arXiv subjects

George Contopoulos

Publications and source records attributed to George Contopoulos.

36 records · Page 2Linked to original sources

arXiv2018

Highlights of chaos research

We describe some highlights in the theory of chaos, that started with Poincare (1899). Generic systems have both ordered and chaotic domains. Chaos appears mainly near un- stable periodic orbits. Large chaotic domains are due to resonance overlap. Two recent developments of the theory of chaos, refer (a) to the analytical formulae (Moser series) in the chaotic domains near unstable periodic orbits, and (b) quantum chaos, described by the Bohmian theory of orbits.

nlin.CD↗

arXiv2018

Global and Local diffusion in the Standard Map

We study the global and the local transport and diffusion in the case of the standard map, by calculating the diffusion exponent $μ$. In the global case we find that the mean diffusion exponent for the whole phase space is either $μ=1$, denoting normal diffusion or $μ=2$ denoting anomalous diffusion (and ballistic motion). The mean diffusion of the whole phase space is normal when no accelerator mode exist and it is anomalous (ballistic) when accelerator mode islands exist even if their area is tiny in the phase space. The local value of the diffusion exponent inside the normal islands of stability is $μ=0$, while inside the accelerator mode islands it is $μ=2$. The local value of the diffusion exponent in the chaotic region outside the islands of stability converges always to the value of 1. The time of convergence can be very long, depending on the distance from the accelerator mode islands and the value of the non linearity parameter $K$. For some values of $K$ the stickiness around the accelerator mode islands is maximum and initial conditions inside the sticky region can be dragged in a ballistic motion for extremely long times of the order of $10^7$ or more but they will finally end up in normal mode diffusion with $μ=1$. We study, in particular, cases with maximum stickiness and cases where normal and accelerator mode islands coexist. We find general analytical solutions of periodic orbits of accelerator type and we give evidence that they are much more numerous than the normal periodic orbits. Thus, we expect that in every small interval $ΔK$ of the non linearity parameter $K$ of the standard map there exist smaller intervals of accelerator mode islands. However, these smaller intervals are in general very small, so that in the majority of the values of $K$ the global diffusion is normal.

nlin.CD↗

arXiv2018

Origin of chaos near three-dimensional quantum vortices: A general Bohmian theory

We provide a general theory for the structure of the quantum flow near 3-d nodal lines, i.e. one-dimensional loci where the 3-d wavefunction becomes equal to zero. In suitably defined co- ordinates (co-moving with the nodal line) the generic structure of the flow implies the formation of 3-d quantum vortices. We show that such vortices are accompanied by nearby invariant lines of the co-moving quantum flow, called X-lines, which are normally hyperbolic. Furthermore, the stable and unstable manifolds of the X-lines produce chaotic scatterings of nearby quantum (Bohmian) trajectories, thus inducing an intricate form of the quantum current in the neighborhood of each 3-d quantum vortex. Generic formulas describing the structure around 3-d quantum vortices are provided, applicable to an arbitrary choice of 3-d wavefunction. We also give specific numerical examples, as well as a discussion of the physical consequences of chaos near 3-d quantum vortices.

quant-ph↗

arXiv2017

Integrals of motion in 3-d Bohmian Trajectories

Chaos in Bohmian Quantum Mechanics is an open field of research. In general, most of the 3-d Bohmian trajectories are free to wander around the 3-d space. However there are cases where the evolution of the trajectories is dictated by exact or approximate integrals of motion. A first case corresponds to partial integrability, where the trajectories (ordered and chaotic) evolve on certain integral surfaces. A second case corresponds to ordered trajectories. In this paper we extend our previous work in 3-d Bohmian Chaos by using both forms of integrability and discuss their physical implications.

quant-ph↗

arXiv2017

Chaos in de Broglie - Bohm quantum mechanics and the dynamics of quantum relaxation

We discuss the main mechanisms generating chaotic behavior of the quantum trajectories in the de Broglie - Bohm picture of quantum mechanics, in systems of two and three degrees of freedom. In the 2D case, chaos is generated via multiple scatterings of the trajectories with one or more `nodal point - X-point complexes'. In the 3D case, these complexes form foliations along `nodal lines' accompanied by `X-lines'. We also identify cases of integrable or partially integrable quantum trajectories. The role of chaos is important in interpreting the dynamical origin of the `quantum relaxation' effect, i.e. the dynamical emergence of Born's rule for the quantum probabilities, which has been proposed as an extension of the Bohmian picture of quantum mechanics. In particular, the local scaling laws characterizing the chaotic scattering phenomena near X-points, or X-lines, are related to the global rate at which the quantum relaxation is observed to proceed. Also, the degree of chaos determines the rate at which nearly-coherent initial wavepacket states lose their spatial coherence in the course of time.

