The speed of Arnold diffusion

A detailed numerical study is presented of the slow diffusion (Arnold diffusion) taking place around resonance crossings in nearly integrable Hamiltonian systems of three degrees of freedom in the so-called `Nekhoroshev regime'. The aim is to construct estimates regarding the speed of diffusion based on the numerical values of a truncated form of the so-called remainder of a normalized Hamiltonian function, and to compare them with the outcomes of direct numerical experiments using ensembles of orbits. In this comparison we examine, one by one, the main steps of the so-called analytic and geometric parts of the Nekhoroshev theorem. We are led to two main results: i) We construct in our concrete example a convenient set of variables, proposed first by Benettin and Gallavotti (1986), in which the phenomenon of Arnold diffusion in doubly resonant domains can be clearly visualized. ii) We determine, by numerical fitting of our data the dependence of the local diffusion coefficient "D" on the size "||R_{opt}||" of the optimal remainder function, and we compare this with a heuristic argument based on the assumption of normal diffusion. We find a power law "D\propto ||R_{opt}||^{2(1+b)}", where the constant "b" has a small positive value depending also on the multiplicity of the resonance considered.Comment: 39 pages, 11 figure

Analytical description of the structure of chaos

We consider analytical formulae that describe the chaotic regions around the main periodic orbit $(x=y=0)$ of the H\'{e}non map. Following our previous paper (Efthymiopoulos, Contopoulos, Katsanikas $2014$) we introduce new variables $(\xi, \eta)$ in which the product $\xi\eta=c$ (constant) gives hyperbolic invariant curves. These hyperbolae are mapped by a canonical transformation $\Phi$ to the plane $(x,y)$, giving "Moser invariant curves". We find that the series $\Phi$ are convergent up to a maximum value of $c=c_{max}$. We give estimates of the errors due to the finite truncation of the series and discuss how these errors affect the applicability of analytical computations. For values of the basic parameter $\kappa$ of the H\'{e}non map smaller than a critical value, there is an island of stability, around a stable periodic orbit $S$, containing KAM invariant curves. The Moser curves for $c \leq 0.32$ are completely outside the last KAM curve around $S$, the curves with $0.32<c<0.41$ intersect the last KAM curve and the curves with $0.41\leq c< c_{max} \simeq 0.49$ are completely inside the last KAM curve. All orbits in the chaotic region around the periodic orbit $(x=y=0)$, although they seem random, belong to Moser invariant curves, which, therefore define a "structure of chaos". Orbits starting close and outside the last KAM curve remain close to it for a stickiness time that is estimated analytically using the series $\Phi$. We finally calculate the periodic orbits that accumulate close to the homoclinic points, i.e. the points of intersection of the asymptotic curves from $x=y=0$, exploiting a method based on the self-intersections of the invariant Moser curves. We find that all the computed periodic orbits are generated from the stable orbit $S$ for smaller values of the H\'{e}non parameter $\kappa$, i.e. they are all regular periodic orbits.Comment: 22 pages, 9 figure

Resonant normal form and asymptotic normal form behavior in magnetic bottle Hamiltonians

We consider normal forms in `magnetic bottle' type Hamiltonians of the form $H=\frac{1}{2}(\rho^2_\rho+\omega^2_1\rho^2) +\frac{1}{2}p^2_z+hot$ (second frequency $\omega_2$ equal to zero in the lowest order). Our main results are: i) a novel method to construct the normal form in cases of resonance, and ii) a study of the asymptotic behavior of both the non-resonant and the resonant series. We find that, if we truncate the normal form series at order $r$, the series remainder in both constructions decreases with increasing $r$ down to a minimum, and then it increases with $r$. The computed minimum remainder turns to be exponentially small in $\frac{1}{\Delta E}$, where $\Delta E$ is the mirror oscillation energy, while the optimal order scales as an inverse power of $\Delta E$. We estimate numerically the exponents associated with the optimal order and the remainder's exponential asymptotic behavior. In the resonant case, our novel method allows to compute a `quasi-integral' (i.e. truncated formal integral) valid both for each particular resonance as well as away from all resonances. We applied these results to a specific magnetic bottle Hamiltonian. The non resonant normal form yields theorerical invariant curves on a surface of section which fit well the empirical curves away from resonances. On the other hand the resonant normal form fits very well both the invariant curves inside the islands of a particular resonance as well as the non-resonant invariant curves. Finally, we discuss how normal forms allow to compute a critical threshold for the onset of global chaos in the magnetic bottle.Comment: 20 pages, 7 figure

