Separatrix Map Research Articles

Dynamics near the invariant manifolds after a Hamiltonian-Hopf bifurcation

We consider a one parameter family of 2-DOF Hamiltonian systems having an equilibrium point that undergoes a Hamiltonian-Hopf bifurcation. We briefly review the well-established normal form theory in this case. Then we focus on the invariant manifolds when there are homoclinic orbits to the complex-saddle equilibrium point, and we study the behavior of the splitting of the 2D invariant manifolds. The symmetries of the normal form are used to reduce the dynamics around the invariant manifolds to the dynamics of a family of area-preserving near-identity Poincaré maps that can be extended analytically to a suitable neighborhood of the separatrices. This allows, in particular, to use well-known results for area-preserving maps and derive an explicit upper bound of the splitting of separatrices for the Poincaré map. We illustrate the results in a concrete example. Different Poincaré sections are used to visualize the dynamics near the 2D invariant manifolds. Last section deals with the derivation of a separatrix map to study the chaotic dynamics near the 2D invariant manifolds.

Quasiperiodic perturbations of heteroclinic attractor networks.

We consider heteroclinic attractor networks motivated by models of competition between neural populations during binocular rivalry. We show that gamma distributions of dominance times observed experimentally in binocular rivalry and other forms of bistable perception, commonly explained by means of noise in the models, can be achieved with quasiperiodic perturbations. For this purpose, we present a methodology based on the separatrix map to model the dynamics close to heteroclinic networks with quasiperiodic perturbations. Our methodology unifies two different approaches, one based on Melnikov integrals and the other one based on variational equations. We apply it to two models: first, to the Duffing equation, which comes from the perturbation of a Hamiltonian system and, second, to a heteroclinic attractor network for binocular rivalry, for which we develop a suitable method based on Melnikov integrals for non-Hamiltonian systems. In both models, the perturbed system shows chaotic behavior, while dominance times achieve good agreement with gamma distributions. Moreover, the separatrix map provides a new (discrete) model for bistable perception which, in addition, replaces the numerical integration of time-continuous models and, consequently, reduces the computational cost and avoids numerical instabilities.

A Second Order Expansion of the Separatrix Map for Trigonometric Perturbations of a Priori Unstable Systems

In this paper we study a so-called separatrix map introduced by Zaslaskii-Filonenko [ZF68] and studied by Treschev and Piftankin [Tre98, Tre02, Pif06, PT07]. We derive a second order expansion of this map for trigonometric perturbations. In [CK15, GK15], and [KZZ15], applying the results of the present paper, we describe a class of nearly integrable deterministic systems with stochastic diffusive behavior.

Capture and long-term suspension of aerosols in an open cavity flow

The motion of non-Brownian aerosols in an open laminar cavity flow is investigated by means of numerical and asymptotic methods. The cavity is a rectangular domain where air enters through an inlet located on the top, and exits in the horizontal direction through symmetric apertures near the bottom wall. While such flows are generally used to provide a clean (particle-free) environment, the introduction of any object inside the cavity can affect the flow topology and, in return, can lead to large aerosol residence times. In addition, it is known that the unsteadiness of the flow, though weak, significantly delays the exit of aerosols from the cavity. The goal of this paper is to analyze the conditions under which these phenomena occur, for heavy inertial particles with a small response time and a finite free-fall velocity, in a laminar cavity flow. A two-dimensional situation is considered and investigated by means of numerical simulations and perturbation methods, at moderate cavity Reynolds numbers. It is observed that any obstacle placed inside the cavity (along the symmetry axis) creates large quasi-steady triangular recirculation cells stretched by the outward flow, provided the Reynolds number is not too large. In particular, streamlines near the floor are reversed, so that deposited particles are swept inward and remain in the cavity. The probability of trapping has been calculated asymptotically, for inertia-free sedimenting particles, and compared to numerical results. Trapping in the wake of obstacles is observed to persist when the flow is slightly unsteady. In addition, it is shown that the small but finite unsteadiness of the flow can lead to the temporary trapping of a significant portion of particles by large recirculation cells near the upper corners of the cavity. This phenomenon takes place in spite of gravity and centrifuge effects due to the curvature of the cell, and leads to a long-term suspension which significantly increases the residence time of aerosols. By making use of separatrix map methods, the critical aerosol diameter below which this phenomenon occurs has been obtained and compared to numerical simulation results.

