Chaotic layer of a pendulum under low-and medium-frequency perturbations

The amplitude of the separatrix map and the size of a pendulum chaotic layer are studied numerically and analytically as functions of the adiabaticity parameter at low and medium perturbation frequencies. Good agreement between the theory and numerical experiment is found at low frequencies. In the medium-frequency range, the efficiency of using resonance invariants of separatrix mapping is high. Taken together with the known high-frequency asymptotics, the results obtained in this work reconstruct the chaotic layer pattern throughout the perturbation frequency range.     

Geometry of a chaotic layer

An analysis is made of the dependence of the geometric shape of the chaotic layer near the separatrix of a nonlinear resonance of a Hamiltonian system on the parameters of this system. A separatrix algorithmic mapping, which describes the motion near the separatrix in the presence of an asymmetric perturbation having an arbitrary degree of asymmetry. The separatrix algorithmic mapping is an algorithm containing conditional transfer instructions, is considered. An analytic procedure is derived to reduce the separatrix algorithmic mapping to the unified surface of the cross section of the initial Hamiltonian system (mapping synchronization procedure). It is observed that in the case of the high-frequency perturbation λ → +∞ (where λ is the ratio of the perturbation frequency to the frequency of small phase oscillations at resonance), the chaotic layer is subjected to strong bending in the sense that during motion near the separatrix theamplitude of the energy deviations relative to the unperturbed separatrix value is much larger than the layer width. However, the synchronized separatrix algorithmic mapping ensures an accurate representation of the phase portrait of the layer for both low and high values of the parameter λ provided that the amplitude of the perturbation is fairly small. This is demonstrated by comparing the phase portraits obtained using the synchronized separatrix algorithmic mapping with the results of direct numerical integrations of the initial Hamiltonian system.     

Structure of the stochastic layer in a driven pendulum

An analysis of the stochastic layer in a driven pendulum is extended to the case when the separatrix map contains both single-and double-frequency harmonics. Resonance invariants of the first three orders are found for the double-frequency harmonic. Combined with the previously known single-frequency invariants, they can be used to obtain further information about the layer, in particular, to examine the neighborhoods of zeros of Melnikov integrals.     

Suppression of dynamic chaos in Hamiltonian systems

The paper describes the results of a recent numerical study on the canonical mapping with a sawtooth force. The dynamic effects of the formation of invariant resonance structures of various orders, whose presence prevents the development of global chaos and restricts momentum diffusion in the phase space, are discussed. The dynamic situation near an integer resonance separatrix in the neighborhood of the critical state is studied, and the conditions responsible for the stability of this separatrix in the critical state are determined. Along with the mapping, the related continuous Hamiltonian system is considered. For this system, the separatrix mapping and the Mel’nikov-Arnold integral are introduced, whose analysis facilitates understanding the reasons responsible for the unusual dynamics. This dynamics is shown to be preserved under substantial saw shape changes. Relevant new problems and open questions are formulated.     

Contribution from the secondary harmonics of a disturbance to the separatrix map of the Hamiltonian system

The special role of low-frequency secondary harmonics with frequencies that are sums of and differences between primary frequencies entering into the Hamiltonian in explicit form has been already discussed in the literature. These harmonics are of the second order of smallness and constitute a minor fraction of the disturbance. Nevertheless, under certain conditions, their contribution to the amplitude of the separatrix map of the system may be several orders of magnitude higher than the contributions from primary harmonics and, thereby, govern the formation of dynamic chaos. This work generalizes currently available theoretical and numerical data on this issue. The role of secondary harmonics is demonstrated with a pendulum the disturbance of which in the Hamiltonian is represented by two asymmetric closely spaced high-frequency harmonics. An analytical expression for the contribution of the secondary harmonics to the separatrix map amplitude for this system is derived, and the range of very low secondary frequencies not studied earlier is considered using this equation. The domains where the separatrix map amplitude linearly grows with frequency and the chaotic layer size is frequency-independent are indicated. Theoretical predictions are compared with numerical data.     

