Christos Efthymiopoulos

Invariant manifolds, phase correlations of chaotic orbits and the spiral structure of galaxies

In the presence of a strong m = 2 component in a rotating galaxy, the phase-space structure near corotation is shaped to a large extent by the invariant manifolds of the short-period family of unstable periodic orbits terminating at L 1 or L 2 . The main effect of these manifolds is to create robust phase correlations among a number of chaotic orbits large enough to support a spiral density wave outside corotation. The phenomenon is described theoretically by soliton-like solutions of a Sine-Gordon equation. Numerical examples are given in an N-body simulation of a barred spiral galaxy. In these examples, we demonstrate how the projection of unstable manifolds in configuration space reproduces essentially the entire observed bar-spiral pattern.

The coalescence of invariant manifolds and the spiral structure of barred galaxies

In a previous paper (Voglis et al. 2006a, paper I) we demonstrated that, in a rotating galaxy with a strong bar, the unstable asymptotic manifolds of the short period family of unstable periodic orbits around the Lagrangian points L1 or L2 create correlations among the apocentric positions of many chaotic orbits, thus supporting a spiral structure beyond the bar. In the present paper we present evidence that the unstable manifolds of all the families of unstable periodic orbits near and beyond corotation contribute to the same phenomenon. Our results refer to a N-Body simulation, a number of drawbacks of which, as well as the reasons why these do not significantly affect the main results, are discussed. We explain the dynamical importance of the invariant manifolds as due to the fact that they produce a phenomenon of ‘stickiness’ slowing down the rate of chaotic escape in an otherwise non-compact region of the phase space. We find a stickiness time of order 100 dynamical periods, which is sufficient to support a long-living spiral structure. Manifolds of different families become important at different ranges of values of the Jacobi constant. The projections of the manifolds of all the different families in the configuration space produce a pattern due to the ‘coalescence’ of the invariant manifolds. This follows closely the maxima of the observed m = 2 component near and beyond corotation. Thus, the manifolds support both the outer edge of the bar and the spiral arms.

Nonconvergence of formal integrals: II. Improved estimates for the optimal order of truncation

We investigate the asymptotic properties of formal integral series in the neighbourhood of an elliptic equilibrium in nonlinear 2 DOF Hamiltonian systems. In particular, we study the dependence of the optimal order of truncation Nopt on the distance ρ from the elliptic equilibrium, by numerical and analytical means. The function Nopt(ρ) determines the region of Nekhoroshev stability of the orbits and the time of practical stability. We find that the function Nopt(ρ) decreases by abrupt steps. The decrease is roughly approximated with an average power law Nopt = O(ρ −a ), with a � 1. We find an analytical explanation of this behaviour by investigating the accumulation of small divisors in both the normal form algorithm via Lie series and in the direct construction of first integrals. Precisely, we find that the series exhibit an apparent radius of convergence that tends to zero by abrupt steps as the order of the series tends to infinity. Our results agree with those obtained by Servizi G et al (1983 Phys. Lett. A 95 11) for a conservative map of the plane. Moreover, our analytical considerations allow us to explain the results of our previous paper (Contopoulos G et al 2003 J. Phys. A: Math. Gen. 36 8639), including in particular the different behaviour observed for low-order and higher order resonances.

Invariant manifolds and the response of spiral arms in barred galaxies

The unstable invariant manifolds of the short-period family of periodic orbits around the unstable Lagrangian points L1 and L2 of a barred galaxy define loci in the configuration space which tak e the form of a trailing spiral pattern. In previous works we have explored the association of such a pattern to the observed spiral patt ern in N-body models of barred-spiral galaxies and found it to be quite relevant. Our aims in the present paper are: a) to investigat e this association in the case of the self-consistent models of Kaufmann & Contopoulos (1996) which provide an approximation of real barred-spiral galaxies. b) to examine the dynamical role played by each of the non-axisymmetric components of the potential, i.e. the bar and the spiral perturbation, and their consequences on the form of the invariant manifolds, and c) to examine the relation of ‘r esponse’ models of barred-spiral galaxies with the theory o f the invariant manifolds. Our method relies on calculating the invariant manifolds for values of the Jacobi constant close to its value for L1 and L2. Our main results are the following: a) The invariant manifolds yield the correct form of the imposed spiral pattern provided that their calculation is done with the spiral potential term tur ned on. We provide a theoretical model explaining the form of the invariant manifolds that supports the spiral structure. The azimuthal displacement of the Lagrangian points with respect to the bar’s major axis is a crucial parameter in this modeling. When this is taken into account, the manifolds necessarily develop in a spiral-l ike domain of the configuration space, delimited from below by the bound ary of a banana-like non-permitted domain, and from above either by rotational KAM tori or by cantori forming a stickiness zone. On the contrary, if the whole non-axisymmetric perturbation is artificially ‘aligned’ with the bar (i.e. there is no azimuthal shift of th e Lagrangian manifolds), the manifolds support a ring rather than a spiral structure. b) We construct ‘spiral response’ models on the b asis of the theory of the invariant manifolds and examine the connection of the latter to the ‘response’ models (Patsis 2006) used to fi t real barred-spiral galaxies, explaining how are the manif olds related to a number of morphological features seen in such models.

Nodal points and the transition from ordered to chaotic Bohmian trajectories

We explore the transition from order to chaos for the Bohmian trajectories of a simple quantum system corresponding to the superposition of three stationary states in a 2D harmonic well with incommensurable frequencies. We study in particular the role of nodal points in the transition to chaos. Our main findings are (a) a proof of the existence of bounded domains in configuration space which are devoid of nodal points, (b) an analytical construction of formal series representing regular orbits in the central domain as well as a numerical investigation of its limits of applicability, (c) a detailed exploration of the phase-space structure near the nodal point. In this exploration we use an adiabatic approximation and we draw the flow chart in a moving frame of reference centered at the nodal point. We demonstrate the existence of a saddle point (called X-point) in the vicinity of the nodal point which plays a key role in the manifestation of exponential sensitivity of the orbits. One of the invariant manifolds of the X-point continues as a spiral terminating at the nodal point. We find cases of Hopf bifurcation at the nodal point and explore the associated phase space structure of the nodal point—X-point complex. We finally demonstrate the mechanism by which this complex generates chaos. Numerical examples of this mechanism are given for particular chaotic orbits, and a comparison is made with previous related works in the literature.

Trojan resonant dynamics, stability, and chaotic diffusion, for parameters relevant to exoplanetary systems

The possibility that giant extrasolar planets could have small Trojan co-orbital companions has been examined in the literature from both viewpoints of the origin and dynamical stability of such a configuration. Here we aim to investigate the dynamics of hypothetical small Trojan exoplanets in domains of secondary resonances embedded within the tadpole domain of motion. To this end, we consider the limit of a massless Trojan companion of a giant planet. Without other planets, this is a case of the elliptic restricted three body problem (ERTBP). The presence of additional planets (hereafter referred to as the restricted multi-planet problem, RMPP) induces new direct and indirect secular effects on the dynamics of the Trojan body. The paper contains a theoretical and a numerical part. In the theoretical part, we develop a Hamiltonian formalism in action-angle variables, which allows us to treat in a unified way resonant dynamics and secular effects on the Trojan body in both the ERTBP or the RMPP. In both cases, our formalism leads to a decomposition of the Hamiltonian in two parts,

Special Features of Galactic Dynamics

H=H_b+H_{sec}

COMBINING PARTICLE ACCELERATION AND CORONAL HEATING VIA DATA-CONSTRAINED CALCULATIONS OF NANOFLARES IN CORONAL LOOPS

H=Hb+Hsec.