Manifolds Of Unstable Periodic Orbits Research Articles

Impact of chaos on precursors of quantum criticality.

Excited-state quantum phase transitions (ESQPTs) are critical phenomena that generate singularities in the spectrum of quantum systems. For systems with a classical counterpart, these phenomena have their origin in the classical limit when the separatrix of an unstable periodic orbit divides phase space into different regions. Using a semiclassical theory of wave propagation based on the manifolds of unstable periodic orbits, we describe the quantum states associated with an ESQPT for the quantum standard map: a paradigmatic example of a kicked quantum system. Moreover, we show that finite-size precursors of ESQPTs shrink as chaos increases due to the disturbance of the system. This phenomenon is explained through destructive interference between principal homoclinic orbits.

Support vector machines for learning reactive islands.

We develop a machine learning framework that can be applied to data sets derived from the trajectories of Hamilton's equations. The goal is to learn the phase space structures that play the governing role for phase space transport relevant to particular applications. Our focus is on learning reactive islands in two degrees-of-freedom Hamiltonian systems. Reactive islands are constructed from the stable and unstable manifolds of unstable periodic orbits and play the role of quantifying transition dynamics. We show that the support vector machines are an appropriate machine learning framework for this purpose as it provides an approach for finding the boundaries between qualitatively distinct dynamical behaviors, which is in the spirit of the phase space transport framework. We show how our method allows us to find reactive islands directly in the sense that we do not have to first compute unstable periodic orbits and their stable and unstable manifolds. We apply our approach to the Hénon-Heiles Hamiltonian system, which is a benchmark system in the dynamical systems community. We discuss different sampling and learning approaches and their advantages and disadvantages.

Semiclassical theory of long time propagation in quantum chaos. Second part

We develop a semiclassical theory of wave propagation based on invariant Lagrangian manifolds existing in conservative Hamiltonian systems with chaotic dynamics. They are stable and unstable manifolds of unstable periodic orbits, and their intersections consist of homoclinic and heteroclinic orbits. For arbitrarily long times, we find matrix elements of the evolution operator between wave functions constructed in the neighbourhood of short unstable periodic orbits, in terms of canonical invariants of homoclinic and heteroclinic orbits. We verify the accuracy of these expressions by computing millions of homoclinic orbits and thousands of heteroclinic ones in the hyperbola billiard. This second part in which we describe off-diagonal matrix elements serves as complement to a previous article (first part) where we have described diagonal matrix elements.

Semiclassical theory of long time propagation in quantum chaos. First part

We develop a semiclassical theory of wave propagation based on invariant Lagrangian manifolds existing in conservative Hamiltonian systems with chaotic dynamics. They are stable and unstable manifolds of unstable periodic orbits, and their intersections consist of homoclinic and heteroclinic orbits. For arbitrary long times, we find matrix elements of the evolution operator between wave functions constructed in the neighbourhood of short unstable periodic orbits, in terms of canonical invariants of homoclinic and heteroclinic orbits. We verify the accuracy of these expressions by computing millions of homoclinic orbits and thousands of heteroclinic ones in the hyperbola billiard. Here we describe diagonal matrix elements while in another article named second part we will describe off-diagonal matrix elements.

Phase space structures causing the reaction rate decrease in the collinear hydrogen exchange reaction

The collinear hydrogen exchange reaction is a paradigm system for understanding chemical reactions. It is the simplest imaginable atomic system with 2 degrees of freedom modeling a chemical reaction, yet it exhibits behaviour that is still not well understood—the reaction rate decreases as a function of energy beyond a critical value. Using lobe dynamics we show how invariant manifolds of unstable periodic orbits guide trajectories in phase space. From the structure of the invariant manifolds we deduce that insufficient transfer of energy between the degrees of freedom causes a reaction rate decrease. In physical terms this corresponds to the free hydrogen atom repelling the whole molecule instead of only one atom from the molecule. We further derive upper and lower bounds of the reaction rate, which are desirable for practical reasons.

