orbits in a rotating frame, which means that the relative configuration is repeated in space. We start the study with the family of symmetric periodic orbits with nearly circular orbits of the two planets. Along this family the ratio of the periods of the two planets varies, and passes through rational values, which correspond to resonances. At these resonant points we have bifurcations of families of resonant elliptic periodic orbits. There are three topologically different resonances: (1) the resonances (n + 1):n, (2:1, 3:2, ...), (2) the resonances (2n + 1):(2n − 1t), (3:1, 5:3, ...) and (3) all other resonan topology at each one of the above three types of resonances is studied, for different values of the sum and of the ratio of the planetary masses. Both symmetric and asymmetric resonant elliptic periodic orbits exist. In general, the symmetric elliptic families bifurcate from the circular family, and the asymmetric elliptic families bifurcate from the symmetric elliptic families. The results are compared with the position of some observed extrasolar planetary systems. In some cases (e.g., Gliese 876) the observed system lies, with a very good accuracy, on the stable part of a family of resonant periodic orbits.