quant-ph↗

arXiv2017

Partial Integrability of 3-d Bohmian Trajectories

In this paper we study the integrability of 3-d Bohmian trajectories of a system of quantum harmonic oscillators. We show that the initial choice of quantum numbers is responsible for the existence (or not) of an integral of motion which confines the trajectories on certain invariant surfaces. We give a few examples of orbits in cases where there is or there is not an integral and make some comments on the impact of partial integrability in Bohmian Mechanics. Finally, we make a connection between our present results for the integrability in the 3-d case and analogous results found in the 2-d and 4-d cases.

quant-ph↗

arXiv2016

Origin of chaos in 3-d Bohmian trajectories

We study the 3-d Bohmian trajectories of a quantum system of three harmonic oscillators. We focus on the mechanism responsible for the generation of chaotic trajectories. We demonstrate the existence of a 3-d analogue of the mechanism found in earlier studies of 2-d systems, based on moving 2-d `nodal point - X-point complexes'. In the 3-d case, we observe a foliation of nodal point - X-point complexes, forming a `3-d structure of nodal and X-points'. Chaos is generated when the Bohmian trajectories are scattered at one or more close encounters with such a structure.

quant-ph↗

arXiv2016

Analytical forms of chaotic spiral arms

We develop an analytical theory of chaotic spiral arms in galaxies. This is based on the Moser theory of invariant manifolds around unstable periodic orbits. We apply this theory to the chaotic spiral arms, that start from the neighborhood of the Lagrangian points L1 and L2 at the end of the bar in a barred-spiral galaxy. The series representing the invariant manifolds starting at the Lagrangian points L1, L2, or unstable periodic orbits around L1 and L2, yield spiral patterns in the configuration space. These series converge in a domain around every Lagrangian point, called "Moser domain" and represent the orbits that constitute the chaotic spiral arms. In fact, these orbits are not only along the invariant manifolds, but also in a domain surrounding the invariant manifolds. We show further that orbits starting outside the Moser domain but close to it converge to the boundary of the Moser domain, which acts as an attractor. These orbits stay for a long time close to the spiral arms before escaping to infinity.

nlin.CD↗

arXiv2014

Non-Linear Effects in Non-Kerr spacetimes

There is a chance that the spacetime around massive compact objects which are expected to be black holes is not described by the Kerr metric, but by a metric which can be considered as a perturbation of the Kerr metric. These non-Kerr spacetimes are also known as bumpy black hole spacetimes. We expect that, if some kind of a bumpy black hole exists, the spacetime around it should possess some features which will make the divergence from a Kerr spacetime detectable. One of the differences is that these non-Kerr spacetimes do not posses all the symmetries needed to make them integrable. We discuss how we can take advantage of this fact by examining EMRIs into the Manko-Novikov spacetime.

gr-qc↗

arXiv2013

Mind the Resonances: Final stages of accretion into bumpy black holes

In this article we discuss a possible way of testing the Kerr black hole hypothesis by taking advantage of phenomena correlated with chaotic motion in the final stages of an accretion disk around a bumpy black hole. We anticipate that these phenomena should have an imprint in the electromagnetic spectrum coming from the accretion disk.

gr-qc↗

arXiv2012

Periodic Orbits and Escapes in Dynamical Systems

We study the periodic orbits and the escapes in two different dynamical systems, namely (1) a classical system of two coupled oscillators, and (2) the Manko-Novikov metric (1992) which is a perturbation of the Kerr metric (a general relativistic system). We find their simple periodic orbits, their characteristics and their stability. Then we find their ordered and chaotic domains. As the energy goes beyond the escape energy, most chaotic orbits escape. In the first case we consider escapes to infinity, while in the second case we emphasize escapes to the central "bumpy" black hole. When the energy reaches its escape value a particular family of periodic orbits reaches an infinite period and then the family disappears (the orbit escapes). As this family approaches termination it undergoes an infinity of equal period and double period bifurcations at transitions from stability to instability and vice versa. The bifurcating families continue to exist beyond the escape energy. We study the forms of the phase space for various energies, and the statistics of the chaotic and escaping orbits. The proportion of these orbits increases abruptly as the energy goes beyond the escape energy.

nlin.CD↗

arXiv2011

Chaotic Spiral Galaxies

We study the role of asymptotic curves in supporting the spiral structure of a N-body model simulating a barred spiral galaxy. Chaotic orbits with initial conditions on the unstable asymptotic curves of the main unstable periodic orbits follow the shape of the periodic orbits for an initial interval of time and then they are diffused outwards supporting the spiral structure of the galaxy. Chaotic orbits having small deviations from the unstable periodic orbits, stay close and along the corresponding unstable asymptotic manifolds, supporting the spiral structure for more than 10 rotations of the bar. Chaotic orbits of different Jacobi constants support different parts of the spiral structure. We also study the diffusion rate of chaotic orbits outwards and find that chaotic orbits that support the outer parts of the galaxy are diffused outwards more slowly than the orbits supporting the inner parts of the spiral structure.

astro-ph.GA↗

arXiv2010

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.