Return Maps, Dynamical Consequences and Applications

After reviewing some general settings for return maps in problems reducible to 2D symplectic maps, details on the construction of return maps are presented. Different forms of such maps close to splitted separatrices (separatrix maps) are introduced, taking into account the size and shape of the splitting function and also the return time to the domains of interest. Then it is shown how to derive approximations by suitable standard-like maps. Dynamical consequences concerning the existence of invariant rotational curves (IRC) are derived. An application is made to theoretically estimate the location of the outermost IRC in the Sitnikov problem, which is in good agreement with numerical data. To compare with the cases which are approximated by the classical standard map, some details on the properties of the standard-like map with two harmonic terms are included. Finally a method to estimate the amount of chaos depending on the form of the separatrix map is introduced. Except otherwise stated all the systems we consider are assumed to be analytic, despite several of the properties we study are no longer analytic.

CHAOTIC ZONES AROUND GRAVITATING BINARIES

The extent of the continuous zone of chaotic orbits of a small-mass tertiary around a system of two gravitationally bound primaries (a double star, a double black hole, a binary asteroid, etc.) is estimated analytically, in function of the tertiary's orbital eccentricity. The separatrix map theory is used to demonstrate that the central continuous chaos zone emerges (above a threshold in the primaries mass ratio) due to overlapping of the orbital resonances corresponding to the integer ratios p:1 between the tertiary and the central binary periods. In this zone, the unlimited chaotic orbital diffusion of the tertiary takes place, up to its ejection from the system. The primaries mass ratio, above which such a chaotic zone is universally present at all initial eccentricities of the tertiary, is estimated. The diversity of the observed orbital configurations of biplanetary and circumbinary exosystems is shown to be in accord with the existence of the primaries mass parameter threshold.

Stable and unstable manifolds, and the structure of magnetic footprints

The generic analytic formulae of the stable and unstable manifolds near the separatrix and on the divertor plates in tokamaks in the presence of resonant magnetic perturbations are derived. The stable (unstable) manifolds on the divertor plates describe the form of the magnetic footprint boundaries. They are derived using the separatrix mapping. The analytical formulae depend on a few of the equilibrium plasma parameters and the magnetic perturbations.

Lyapunov exponents in resonance multiplets

The problem of estimating the maximum Lyapunov exponents of the motion in a multiplet of interacting nonlinear resonances is considered for the case when the resonances have comparable strength. The corresponding theoretical approaches are considered for the multiplets of two, three, and infinitely many resonances (i.e., doublets, triplets, and “infinitets”). The analysis is based on the theory of separatrix and standard maps. A “multiplet separatrix map” is introduced, valid for description of the motion in the resonance multiplet under certain conditions. In numerical experiments it is shown that, at any given value of the adiabaticity parameter (which controls the degree of interaction/overlap of resonances in the multiplet), the value of the maximum Lyapunov exponent in the multiplet of equally-spaced equally-sized resonances is minimal in the doublet case and maximal in the infinitet case. This is consistent with the developed theory.

Poincaré recurrences and Ulam method for the Chirikov standard map

We study numerically the statistics of Poincar\'e recurrences for the Chirikov standard map and the separatrix map at parameters with a critical golden invariant curve. The properties of recurrences are analyzed with the help of a generalized Ulam method. This method allows to construct the corresponding Ulam matrix whose spectrum and eigenstates are analyzed by the powerful Arnoldi method. We also develop a new survival Monte Carlo method which allows us to study recurrences on times changing by ten orders of magnitude. We show that the recurrences at long times are determined by trajectory sticking in a vicinity of the critical golden curve and secondary resonance structures. The values of Poincar\'e exponents of recurrences are determined for the two maps studied. We also discuss the localization properties of eigenstates of the Ulam matrix and their relation with the Poincar\'e recurrences.

Stochastic dynamics of electric dipole in external electric fields: A perturbed nonlinear pendulum approach

The motion of a dipole in external electric fields is considered in the framework of nonlinear pendulum dynamics. A stochastic layer is formed near the separatrix of the dipole pendulum in a restoring static electric field under the periodic perturbation by plane-polarized electric fields. The width of the stochastic layer depends on the direction of the forcing field variation, and this width can be evaluated as a function of perturbation frequency, amplitude, and duration. A numerical simulation of the approximate stochastic layer width of a perturbed pendulum yields a multi-peak frequency spectrum. It is described well enough at high perturbation amplitudes by an analytical estimation based on the separatrix map with an introduced expression of the most effective perturbation phase. The difference in the fractal dimensions of the phase spaces calculated geometrically and using the time-delay reconstruction is attributed to the predominant development of periodic and chaotic orbits, respectively. The correlation of the stochastic layer width with the phase space fractal dimensions is discussed.

