arXiv subjects

G. Contopoulos

Publications and source records attributed to G. Contopoulos.

At least 19 recordsLinked to original sources

arXiv2025

Integrals and chaos in generalized H\'{e}non-Heiles Hamiltonians

We study the approximate (formal) integrals of motion in the Hamiltonian $ H = \frac{1}{2}\left( \dot{x}^2 + \dot{y}^2 + x^2 + y^2 \right) + \epsilon\,\left( xy^2 + \alpha x^3\right)$ which is an extension of the usual H\'{e}non-Heiles Hamiltonian that has $\alpha = -1/3$. We compare the theoretical surfaces of section (at $y=0$) with the exact surfaces of section calculated by integrating numerically many orbits. For small $\epsilon$, the invariant curves of the theoretical and the exact surfaces of section are close to each other, but for large $\epsilon$ there are differences. The most important is the appearance of chaos in the exact case, which becomes dominant as $\epsilon$ approaches the escape perturbation for $\alpha<0$. We study in particular the cases $\alpha = 1/3$, which represents an integrable system, and $\alpha = 0$. Finally we examine the generation of chaos through the resonance overlap mechanism in the case $\alpha=-1/3$ (the original H\'{e}non-Heiles system) by showing both the homoclinic and the heteroclinic intersection of the asymptotic curves of the unstable periodic orbits.

nlin.CD↗

arXiv2022

Perturbed precessing ellipses as the building blocks of spiral arms in a barred galaxy with two pattern speeds

Observations and simulations of barred spiral galaxies have shown that, in general, the spiral arms rotate at a different pattern speed to that of the bar. The main conclusion from the bibliography is that the bar rotates faster than the spiral arms with a double or even a triple value of angular velocity. The theory that prevails in explaining the formation of the spiral arms in the case of a barred spiral galaxy with two pattern speeds is the manifold theory, where the orbits that support the spiral density wave are chaotic, and are related to the manifolds emanating from the Lagrangian points L_1 and L_2 at the end of the bar. In the present study, we consider an alternative scenario in the case where the bar rotates fast enough in comparison with the spiral arms and the bar potential can be considered as a perturbation of the spiral potential. In this case, the stable elliptical orbits that support the spiral density wave (in the case of grand design galaxies) are transformed into quasiperiodic orbits (or 2D tori) with a certain thickness. The superposition of these perturbed preccesing ellipses for all the energy levels of the Hamiltonian creates a slightly perturbed symmetrical spiral density wave.

astro-ph.GA↗

arXiv2021

Precessing ellipses as the building blocks of spiral arms

Stable periodic orbits in spiral galactic models that form families of precessing ellipses can create spiral density waves similar to those that are observed in real grand-design galaxies. We study the range in parameter space for which the amplitude of the spiral perturbation, the pattern speed, and the pitch angle collaborate so as to lead to the creation of density waves that are supported by precessing ellipses and their surrounding matter in ordered motion. Quantitative estimates lead to a correlation between the pitch angle and the amplitude of the spiral perturbation and also between the pitch angle and the pattern speed of the spiral arms. These correlations can be regarded as an orbital analog of a nonlinear dispersion relation in density wave theory.

nlin.CD↗

arXiv2019

Manifold spirals in barred galaxies with multiple pattern speeds

In the manifold theory of spiral structure in barred galaxies, the usual assumption is that the spirals rotate with the same pattern speed as the bar. Here we generalize the manifold theory under the assumption that the spirals rotate with different pattern speed than the bar. More generally, we consider the case when one or more modes, represented by the potentials V_2, V_3, \ldots, co-exist in the galactic disc in addition to the bar's mode V_{bar}, but rotate with pattern speeds \Omega_2, \Omega_3, \ldots incommensurable between themselves and with \Omega_{bar}. Through a perturbative treatment (assuming that V_2,V_3... are small with respect to V_{bar}) we then show that the unstable Lagrangian points L_1, L_2 of the pure bar model (V_{bar},\Omega_{bar}) are `continued' in the full model as periodic orbits, when we have one extra pattern speed different from \Omega_{bar}, or as epicyclic `Lissajous-like' unstable orbits, when we have more than one extra pattern speeds. As an example we compute the generalized orbits GL_1, GL_2 and their manifolds in a Milky-way like model with bar and spiral pattern speeds assumed different. We find that the manifolds produce a time-varying morphology consisting of segments of spirals or `pseudorings'. These structures are repeated after a period equal to half the relative period of the imposed spirals with respect to the bar. Along one period, the manifold-induced time-varying structures are found to continuously support at least some part of the imposed spirals, except at short intervals around those times at which the relative phase of the imposed spirals with respect to the bar becomes equal to \pm\pi/2. A connection of these effects to the phenomenon of recurrent spirals is discussed.