Topological classification of Morse–Smale diffeomorphisms on 3-manifolds

The topological classification of even the simplest Morse–Smale diffeomorphisms on 3-manifolds does not fit into the concept of singling out a skeleton consisting of stable and unstable manifolds of periodic orbits. The reason for this lies primarily in the possibility of “wild” behavior of separatrices of saddle points. Another difference between Morse–Smale diffeomorphisms in dimension 3 and their surface analogues lies in the variety of heteroclinic intersections: a connected component of such an intersection may not be only a point, as in the 2-dimensional case, but also a curve, compact or noncompact. The problem of topological classification of Morse–Smale cascades on 3-manifolds either without heteroclinic points (gradient-like cascades) or without heteroclinic curves was solved in a series of papers from 2000 to 2016 by C. Bonatti, V. Grines, F. Laudenbach, V. Medvedev, E. Pecou, and O. Pochinka. The present article is devoted to completing the topological classification of the set MS(M3) of orientation-preserving Morse–Smale diffeomorphisms f on a smooth closed orientable 3-manifold M3. The complete topological invariant for a diffeomorphism f∈MS(M3) is the equivalence class of its scheme Sf which contains information on the periodic data and the topology of embedding of 2-dimensional invariant manifolds of the saddle periodic points of f into the ambient manifold.

Efficient manifolds tracing for planar maps.

In this work, we introduce an exact calculation method and an approximation technique for tracing the invariant manifolds of unstable periodic orbits of planar maps. The exact method relies on an adaptive refinement procedure that prevents redundant calculations occurring in other approaches, and the approximated method relies on a novel interpolation approach based on normal displacement functions. The resulting approximated manifold is precise when compared to the exact one, and its relative computational cost falls like the inverse of the manifold length. To present the tracing method, we obtain the invariant manifolds of the Chirikov-Taylor map, and as an application we illustrate the transition from homoclinic to heteroclinic chaos in the Duffing oscillator that leads from localized chaos to global chaotic motion.

A Simple Financial Market Model with Heterogeneous Speculators: 2-dimensional Discontinuous System

This paper aims at a financial market model with interacting chartists and fundamentalists and chase sell, the model dynamics is driven by a two-dimensional discontinuous piecewise linear map. The parameter space is partitioned into regions depending on the types of the fixed points at the two sides of the border. For the fixed points of both sides are attractors (regular/ spiral/flip), the existence and coexistence of periodic points are studied analytically and numerically. The existence of chaotic orbit is explained by using the theory of homoclinic intersection for stable and unstable manifold of periodic orbit. The basins of coexisting multi-attractors are presented. The results can deep our knowledge of both financial market and dynamical system.

Fractal boundaries in chaotic hamiltonian systems

Fractal structures are typically present in the dynamics of chaotic orbits in non-integrable open Hamiltonian systems and result from the extremely complicated nature of the invariant manifolds of unstable periodic orbits. Exit basins, the set of initial conditions leading to orbits escaping through a given exit, have very frequently fractal boundaries. In this work we analyze exit basin boundaries in a dynamical system of physical interest, namely the motion of charged particles in a magnetized plasma subjected to electrostatic drift waves, and characterize in a quantitative way the fractality of these structures and their observable consequences, as the final-state uncertainty.

Spatial approaches to moons from resonance relative to invariant manifolds

In this study, the final approach to a moon or other body from resonance is explored and compared to the invariant manifolds of unstable periodic orbits. It is shown that the stable manifolds of planar Lyapunov orbits can act as a guide for the periods or resonances that are required for the final approach in both the planar and spatial problems. Previously developed techniques for the planar problem are expanded for use with resonances and used for comparison with trajectories approaching a moon from these resonances. A similar technique is then used for exploring the relationship of invariant manifolds to approach trajectories in the spatial problem. It is shown that the invariant manifolds of unstable periodic orbits provide insight into the trajectory design, and they can be used as a guide to the more direct approach trajectories.

A greedy global search algorithm for connecting unstable periodic orbits with low energy cost.

