### Pictures archive

**Sticky Orbits and Boxy Structures in Galaxies**

A significant role for the morphology of 3D bars play the chaotic orbits that are found to be sticky to the rotational tori (Vrahatis et al. 1997) around the stable periodic orbits belonging to the families x1v1 and x1v1' (Skokos et al. 2002). These families are introduced in the system at the vertical 2:1 resonance and are considered as the backbone of the boxy/peanut (b/p) bulges observed in edge-on disk galaxies. The sticky chaotic orbits support simultaneously the b/p morphology in edge-on views and boxy features in face-on views. The figure at the top (a) summarizes the contribution of all significant orbits in a 3D Ferrers bar model that participate in enhancing b/p bilges at a characteristic energy (Patsis & Katsanikas 2014a). It gives the (z,p_{z}) projection of the 4D phase space of x1 orbits perturbed in the p_{z} direction, at Ej=-0.41. We indicate with black and grey "*" symbols the location of the x1v1 and x1v1' periodic orbits respectively. Red and green "x" symbols indicate the two branches of x1v2 (Skokos et al. 2002). The "▲" at (z,p_{z}=(0.1,0.42) marks the location of the orbits of the x1mul2 family that also contribute to the reinforcement of a b/p morphology of the galactic bulge.

A characteristic examples of a sticky chaotic orbit that support simultaneously boxy edge-on and face-on morphologies are given in (b).

These sticky chaotic orbits exist in a certain range of energies. The face-on views of two more peanut-supporting orbits are given in (c) and (d) (Patsis & Katsanikas 2014b).

References:

Patsis P.A., Katsanikas M., 2014a, MNRAS 445, 3525 (arxiv)

Patsis P.A., Katsanikas M., 2014b, MNRAS 445, 3546 (arxiv)

Skokos Ch., Patsis P.A., Athanassoula E., 2002, MNRAS 333, 847

Vrahatis M., Isliker H., Bountis T.C., 1997, IJBC 7, 2707

For a detailed study of the phase space in 3D rotating Hamiltonian systems see:

Katsanikas M., Patsis P.A., Contopoulos G., 2013, IJBC 23, 1330005

Katsanikas M., Patsis P.A., Contopoulos G., 2011, IJBC 21, 2321

Katsanikas M., Patsis P.A., 2011, IJBC 21, 467

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View previous Pictures of the Month here...

**The Speed of Arnold Diffusion**

The phenomenon of Arnold diffusion is important in a number of applications in solar system dynamics and plasma physics. The above figure, published in Efthymiopoulos and Harsoula, Physica D, 2013 (in press), shows a visualization of Arnold diffusion in a set of variables arising after a so-called optimal normal form construction using a special computer-algebraic program. According to theory, at a crossing domain of multiple resonances, the normal form construction allows to express the equations of motion as a Hamiltonian system of two degrees of freedom. This is perturbed, however, by a third degree of freedom appearing in the so-called `remainder' of the normal form. After a very slow drift in this third dimension (corresponding, in the figure, to the long edge of the framed parallelepiped), the orbits by-pass the barriers (invariant tori) of the 2D dynamics, and thus transit from one resonance to another. The timescale for this phenomenon is millions, or billions of characteristic periods of the orbits (see Efthymiopoulos and Harsoula 2013 for details).

**Trajectories of Particles in 3D Dissipative Pulsar Magnetospheres**

We calculate realistic trajectories of particles in 3D dissipative pulsar magnetospheres and compute their Lorentz factor under the influence of both the accelerating electric field E? and curvature radiation-reaction. We determine the radiation intensity as a function of pulsar phase and thus generate realistic γ-ray light curves. Left-hand panel: the distribution of intensity as observed by an observer situated 75o from the rotation axis of an orthogonal rotator. Right-hand panel: associated γ?ray light-curve. Notice that the emission region is closely associated with the magnetospheric current sheet (after Kalapotharakos, Harding, Kazanas & Contopoulos 2012).

**Structures supported by Chaotic Orbits**

Models based on orbital theory indicate that the observed structures in disk galaxies (bars and spirals) can be supported by particles in chaotic motion. Here is an example of a model based on a potential estimated from near-infrared observations of the barred-spiral galaxy NGC 1300 (Kalapotharakos et al. 2010, 403, 83).

(a) The effective potential isocontours of the model. We have one stable (L1S) and five unstable (L4, L5, L1, L1', L2) Lagrangian points indicated with arrows. (b) The colour-scaled density response of a stellar disk after imposing this gravitational field. We observe the appearance of a typical grand design barred spiral morphology. The spirals, and to a large extent the "ansae" type bar are supported by particles in chaotic motion. A typical orbit supporting this structure during a time corresponding to ten bar rotation can be seen here (Patsis, Kalapotharakos, Grosbol 2010, MNRAS 408, 22; Patsis 2012, AApTr 27, 113).