nlin.CD↗

arXiv2016

Analytical study of chaos and applications

We summarize various cases where chaotic orbits can be described analytically. First we consider the case of a magnetic bottle where we have non-resonant and resonant ordered and chaotic orbits. In the sequence we consider the hyperbolic Henon map, where chaos appears mainly around the origin, which is an unstable periodic orbit. In this case the chaotic orbits around the origin are represented by analytic series (Moser series). We find the domain of convergence of these Moser series and of similar series around other unstable periodic orbits. The asymptotic manifolds from the various unstable periodic orbits intersect at homoclinic and heteroclinic orbits that are given analytically. Then we consider some Hamiltonian systems and we find their homoclinic orbits by using a new method of analytic prolongation. An application of astronomical interest is the domain of convergence of the analytical series that determine the spiral structure of barred-spiral galaxies.

nlin.CD↗

arXiv2015

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.

nlin.CD↗

arXiv2015

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.

math-ph↗

arXiv2014

Analytical study of the structure of chaos near unstable points

In a 2D conservative Hamiltonian system there is a formal integral $\Phi$ besides the energy H. This is not convergent near a stable periodic orbit, but it is convergent near an unstable periodic orbit. We explain this difference and we find the convergence radius along the asymptotic curves. In simple mappings this radius is infinite. This allows the theoretical calculation of the asymptotic curves and their intersections at homoclinic points. However in more complex mappings and in Hamiltonian systems the radius of convergence is in general finite and does not allow the theoretical calculation of any homoclinic point. Then we develop a method similar to analytic continuation, applicable in systems expressed in action-angle variables, that allows the calculation of the asymptotic curves to an arbitrary length. In this way we can study analytically the chaotic regions near the unstable periodic orbit and near its homoclinic points.

nlin.CD↗

arXiv2014

Analytical invariant manifolds near unstable points and the structure of chaos

It is known that the asymptotic invariant manifolds around an unstable periodic orbit in conservative systems can be represented by convergent series (Cherry 1926, Moser 1956, 1958, Giorgilli 2001). The unstable and stable manifolds intersect at an infinity of homoclinic points, generating a complicated homoclinic tangle. In the case of simple mappings it was found (Da Silva Ritter et al. 1987) that the domain of convergence of the formal series extends to infinity along the invariant manifolds. This allows in practice to study the homoclinic tangle using only series. However in the case of Hamiltonian systems, or mappings with a finite analyticity domain,the convergence of the series along the asymptotic manifolds is also finite. Here, we provide numerical indications that the convergence does not reach any homoclinic points. We discuss in detail the convergence problem in various cases and we find the degree of approximation of the analytical invariant manifolds to the real (numerical) manifolds as i) the order of truncation of the series increases, and ii) we use higher numerical precision in computing the coefficients of the series. Then we introduce a new method of series composition, by using action-angle variables, that allows the calculation of the asymptotic manifolds up to an a arbitrarily large extent. This is the first case of an analytic development that allows the computation of the invariant manifolds and their intersections in a Hamiltonian system for an extent long enough to allow the study of homoclinic chaos by analytical means.

nlin.CD↗

arXiv2012

Order in de Broglie - Bohm quantum mechanics

A usual assumption in the so-called {\it de Broglie - Bohm} approach to quantum dynamics is that the quantum trajectories subject to typical `guiding' wavefunctions turn to be quite irregular, i.e. {\it chaotic} (in the dynamical systems' sense). In the present paper, we consider mainly cases in which the quantum trajectories are {\it ordered}, i.e. they have zero Lyapunov characteristic numbers. We use perturbative methods to establish the existence of such trajectories from a theoretical point of view, while we analyze their properties via numerical experiments. Using a 2D harmonic oscillator system, we first establish conditions under which a trajectory can be shown to avoid close encounters with a moving nodal point, thus avoiding the source of chaos in this system. We then consider series expansions for trajectories both in the interior and the exterior of the domain covered by nodal lines, probing the domain of convergence as well as how successful the series are in comparison with numerical computations or regular trajectories. We then examine a H\'{e}non - Heiles system possessing regular trajectories, thus generalizing previous results. Finally, we explore a key issue of physical interest in the context of the de Broglie - Bohm formalism, namely the influence of order in the so-called {\it quantum relaxation} effect. We show that the existence of regular trajectories poses restrictions to the quantum relaxation process, and we give examples in which the relaxation is suppressed even when we consider initial ensembles of only chaotic trajectories, provided, however, that the system as a whole is characterized by a certain degree of order.