A method for space mission trajectory design is presented in the form of a greedy global search algorithm. It uses invariant manifolds of unstable periodic orbits and its main advantage is that it performs a global search for the suitable legs of the invariant manifolds to be connected for a preliminary transfer design, as well as the appropriate points of the legs for maneuver application. The designed indirect algorithm bases the greedy choice on the optimality conditions that are assumed for the theoretical minimum transfer cost of a spacecraft when using invariant manifolds. The method is applied to a test case space mission design project in the Earth–Moon system and is found to compare favorably with previous techniques applied to the same project.

Horseshoes for the nearly symmetric heavy top

We prove the existence of horseshoes in the nearly symmetric heavy top. This problem was previously addressed but treated inappropriately due to a singularity of the equations of motion. We introduce an (artificial) inclined plane to remove this singularity and use a Melnikov-type approach to show that there exist transverse homoclinic orbits to periodic orbits on four-dimensional level sets. The price we pay for removing the singularity is that the Hamiltonian system becomes a three-degree-of-freedom system with an additional first integral, unlike the two-degree-of-freedom formulation in the classical treatment. We therefore have to analyze three-dimensional stable and unstable manifolds of periodic orbits in a six-dimensional phase space. A new Melnikov-type technique is developed for this situation. Numerical evidence for the existence of transverse homoclinic orbits on a four-dimensional level set is also given.

Scattering echoes in a three-dimensional waveguide with a ripple cavity

We study the classical and quantum scattering of an electron in a cylindrical waveguide with a ripple cavity. The classical electron scattering is chaotic and governed by a heteroclinic tangle formed by the stable and unstable manifolds of periodic orbits sitting at the edges of the ripple cavity. The quantum scattering consists of time-periodic echoes in the transmitted electron probability. The period of the quantum echoes is related to the development parameter of the classical heteroclinic tangle. We show how the heteroclinic tangle and the quantum echoes are influenced by the diameter of the waveguide and also the angular momentum of the incident electron.

Special features of galactic dynamics: Disc dynamics

This is a tutorial presentation of special features of galactic disc dynamics, which completes our introduction to galactic dynamics initially presented in [30]. The emphasis is on topics where galactic dynamics and celestial mechanics share common starting points and/or methods of approach. We start by giving some definitions and general notions on the link between observations and dynamical modeling of discs. Then we focus on the application of resonant Hamiltonian perturbation theory in disc resonances. By examining in detail the case of the Inner Lindblad resonance, we demonstrate how resonant perturbation theory leads to an orbital theory of spiral structure in normal galaxies. Passing to barred galaxies, the phase space structure and the role of chaos in the corotation region are analyzed. This is accomplished by a summary of the modern theory of invariant manifolds of unstable periodic orbits in the vicinity of L1 or L2, which can interpret the generation of spiral patterns by chaotic orbits beyond corotation. Some additional topics, potentially important for disc dynamics, are briefly commented.

Local unstable manifolds of periodic orbits in thin domains

We compare the dynamics of a system of damped wave equations on a thin domain with the dynamics of the limit system when the thickness of the thin domain goes to 0. More precisely, under the assumption that the problem in the thin domain admits a periodic orbit γ ε close to a hyperbolic periodic orbit γ 0 of the problem in the limit domain, we show that γ ε is also hyperbolic and that the local unstable manifold of the periodic orbit γ ε is close to the local unstable manifold of γ 0 .

Designing Ephemeris Capture Trajectories at Europa Using Unstable Periodic Orbits

Europa's proximity to Jupiter creates a hostile dynamic environment for a spacecraft with limited control authority as is typical with low-thrust missions. The unstable third-body effects are magnified near the polar inclinations where science orbits reside, and plane changes deep within Europa's gravity well are fuel and time expensive. Therefore, designing capture trajectories directly to highly inclined states is desirable but challenging because of the pronounced instability and lack of thrust authority. The approach outlined here confronts this problem by using dynamical systems theory and an extensive preexisting database of restricted three-body problem periodic orbits. The stable manifolds of unstable periodic orbits are used to attract a spacecraft towards Europa. By selecting an appropriate periodic orbit, a mission designer can control important characteristics of the captured state including stability, minimum altitudes, characteristic inclinations, and characteristic radii, among others. Several free parameters are adjusted in the nontrivial mapping from the simplified to a more realistic model until a satisfactory ephemeris capture is found. Although the final capture trajectory is ballistic by design, low thrust is used to target the state that leads to the capture orbit, control the spacecraft after arriving on the unstable quasi-periodic orbit, and begin the spiral down towards the science orbit. Despite the limited control authority and the highly unstable dynamical environment, the method provides structure to the design process and enables the systematic targeting of ephemeris model capture state characteristics.