Width of the chaotic layer: Maxima due to marginal resonances

Modern theoretical methods for estimating the width of the chaotic layer in the presence of prominent marginal resonances are considered in the perturbed pendulum model of nonlinear resonance. The fields of applicability of these methods are explicitly and precisely formulated. The comparative accuracy is investigated in massive and long-run numerical experiments. It is shown that the methods are naturally subdivided in classes applicable for adiabatic and nonadiabatic cases of perturbation. It is explicitly shown that the pendulum approximation of marginal resonance works well in the nonadiabatic case. In this case, the role of marginal resonances in determining the total layer width is demonstrated to diminish with increasing main parameter λ (equal to the ratio of the perturbation frequency to the frequency of small-amplitude phase oscillations on the resonance). Solely the "bending effect" is important in determining the total amplitude of the energy deviations of the near-separatrix motion at λ≳7. In the adiabatic case, it is demonstrated that the geometrical form of the separatrix cell can be described analytically quite easily by means of using a specific representation of the separatrix map. It is shown that the nonadiabatic (and, to some extent, intermediary) case is most actual, in comparison with the adiabatic one, for the physical or technical applications that concern the energy jumps in the near-separatrix chaotic motion.

Asymptotic properties of wall-induced chaotic mixing in point vortex pairs

The purpose of this work is to analyze the flow due to a potential point vortex pair in the vicinity of a symmetry line (or “wall”), in order to understand why the presence of the wall, even far from the vortices, accelerates fluid mixing around the vortex pair. An asymptotic analysis, in the limit of large distances to the wall, allows to approximate the wall effect as a constant translation of the vortex pair parallel to the wall, plus a straining flow which induces a natural blinking vortex mechanism with period half the rotation period. A Melnikov analysis of lagrangian particles, in the frame translating and rotating with the vortices, shows that a homoclinic bifurcation indeed occurs, so that the various separatrices located near the vortex pair (and rotating with it) do not survive when a wall is present. The thickness of the resulting inner stochastic layer is estimated by using the separatrix map method and is shown to scale like the inverse of the squared distance to the wall. This estimation provides a lower-bound to the numerical thickness measured from either Poincaré sections or simulations of lagrangian particles transported by the exact potential velocity field in the laboratory frame. In addition, it is shown that the outer homoclinic cycle, separating the vortices from the external (open) flow, is also perturbed from inside by the rotation of the vortex pair. As a consequence, a stochastic layer is shown to exist also in the vicinity of this cycle, allowing fluid exchange between the vortices and the outer flow. However, the thickness of this outer stochastic zone is observed to be much smaller than the one of the inner stochastic zone near vortices, as soon as the distance to the wall is large enough.

Instability for <em>a priori</em> unstable Hamiltonian systems: A dynamical approach

In this article, we consider an a priori unstable Hamiltonian system with three degrees of freedom, for which we construct a drifting solution with an optimal time of instability. Such a result has been already proved by Berti, Bolle and Biasco using variational arguments, and by Treschev using his separatrix map theory. Our approach is new: it is based on a special type of symbolic dynamics corresponding to the random iteration of a family of twist maps of the annulus, and it gives the first concrete application of this idea introduced by Moeckel in an abstract setting and further studied by Marco. Our method should also be useful in obtaining the optimal time of instability in the more difficult context of a priori stable Hamiltonian systems.

Some remarks on the abundance of stable periodic orbits inside homoclinic lobes

We consider a family F ϵ of area-preserving maps (APMs) with a hyperbolic point H ϵ whose invariant manifolds form a figure-eight and we study the abundance of elliptic periodic orbits visiting homoclinic lobes (EPL), a domain typically dominated by chaotic behavior. To this end, we use the Chirikov separatrix map ( SM ) as a model of the return to a fundamental domain containing lobes. We obtain an explicit estimate, valid for families F ϵ with central symmetry and close to an integrable limit, of the relative measure of the set of parameters ϵ for which F ϵ has EPL trajectories. To get this estimate we look for EPL of the SM with the lowest possible period. The analytical results are complemented with quantitative numerical studies of the following families F ϵ of APMs: • The SM family, and we compare our analytical results with the numerical estimates. • The standard map (STM) family, and we show how the results referring to the SM model apply to the EPL visiting the lobes that the invariant manifolds of the STM hyperbolic fixed point form. • The conservative Hénon map family, and we estimate the number of a particular type of symmetrical EPL related to the separatrices of the 4-periodic resonant islands. The results obtained can be seen as the quantitative analogs to those in Simó and Treschev (2008) [9], although here we deal with the a priori stable situation instead.

LYAPUNOV AND DIFFUSION TIMESCALES IN THE SOLAR NEIGHBORHOOD

We estimate the Lyapunov times (characteristic times of predictability of motion) in Quillen's models for the dynamics in the solar neighborhood. These models take into account perturbations due to the Galactic bar and spiral arms. For estimating the Lyapunov times, an approach based on the separatrix map theory is used. The Lyapunov times turn out to be typically of the order of 10 Galactic years. We show that only in a narrow range of possible values of the problem parameters the Galactic chaos is adiabatic; usually it is not slow. We also estimate the characteristic diffusion times in the chaotic domain. In a number of models, the diffusion times turn out to be small enough to permit migration of the Sun from the inner regions of the Milky Way to its current location. Moreover, due to the possibility of ballistic flights inside the chaotic layer, the chaotic mixing might be even far more effective and quicker than in the case of normal diffusion. This confirms the dynamical possibility of Minchev and Famaey's migration concept.