Periodic orbits in a near Yang-Mills potential

We consider the orbits in the Yang-Mills (YM) potential V=1/2 x2 y2 and in the potentials of the general form Vg=1/2 [{\alpha} (x2 +y2)+x2 y2]. The stable period-9 (number of intersection with the x-axis, with ) orbit found in the YM potential is a bifurcation of a basic period-9 orbit of the Vg potential for a value of {\alpha} slightly above zero. This basic period-9 family and its bifurcations exist only up to a maximum value of {\alpha}={\alpha}max. We calculate the Henon stability index of these orbits. The pattern of the stability diagram is the same for all the symmetric orbits of odd periods 3,5,7,9 and 11. We also found the stability diagrams for asymmetric orbits of period 2,3,4,5 which have again the same pattern. All these orbits are unstable for {\alpha}=0 (YM potential). These new results indicate that in the YM potential the only stable orbits are those of period-9 and some orbits with multiples of 9 periods.Comment: 11 pages, 11figure

Building CX peanut-shaped disk galaxy profiles. The relative importance of the 3D families of periodic orbits bifurcating at the vertical 2:1 resonance

We present and discuss the orbital content of a rather unusual rotating barred galaxy model, in which the three-dimensional (3D) family, bifurcating from x1 at the 2:1 vertical resonance with the known "frown-smile" side-on morphology, is unstable. Our goal is to study the differences that occur in the phase space structure at the vertical 2:1 resonance region in this case, with respect to the known, well studied, standard case, in which the families with the frown-smile profiles are stable and support an X-shaped morphology. The potential used in the study originates in a frozen snapshot of an $N$-body simulation in which a fast bar has evolved. We follow the evolution of the vertical stability of the central family of periodic orbits as a function of the energy (Jacobi constant) and we investigate the phase space content by means of spaces of section. The two bifurcating families at the vertical 2:1 resonance region of the new model change their stability with respect to that of most studied analytic potentials. The structure in the side-on view that is directly supported by the trapping of quasi-periodic orbits around 3D stable periodic orbits has now an infinity symbol (i.e. $\infty$-type) profile. However, the available sticky orbits can reinforce other types of side-on morphologies as well. In the new model, the dynamical mechanism of trapping quasi-periodic orbits around the 3D stable periodic orbits that build the peanut, supports the $\infty$-type profile. The same mechanism in the standard case supports the X shape with the frown-smile orbits. Nevertheless, in both cases (i.e. in the new and in the standard model) a combination of 3D quasi-periodic orbits around the stable x1 family with sticky orbits can support a profile reminiscent of the shape of the orbits of the 3D unstable family existing in each model.Comment: 8 pages, 8 figure

Asymptotic Orbits in Barred Spiral Galaxies

We study the formation of the spiral structure of barred spiral galaxies, using an $N$-body model. The evolution of this $N$-body model in the adiabatic approximation maintains a strong spiral pattern for more than 10 bar rotations. We find that this longevity of the spiral arms is mainly due to the phenomenon of stickiness of chaotic orbits close to the unstable asymptotic manifolds originated from the main unstable periodic orbits, both inside and outside corotation. The stickiness along the manifolds corresponding to different energy levels supports parts of the spiral structure. The loci of the disc velocity minima (where the particles spend most of their time, in the configuration space) reveal the density maxima and therefore the main morphological structures of the system. We study the relation of these loci with those of the apocentres and pericentres at different energy levels. The diffusion of the sticky chaotic orbits outwards is slow and depends on the initial conditions and the corresponding Jacobi constant.Comment: 17 pages, 24 figure