Intersections of stable and unstable manifolds: the skeleton of Lagrangian chaos

Abstract We study Hamiltonian chaos generated by the dynamics of passive tracers moving in a two-dimensional fluid flow and describe the complex structure formed in a chaotic layer that separates a vortex region from the shear flow. The stable and unstable manifolds of unstable periodic orbits are computed. It is shown that their intersections in the Poincare map as an invariant set of homoclinic points constitute the backbone of the chaotic layer. Special attention is paid to the finite time properties of the chaotic layer. In particular, finite time Lyapunov exponents are computed and a scaling law of the variance of their distribution is derived. Additionally, the box counting dimension as an effective dimension to characterize the fractal properties of the layer is estimated for different duration times of simulation. Its behavior in the asymptotic time limit is discussed. By computing the Lyapunov exponents and by applying methods of symbolic dynamics, the formation of the layer as a function of the external forcing strength, which in turn represents the perturbation of the originally integrable system, is characterized. In particular, it is shown that the capture of KAM tori by the layer has a remarkable influence on the averaged Lyapunov exponents.

Dramatic shape sensitivity of directional emission patterns from similarly deformed cylindrical polymer lasers

Recent experiments on similarly shaped polymer micro-cavity lasers show a dramatic difference in the far-field emission patterns. We show for different deformations of the ellipse, quadrupole and hexadecapole that the large differences in the far-field emission patterns is explained by the differing ray dynamics corresponding to each shape. Analyzing the differences in the appropriate phase space for ray motion, it is shown that the differing geometries of the unstable manifolds of periodic orbits are the decisive factors in determining the far-field pattern. Surprisingly, we find that strongly chaotic ray dynamics is compatible with highly directional emission in the far-field.

Computing unstable manifolds of periodic orbits in delay differential equations

We present the first algorithm for computing unstable manifolds of saddle-type periodic orbits with one unstable Floquet multiplier in systems of autonomous delay differential equations (DDEs) with one fixed delay. Specifically, we grow the one-dimensional unstable manifold W u( q) of an associated saddle fixed point q of a Poincaré map defined by a suitable Poincaré section Σ. Starting close to q along the linear approximation to W u( q) given by the associated eigenfunction, our algorithm grows the manifold as a sequence of points, where the distance between points is governed by the curvature of the one-dimensional intersection curve W u( q)∩ Σ of W u( q) with Σ. Our algorithm makes it possible to study global bifurcations in DDEs. We illustrate this with the break-up of an invariant torus and a subsequent crisis bifurcation to chaos in a DDE model of a semiconductor laser with phase-conjugate feedback.

Homoclinic chaos for ray optics in a fiber

In the geometrical optics approximation, stable and unstable manifolds of periodic orbits, invariant tori, and hyperbolic invariant manifolds are shown to exist and produce trapping of bundles of light rays near the axis of a translation-invariant, axisymmetric optical fiber whose squared refractive index is a parabolic function of squared radius. Periodic symmetry-breaking perturbations in the refractive index are shown to destroy this ray trapping and to produce homoclinic tangles, through which nearby trapped rays may escape, and untrapped rays may become at least temporarily trapped. Melnikov's technique is used to prove that the perturbations cause the stable and unstable manifolds of the unperturbed periodic orbits, invariant tori, and hyperbolic invariant manifolds to develop transverse intersections and therefore, to form homoclinic tangles. These tangles imply either homoclinic chaos by the Smale horseshoe mechanism and the Poincaré-Birkhoff-Smale theorem, or Arnol'd diffusion. In both situations, lobe dynamics will dominate phase space transport, seen here as a flux of (initially) untrapped rays passing through the trapping region.