quant-ph↗

arXiv2012

Instabilities and stickiness in a 3D rotating galactic potential

We study the dynamics in the neighborhood of simple and double unstable periodic orbits in a rotating 3D autonomous Hamiltonian system of galactic type. In order to visualize the four dimensional spaces of section we use the method of color and rotation. We investigate the structure of the invariant manifolds that we found in the neighborhood of simple and double unstable periodic orbits in the 4D spaces of section. We consider orbits in the neighborhood of the families x1v2, belonging to the x1 tree, and the z-axis (the rotational axis of our system). Close to the transition points from stability to simple instability, in the neighborhood of the bifurcated simple unstable x1v2 periodic orbits we encounter the phenomenon of stickiness as the asymptotic curves of the unstable manifold surround regions of the phase space occupied by rotational tori existing in the region. For larger energies, away from the bifurcating point, the consequents of the chaotic orbits form clouds of points with mixing of color in their 4D representations. In the case of double instability, close to x1v2 orbits, we find clouds of points in the four dimensional spaces of section. However, in some cases of double unstable periodic orbits belonging to the z-axis family we can visualize the associated unstable eigensurface. Chaotic orbits close to the periodic orbit remain sticky to this surface for long times (of the order of a Hubble time or more). Among the orbits we studied we found those close to the double unstable orbits of the x1v2 family having the largest diffusion speed.

nlin.CD↗

arXiv2011

Wavepacket approach to particle diffraction by thin targets: Quantum trajectories and arrival times

We develop a wavepacket approach to the diffraction of charged particles by a thin material target and we use the de Broglie-Bohm quantum trajectories to study various phenomena in this context. We find the form of the separator, i.e.the limit between the domains of prevalence of the ingoing and outgoing quantum flow. The structure of the quantum-mechanical currents in the neighborhood of the separator implies the formation of an array of \emph{quantum vortices} (nodal point - X point complexes). We show how the deformation of the separatior near Bragg angles explains the emergence of a diffraction pattern by the de Broglie - Bohm trajectories. We calculate the arrival time distributions for particles scattered at different angles. The predictions of the de Broglie - Bohm theory for $\Delta T$ turn to be different from estimates of the same quantity using other theories on time observables like the sum-over-histories or the Kijowski approach. We propose an experimental setup aiming to test such predictions. Finally, we explore the semiclassical limit of short wavelength and short quantum coherence lengths, and demonstrate how, in this case, results with the de Broglie - Bohm trajectories are similar to the classical results of Rutherford scattering.

quant-ph↗

arXiv2011

Orbits in a non-Kerr Dynamical System

We study the orbits in a Manko-Novikov type metric (MN) which is a perturbed Kerr metric. There are periodic, quasi-periodic, and chaotic orbits, which are found in configuration space and on a surface of section for various values of the energy E and the z-component of the angular momentum Lz. For relatively large Lz there are two permissible regions of non-plunging motion bounded by two closed curves of zero velocity (CZV), while in the Kerr metric there is only one closed CZV of non-plunging motion. The inner permissible region of the MN metric contains mainly chaotic orbits, but it contains also a large island of stability. We find the positions of the main periodic orbits as functions of Lz and E, and their bifurcations. Around the main periodic orbit of the outer region there are islands of stability that do not appear in the Kerr metric. In a realistic binary system, because of the gravitational radiation, the energy E and the angular momentum Lz of an inspiraling compact object decrease and therefore the orbit of the object is non-geodesic. In fact in an EMRI system the energy E and the angular momentum Lz decrease adiabatically and therefore the motion of the inspiraling object is characterized by the fundamental frequencies which are drifting slowly in time. In the Kerr metric the ratio of the fundamental frequencies changes strictly monotonically in time. However, in the MN metric when an orbit is trapped inside an island the ratio of the fundamental frequencies remains constant for some time. Hence, if such a phenomenon is observed this will indicate that the system is non integrable and therefore the central object is not a Kerr black hole.

gr-qc↗

arXiv2011

Quantum vortices and trajectories in particle diffraction

We investigate the phenomenon of the diffraction of charged particles by thin material targets using the method of the de Broglie-Bohm quantum trajectories. The particle wave function can be modeled as a sum of two terms $\psi=\psi_{ingoing}+\psi_{outgoing}$. A thin separator exists between the domains of prevalence of the ingoing and outgoing wavefunction terms. The structure of the quantum-mechanical currents in the neighborhood of the separator implies the formation of an array of \emph{quantum vortices}. The flow structure around each vortex displays a characteristic pattern called `nodal point - X point complex'. The X point gives rise to stable and unstable manifolds. We find the scaling laws characterizing a nodal point-X point complex by a local perturbation theory around the nodal point. We then analyze the dynamical role of vortices in the emergence of the diffraction pattern. In particular, we demonstrate the abrupt deflections, along the direction of the unstable manifold, of the quantum trajectories approaching an X-point along its stable manifold. Theoretical results are compared to numerical simulations of quantum trajectories. We finally calculate the {\it times of flight} of particles following quantum trajectories from the source to detectors placed at various scattering angles $\theta$, and thereby propose an experimental test of the de Broglie - Bohm formalism.