Dynamics in chaotic zones of area preserving maps: Close to separatrix and global instability zones

The purpose of this paper is to study phenomena in chaotic zones of area preserving maps using simpler models which are easier to analyse theoretically and numerically. First of all the study of the dynamics in a neighbourhood of the separatrices of a resonant zone is carried out. The well-known separatrix map, defined on a figure eight when needed, is used to determine the location of rotational invariant curves (r.i.c.) inside and outside the resonance. The interest in this part is on a quantitative description of the dynamics in a neighbourhood of the separatrices: to produce theoretical estimates of the width of the stochastic zone, distance to the r.i.c., existence of tiny islands close to the separatrices, …In every one of the studied items one has tried to complement the limit analytic study with realistic numerical simulations, describing the analogy when possible. After this study, we focus on the formation of larger domains without r.i.c. (e.g. Birkhoff domains). To this end we introduce the biseparatrix map model. Although this is a qualitative model, the mechanism of destruction of the “last” r.i.c., and hence the process of creation of zones without r.i.c., is clarified by means of this simple model. Several numerical examples illustrate the results obtained and are used as a test of the theoretical quantitative predictions.

The Kepler map in the three-body problem

The Kepler map was derived by Petrosky (1986) and Chirikov and Vecheslavov (1986) as a tool for description of the long-term chaotic orbital behaviour of the comets in nearly parabolic motion. It is a two-dimensional area-preserving map, describing the motion of a comet in terms of energy and time. Its second equation is based on Kepler’s third law, hence the title of the map. Since 1980s the Kepler map has become paradigmatic in a number of applications in celestial mechanics and atomic physics. It represents an important kind of general separatrix maps. Petrosky and Broucke (1988) used refined methods of mathematical physics to derive analytical expressions for its single parameter. These methods became available only in the second half of the 20th century, and it may seem that the map is inherently a very modern mathematical tool. With the help of the Jacobi integral I show that the Kepler map, including analytical formulae for its parameter, can be derived by quite elementary methods. The prehistory and applications of the Kepler map are considered and discussed.

Separatrix Map Analysis for Fractal Scatterings in Weak Interactions of Solitary Waves

Previous studies have shown that fractal scatterings in weak interactions of solitary waves in the generalized nonlinear Schrödinger (NLS) equations are described by a universal second‐order separatrix map. In this paper, this separatrix map is analyzed in detail, and hence a complete characterization of fractal scatterings in these weak interactions is obtained. In particular, scaling laws of these fractals are derived analytically for different initial conditions, and these laws are confirmed by direct numerical simulations. In addition, an analytical criterion for the occurrence of fractal scatterings is given explicitly.

Adiabatic chaos in the Prometheus-Pandora system

The chaotic orbital motion of Prometheus and Pandora, the 16th and 17th satellites of Saturn, is studied. Chaos in their orbital motion, as found by Goldreich & Rappaport and Renner & Sicardy, is due to interaction of resonances in the resonance multiplet corresponding to the 121:118 commensurability of the mean motions of the satellites. It is shown rigorously that the system moves in adiabatic regime. The Lyapunov time (the ‘time horizon of predictability’ of the motion) is calculated analytically and compared to the available numerical–experimental estimates. For this purpose, a method of analytical estimation of the maximum Lyapunov exponent in the perturbed pendulum model of non-linear resonance is applied. The method is based on the separatrix map theory. An analytical estimate of the width of the chaotic layer is made as well, based on the same theory. The ranges of chaotic diffusion in the mean motion are shown to be almost twice as big compared to previous estimates for both satellites.

A universal separatrix map for weak interactions of solitary waves in generalized nonlinear Schrödinger equations

It is known that weak interactions of two solitary waves in generalized nonlinear Schrödinger (NLS) equations exhibit fractal dependence on initial conditions, and the dynamics of these interactions is governed by a universal two-degree-of-freedom ODE system [Y. Zhu J. Yang, Universal fractal structures in the weak interaction of solitary waves in generalized nonlinear Schrödinger equations, Phys. Rev. E 75 (2007) 036605]. In this paper, this ODE system is analyzed comprehensively. Using asymptotic methods along separatrix orbits, a simple second-order map is derived. This map does not have any free parameters after variable rescalings, and thus is universal for all weak interactions of solitary waves in generalized NLS equations. Comparison between this map’s predictions and direct simulations of the ODE system shows that the map can capture the fractal-scattering phenomenon of the ODE system very well both qualitatively and quantitatively.