astro-ph.IM↗

arXiv2010

An observable signature of a background deviating from Kerr

By detecting gravitational wave signals from extreme mass ratio inspiraling sources (EMRIs) we will be given the opportunity to check our theoretical expectations regarding the nature of supermassive bodies that inhabit the central regions of galaxies. We have explored some qualitatively new features that a perturbed Kerr metric induces in its geodesic orbits. Since a generic perturbed Kerr metric does not possess all the special symmetries of a Kerr metric, the geodesic equations in the former case are described by a slightly nonintegrable Hamiltonian system. According to the Poincaré-Birkhoff theorem this causes the appearance of the so-called Birkhoff chains of islands on the corresponding surfaces of section in between the anticipated KAM curves of the integrable Kerr case, whenever the intrinsic frequencies of the system are at resonance. The chains of islands are characterized by finite width, i.e. there is a finite range of initial conditions that correspond to a particular resonance and consequently to a constant rational ratio of intrinsic frequencies. Thus while the EMRI changes adiabatically by radiating energy and angular momentum, by monitoring the frequencies of a signal we can look for a transient pattern, in the form of a plateau, in the evolution of their ratio. We have shown that such a plateau is anticipated to be apparent in a quite large fraction of possible orbital characteristics if the central gravitating source is not a Kerr black hole. Moreover the plateau in the ratio of frequencies is expected to be more prominent at specific rational values that correspond to the strongest resonances. This gives a possible observational detection of such non-Kerr exotic objects.

gr-qc↗

arXiv2009

How to observe a non-Kerr spacetime

We present a generic criterion which can be used in gravitational-wave data analysis to distinguish an extreme-mass-ratio inspiral into a Kerr background spacetime from one into a non-Kerr background spacetime. The criterion exploits the fact that when an integrable system, such as the system that describes geodesic orbits in a Kerr spacetime, is perturbed, the tori in phase space which initially corresponded to resonances disintegrate so as to form the so called Birkhoff chains on a surface of section, according to the Poincaré-Birkhoff theorem. The KAM curves of these islands in such a chain share the same ratio of frequencies, even though the frequencies themselves vary from one KAM curve to another inside an island. On the other hand, the KAM curves, which do not lie in a Birkhoff chain, do not share this characteristic property. Such a temporal constancy of the ratio of frequencies during the evolution of the gravitational-wave signal will signal a non-Kerr spacetime which could then be further explored.

gr-qc↗

arXiv2008

Dynamics and chaos in the unified scalar field Cosmology

We study the dynamics of the closed scalar field FRW cosmological models in the framework of the so called Unified Dark Matter (UDM) scenario. Performing a theoretical as well as a numerical analysis we find that there is a strong indication of chaos in agreement with previous studies. We find that a positive value of the spatial curvature is essential for the appearance of chaoticity, though the Lyapunov number seems to be independent of the curvature value. Models that are close to flat exhibit a chaotic behavior after a long time while pure flat models do not exhibit any chaos. Moreover, we find that some of the semiflat models in the UDM scenario exhibit similar dynamical behavior with the Lambda cosmology despite their chaoticity. Finally, we compare the measured evolution of the Hubble parameter derived from the differential ages of passively evolving galaxies with that expected in the semiflat unified scalar field cosmology. Based on a specific set of initial conditions we find that the UDM scalar field model matches well the observational data.

astro-ph↗

arXiv2007

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.

quant-ph↗

arXiv1999

Diffusion and scaling in escapes from two-degree-of-freedom Hamiltonian systems

This paper summarises an investigation of the statistical properties of orbits escaping from three different two-degree-of-freedom Hamiltonian systems which exhibit global stochasticity. Each H=H_{0}+eH', with H_{0} integrable and eH' a nonintegrable correction, not necessarily small. For e below a critical e_{0} escapes are impossible energetically. For somewhat higher values, escape is allowed energetically but many orbits never escape and the escape probability P for a generic orbit ensemble decays exponentially. At or near a critical e_{1}>e_{0}, there is an abrupt qualitative change in behaviour. Above e_{1}, P typically exhibits (1) a rapid evolution towards a nonzero P_{0}(e) followed by (2) a much slower subsequent decay towards zero which, in at least one case, is well fit by a power law P=const x t^{m}, with m=0.35-0.40. In all three cases, P_{0} and the time T required to converge towards P_{0} scales in e-e_{1}, i.e., P_{0}=const x (e-e_{1})^{a} and T=const x (e-e_{1})^{b}, and T also scales in the size r of the region sampled for initial conditions, i.e., T=const x r^{-d}. To within statistical uncertainties, the best fit values of the critical exponents are the same for all three potentials, namely: a=0.5, b=0.4, and d=0.1, and satisfy a-b-d=0. The transitional behaviour observed near e_{1} is attributed to the breakdown of some especially significant KAM tori or cantori. The power law behaviour at late times is interpreted as reflecting intrinsic diffusion of chaotic orbits through cantori surrounding islands of regular orbits.

astro-ph↗