quant-ph↗

arXiv2010

The structure and evolution of confined tori near a Hamiltonian Hopf Bifurcation

We study the orbital behavior at the neighborhood of complex unstable periodic orbits in a 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 this paper we first integrate initial conditions close to the ones of a complex unstable periodic orbit, which is close to the transition point. Then, we plot the consequents of the corresponding orbit in a 4D surface of section. To visualize this surface of section we use the method of color and rotation [Patsis and Zachilas 1994]. We find that the consequents are contained in 2D "confined tori". Then, we investigate 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 observe clouds of points in the 4D surfaces of section. The transition between the two types of orbital behavior is abrupt.

nlin.CD↗

arXiv2009

Origin of chaos near critical points of quantum flow

The general theory of motion in the vicinity of a moving quantum nodal point (vortex) is studied in the framework of the de Broglie - Bohm trajectory method of quantum mechanics. Using an adiabatic approximation, we find that near any nodal point of an arbitrary wavefunction $ψ$ there is an unstable point (called X-point) in a frame of reference moving with the nodal point. We find general formulae for the nodal point - X-point complex as well as necessary and sufficient conditions of validity of the adiabatic approximation. Chaos emerges from the consecutive scattering events of the orbits with nodal point - X-point complexes. A theoretical model is constructed yielding the local value of the Lyapunov characteristic number in a scattering event, which scales as an inverse power of the speed of the nodal point in the rest frame, or proportionally to the size of the nodal point X- point complex. The results of detailed numerical experiments with different wavefunctions possessing multiple moving nodal points are reported. The statistics of the Lyapunov characteristic numbers of the orbits are found and compared to the number of encounter events of each orbit with the nodal point X-point complexes. Various phenomena appearing at first as counter-intuitive find a straightforward explanation.

quant-ph↗

arXiv2008

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 $L_1$ and $L_2$ of a barred galaxy define loci in the configuration space which take the form of a trailing spiral pattern. In the present paper we investigate this association in the case of the self-consistent models of Kaufmann & Contopoulos (1996) which provide an approximation of real barred-spiral galaxies. We also examine the relation of `response' models of barred-spiral galaxies with the theory of the invariant manifolds. Our main results are the following: The invariant manifolds yield the correct form of the imposed spiral pattern provided that their calculation is done with the spiral potential term turned 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-like domain of the configuration space, delimited from below by the boundary of a banana-like non-permitted domain, and from above either by rotational KAM tori or by cantori forming a stickiness zone. We construct `spiral response' models on the basis of the theory of the invariant manifolds and examine the connection of the latter to the `response' models (Patsis 2006) used to fit real barred-spiral galaxies, explaining how are the manifolds related to a number of morphological features seen in such models.

astro-ph↗

arXiv2008

Stickiness in Chaos

We distinguish two types of stickiness in systems of two degrees of freedom (a) stickiness around an island of stability and (b) stickiness in chaos, along the unstable asymptotic curves of unstable periodic orbits. We studied these effects in the standard map with a rather large nonlinearity K=5, and we emphasized the role of the asymptotic curves U, S from the central orbit O and the asymptotic curves U+U-S+S- from the simplest unstable orbit around the island O1. We calculated the escape times (initial stickiness times) for many initial points outside but close to the island O1. The lines that separate the regions of the fast from the slow escape time follow the shape of the asymptotic curves S+,S-. We explained this phenomenon by noting that lines close to S+ on its inner side (closer to O1) approach a point of the orbit 4/9, say P1, and then follow the oscillations of the asymptotic curve U+, and escape after a rather long time, while the curves outside S+ after their approach to P1 follow the shape of the asymptotic curves U- and escape fast into the chaotic sea. All these curves return near the original arcs of U+,U- and contribute to the overall stickiness close to U+,U-. The isodensity curves follow the shape of the curves U+,U- and the maxima of density are along U+,U-. For a rather long time the stickiness effects along U+,U- are very pronounced. However after much longer times (about 1000 iterations) the overall stickiness effects are reduced and the distribution of points in the chaotic sea outside the islands tends to be uniform.

astro-ph↗