**Confined tori near a Hamiltonian Hopf bifurcation**

We have studied the orbital behavior in the neighborhood of complex unstable periodic orbits in a rotating 3D autonomous Hamiltonian system of galactic type. At a transition of a family of periodic orbits from stability to complex instability (also known as Hamiltonian Hopf Bifurcation) the four eigenvalues of the stable periodic orbits move out of the unit circle. Then the periodic orbits become complex unstable. In a recent paper (Katsanikas, Patsis, Contopoulos 2011, to appear in the International Journal of Bifurcation and Chaos) we first integrated initial conditions close to the ones of a complex unstable periodic orbit, which is close to the transition point. Then, we have plotted the consequents of the corresponding orbit in a 4D surface of section. To visualize this surface of section we used the method of color and rotation (Patsis and Zachilas 1994). We found that the consequents are contained in nearly 2D "confined tori". Then, we investigated the structure of the phase space in the neighborhood of complex unstable periodic orbits, which are further away from the transition point. In these cases we observed clouds of points in the 4D surfaces of section. The transition between the two types of orbital behavior is rather abrupt.

**March 2011 **

Example processing and analysis performed on NOAA active region (AR) 9077, observed by the Michelson Doppler Imager (MDI) magnetograph onboard the Solar and Heliospheric Observatory (SoHO). This is the AR that gave the famous “Bastille day flare” on 2000 July 14th. (Right) A white‐light image of the solar disk with the AR bounded by a box (notice the complex of dark sunspots comprising it). (Lower left) The AR in white light, observed nearly simultaneously by the Transition Region and Coronal Explorer (TRACE) extreme ultraviolet (EUV) imager, with the inferred velocity flow field (curves). (Middle left) A patch of the SoHO/MDI photospheric magnetogram containing the AR. The gray curves on top indicate the results of the three‐dimensional magnetic-field extrapolation we applied to the magnetogram. (Upper left) To assess the quality of the extrapolation, we have plotted EUV TRACE observations on top of the same magnetogram. Notice the qualitative agreement between extrapolation results and actual observations (from a successful Marie Curie Actions proposal entitled "Solar Magnetic Eruptions: Understanding and Forecasting (SoME-UFo)" by M. K. Georgoulis, 2010).

**January 2011 **

Nonlinear force-free numerical reconstruction of the global coronal magnetic field corresponding to the photospheric radial field of the so-called `Halloween 2003’ period (Carrington rotation 2009). Field-line color according to the twist parameter \alpha=curlB/B (I. Contopoulos, C. Kalapotharakos, M. Georgoulis 2011, Solar Physics).

**November 2010 **

The picture of this month is from the recent succesful presentation of the PhD Thesis of P. Tsoutsis.

In this thesis, the connection between the theory of invariant manifolds emanating from unstable periodic orbits, and the generation of non-axisymmetric structures in strongly nonlinear models of barrred-spiral galaxies is studied. We first apply the theory of invariant manifolds in the case of the family of unstable short period orbits around the unstable points L1, L2 of a rotating N-body system. We find a direct link between the so called `bundles of preferential directions' of the invariant manifolds (plotted in configuration space) and the density maxima of the bodies of the simulation which produce spiral arms beyond corotation.

A chain of stars, with initial conditions along the invariant manifolds, defines a flow which can be shown to maintain the apocenters of the chaotic orbits correlated for long time intervals. This correlation can be described analytically by a soliton-like solution of an equation giving the positions of the apocenters of the chaotic orbits (Voglis 2003). We further elucidate this mechanism by demonstrating that the invariant manifolds are responsible for stickiness effects, which result in a significant slowing down of the rate of chaotic escapes in an otherwise non-compact region of the phase space. Extending the theory of invariant manifolds to other families of unstable periodic orbits we find a new phenomenon called `coalescence' of the invariant manifolds. Namely, the superposition of all the unstable invariant manifolds of many different unstable periodic orbits (for the same value of the Jacobi constant), enhances the density along the bundles of preferential directions.

This phenomenon becomes efficient precisely in the area where the invariant manifolds are found to support the spiral density maxima of the N-body particles. The above picture shows the coalescence of the invariant manifolds of 7 different unstable periodic orbits (namely the -1:1, -1:1 of multiplicity 2, -2:1, PL1, -4:1, 4:1, 3:1) for the same value of the Jacobi constant.

**October 2010**

Regions of various escape times of chaotic orbits, having initial conditions in a small area near an island of stability (gray color in the right of the figure) in the case of the “standard map”. Red color corresponds to fast escapes (after 1-5 iterations), yellow to slower escapes (after 5-10 iterations), green to even slower escapes (10-100 iterations) and blue to the slowest escapes (after more than 100 iterations). The limits between the various regions of different escape rates coincide with the stable asymptotic curves (black curves) of the unstable periodic orbits lying in this region (see for details and explanation the reference: “Stickiness in Chaos”, G. Contopoulos and M. Harsoula, 2008, Int. J. Bif. Chaos, 18, p.2929).