Formation of a chaotic layer of nonlinear resonance by a two-frequency perturbation

A new effect [V. V. Vecheslavov, Zh. éksp. Teor. Fiz. 109, 2208 (1996) (JETP 82, 1190 (1996)]—the appearance of low-frequency secondary harmonics in the separatrix mapping of a system—is discussed in detail for the example of a pendulum with a two-frequency perturbation. It is shown that there exist regions of values of the perturbation parameters where these harmonics make the main contribution to the formation of the chaotic layer of the fundamental resonance. The results of analytical and numerical determinations of the amplitudes of the secondary harmonics are compared. Pis’ma Zh. éksp. Teor. Fiz. 63, No. 12, 989–994 (25 June 1996)     

Adiabatic invariance and separatrix: Single separatrix crossing

Detailed numerical experiments on the dynamics and statistics of a single crossing of the separatrix of a nonlinear resonance with a time-varying amplitude are described. The results are compared with a simple approximate theory first developed by Timofeev and further improved and generalized by Tennyson and coworkers. The main attention is paid to a new, ballistic, regime of separatrix crossing in which the violation of adiabaticity is maximal. Some unsolved problems and open questions are also discussed.     

Formation of electron depletion layer and parallel electric field in the separatrix region of anti-parallel magnetic reconnection

It is generally accepted that during collisionless magnetic reconnection, electrons flow toward the X line in the separatrix region, and then an electron depletion layer is formed.In this paper, with two-dimensional(2 D) particle-in-cell(PIC)simulation, we investigate the characteristics of the separatrix region during magnetic reconnection.In addition to the electron depletion layer, we find that there still exists an electric field parallel to the magnetic field in the separatrix region.Because a reduced ion-to-electron mass ratio and light speed are usually used in PIC simulation models, we also change these parameters to analyze the characteristics of the separatrix region.It is found that the increase in the ion-to-electron mass ratio makes the electron depletion layer and the parallel electric field more obvious, while the influence of light speed is less pronounced.     

Wave front interaction model of stabilized propagating wave segments

A wave front interaction model is developed to describe the relationship between excitability and the size and shape of stabilized wave segments in a broad class of weakly excitable media. These wave segments of finite size are unstable but can be stabilized by feedback to the medium excitability; they define a separatrix between spiral wave behavior and contracting wave segments. Unbounded wave segments (critical fingers) lie on the asymptote of this separatrix, defining the boundary between excitable and subexcitable media. The model predictions are compared with results from numerical simulations.     

Fine structure of splitting of the separatrix of a nonlinear resonance

This report is a continuation of an analysis, initiated elsewhere V.V. Vecheslavov and B. V. Chirikov, Zh. éksp. Teor. Fiz. 114, 1516 (1998) [JETP 86, 823 (1998)], of the effect of splitting of the separatrix of a nonlinear resonance for the model of standard mapping, based on results of direct measurements of the splitting angle α(K), where K is the system parameter. Measurements were made in the previously used wide range 0.1≳α≳10⁻²⁰⁸ (1⩾K⩾0.0004), but with significantly higher relative (better than 10⁵⁰) and average (∼10⁻⁵⁵) accuracy. This procedure made it possible to substantially refine the effects observed in Ref. 1 and construct qualitatively new empirical dependences providing reliable extrapolation of the data obtained for the angle and the invariant in the intermediate asymptotic limit K≲10⁻² beyond the limits of the investigated region. The results obtained by us can be useful for further development of the theory of separatrix splitting and formation of the stochastic layer of a nonlinear resonance. Zh. éksp. Teor. Fiz. 116, 336–346 (July 1999)     

Probability phenomena due to separatrix crossing

The effect of a small or slow perturbation on a Hamiltonian system with one degree of freedom is considered. It is assumed that the phase portrait ("phase plane") of the unperturbed system is divided by separatrices into several regions and that under the action of the perturbations phase points can cross these separatrices. The probabilistic phenomena are described that arise due to these separatrix crossings, including the scattering of trajectories, random jumps in the values of adiabatic invariants, and adiabatic chaos. These phenomena occur both in idealized problems in classical mechanics and in real physical systems in planetary science and plasma physics contexts.     

Experimental investigation of the role of fluid turbulent stresses and edge plasma flows for intrinsic rotation generation in DIII-D H-mode plasmas

The first measurements of turbulent stresses and flows inside the separatrix of a tokamak H-mode plasma are reported, using a reciprocating multitip Langmuir probe at the DIII-D tokamak. A strong co-current rotation layer at the separatrix is found to precede intrinsic rotation development in the core. The measured fluid turbulent stresses transport toroidal momentum outward against the velocity gradient and thus try to sustain the edge layer. However, large kinetic stresses must exist to explain the net inward momentum transport leading to co-current core plasma rotation. The importance of such kinetic stresses is corroborated by the success of a simple orbit loss model, representing a purely kinetic mechanism, in the prediction of features of the edge corotation layer.     

Chaos healing by separatrix disappearance and quasisingle helicity states of the reversed field pinch

The resilience to chaotic perturbations of one-parameter one-degree-of-freedom Hamiltonian dynamics is shown to increase when its corresponding separatrix vanishes due to a saddle-node bifurcation. This is first highlighted for the magnetic chaos related to quasisingle helicity (QSH) states of the reversed field pinch. It provides a rationale for the confinement improvement of helical structures experimentally found for QSH plasmas; such a feature would not be expected from the classical resonance overlap picture as the separatrix disappearance occurs when the amplitude of the dominant mode increases.     

Transition to chaos in the conservative four-wave parametric interactions

In this work, we analyze the transition from regular to chaotic states in the parametric four-wave interactions. The temporal evolution describing the coupling of two sets of three-waves with quadratic nonlinearity is considered. This system is shown to undergo a chaotic transition via the separatrix chaos scenario, where a soliton-like solution (separatrix) that is found for the integrable (perfect matched) case becomes irregular as a small mismatch is turned on. As the mismatch is increased the separatrix chaotic layer spreads along the phase space, eventually engrossing most part of it. This scenario is typical of low-dimensional Hamiltonian systems.     

Separatrix conservation mechanism for nonlinear resonance in strong chaos

We propose a simple, approximate theory of the fairly general mechanism for the separatrix conservation of nonlinear resonance, which leads to the complete suppression of global diffusion despite the strong local chaos of motion. This theory allows the separatrix splitting angle to be plotted against system parameters and, in particular, yields their values at which the separatrix remains unsplit. We present the results of our numerical experiments confirming theoretical conclusions for a certain class of dynamical Hamiltonian systems. New features of chaos suppression have been found in such systems. In conclusion, we discuss the range of application of the proposed theory.     

The width of the exponentially narrow stochastic layers

Exponentially small splitting of the separatrix has been calculated for a high frequency large amplitude perturbation and the correspondent correction to the width of the stochastic layer is obtained. The result can be applied to the large amplitude perturbation.     

Detecting level crossings without looking at the spectrum

In many physical systems it is important to be aware of the crossings and avoided crossings which occur when eigenvalues of a physical observable are varied using an external parameter. We have discovered a powerful algebraic method of finding such crossings via a mapping to the problem of locating the roots of a polynomial in that parameter. We demonstrate our method on atoms and molecules in a magnetic field, where it has implications in the search for Feshbach resonances. In the atomic case our method allows us to point out a new class of invariants of the Breit-Rabi Hamiltonian of magnetic resonance. In the case of molecules, it enables us to find curve crossings with practically no knowledge of the corresponding Born-Oppenheimer potentials.     

On the estimate of the stochastic layer width for a model of tracer dynamics

An analytical estimate of the width of the generated chaotic layer in a time-periodically driven stream function model for the motion of passive tracers is discussed. It is based essentially on the method of the separatrix map and the use of the Melnikov theory. Energy-time variables are used to derive lower bounds for the half width of the layer. In order to perform a comparison with numerical simulations, the results are transformed into space variables. The analytic results of the layer thickness in both parallel and perpendicular directions to the shear flow are compared with numerical computations and some systematic deviations are discussed.     

Formation of inner structure of a reconnection separatrix region

We present multipoint spacecraft observations at the dayside magnetopause of a magnetic reconnection separatrix region. This region separates two plasmas with significantly different temperatures and densities, at a large distance from the X line. We identify which terms in the generalized Ohm's law balance the observed electric field throughout the separatrix region. The electric field inside a thin approximately c/omega pi Hall layer is balanced by the j x B/ne term while other terms dominate elsewhere. On the low density side of the region we observe a density cavity which forms due to the escape of magnetospheric electrons along the newly opened field lines. The perpendicular electric field inside the cavity constitutes a potential jump of several kV. The observed potential jump and field aligned currents can be responsible for strong aurora.     

Diffusive transport enhancement and escape processes in frictionless nonlinear oscillators with noise

The time-dependent escape rates and evolution of a distribution density are considered for a Hamiltonian many-dimensional nonlinear oscillator with external noise. The Hamiltonian dynamics is assumed to be nearly integrable and is described in terms of isolated nonlinear resonances. In case of a small angle between the resonant oscillations and the resonance line, a dynamic enhancement of diffusion occurs inside the separatrix, leading to a strongly enhanced growth of distribution tails and escape rates even when the resonances are relatively narow. The underlying mechanism of the phenomenon is essentially many-dimensional.