Separatrix Crossings Research Articles

Breakdown of autoresonance due to separatrix crossing in dissipative systems: From Josephson junctions to the three-wave problem

Optimal energy amplification via autoresonance in dissipative systems subjected to separatrix crossings is discussed through the universal model of a damped driven pendulum. Analytical expressions of the autoresonance responses and forces as well as the associated adiabatic invariants for the phase space regions separated by the underlying separatrix are derived from the energy-based theory of autoresonance. Additionally, applications to a single Josephson junction, topological solitons in Frenkel-Kontorova chains, as well as to the three-wave problem in dissipative media are discussed in detail from the autoresonance analysis.

Periodic orbits and stability islands in chaotic seas created by separatrix crossings in slow-fast systems

We consider a 2 d.o.f. natural Hamiltonian system with one degree of freedomcorresponding to fast motion and the other one corresponding to slow motion. The Hamiltonianfunction is the sum of potential and kinetic energies, the kinetic energy being a weighted sumof squared momenta. The ratio of time derivatives of slow and fast variables is of order$\epsilon $«$ 1$. At frozen values of the slow variables there is a separatrix on the phaseplane of the fast variables and there is a region in the phase space (the domain ofseparatrix crossings) where the projections of phase points onto the plane of the fastvariables repeatedly cross the separatrix in the process of evolution of the slowvariables. Under a certain symmetry condition we prove the existence of many, of order$1/\epsilon$, stable periodic trajectories in the domain of the separatrix crossings. Each ofthese trajectories is surrounded by a stability island whose measure is estimated frombelow by a value of order $\epsilon$. Thus, the total measure of the stability islands isestimated from below by a value independent of $\epsilon$. We find the location of stableperiodic trajectories and an asymptotic formula for the number of these trajectories. Asan example, we consider the problem of motion of a charged particle in the parabolicmodel of magnetic field in the Earth magnetotail.

Stability islands in domains of separatrix crossings in slow-fast Hamiltonian systems

We consider a two-degrees-of-freedom Hamiltonian system with one degree of freedom corresponding to fast motion and the other corresponding to slow motion. The ratio of typical velocities of changes of the slow and fast variables is the small parameter ɛ of the problem. At frozen values of the slow variables, there is a separatrix on the phase plane of the fast variables, and there is a region in the phase space (the domain of separatrix crossings) where the projections of phase points onto the plane of the fast variables repeatedly cross the separatrix in the process of evolution of the slow variables. Under a certain symmetry condition, we prove the existence of many (of order 1/ɛ) stable periodic trajectories in the domain of separatrix crossings. Each of these trajectories is surrounded by a stability island whose measure is estimated from below by a value of order ɛ. So, the total measure of the stability islands is estimated from below by a value independent of ɛ. The proof is based on an analysis of asymptotic formulas for the corresponding Poincaré map.

On the absence of stable periodic orbits in domains of separatrix crossings in nonsymmetric slow-fast Hamiltonian systems

We consider a two degree of freedom Hamiltonian system with one degree of freedom corresponding to fast motion and the other corresponding to slow motion. We assume that at frozen values of the slow variables there is a separatrix on the phase plane of the fast variables and there is a region in the phase space (the domain of separatrix crossings) where projections of phase points onto the plane of the fast variables repeatedly cross the separatrix in the process of evolution of the slow variables. Under rather general conditions, we prove that there are no stable periodic trajectories of any prescribed period inside the domain of separatrix crossings, except maybe for periodic trajectories passing anomalously close to the saddle point.

Universality in nonadiabatic behavior of classical actions in nonlinear models of Bose-Einstein condensates

We discuss the dynamics of approximate adiabatic invariants in several nonlinear models being related to the physics of Bose-Einstein condensates (BECs). We show that the nonadiabatic dynamics in Feshbach resonance passage, nonlinear Landau-Zener (NLZ) tunneling, and BEC tunneling oscillations in a double well can be considered within a unifying approach based on the theory of separatrix crossings. The separatrix crossing theory was applied previously to some problems of classical mechanics, plasma physics, and hydrodynamics, but has not been used in the rapidly growing BEC-related field yet. We derive explicit formulas for the change in the action in several models. Extensive numerical calculations support the theory and demonstrate its universal character. We also discovered a nonlinear phenomenon in the NLZ model which we propose to call separated adiabatic tunneling.

Understanding the behavior of Prometheus and Pandora

We revisit the dynamics of Prometheus and Pandora, two small moons flanking Saturn's F ring. Departures of their orbits from freely precessing ellipses result from mutual interactions via their 121:118 mean motion resonance. Motions are chaotic because the resonance is split into four overlapping components. Orbital longitudes were observed to drift away from predictions based on Voyager ephemerides. A sudden jump in mean motions took place close to the time at which the orbits' apses were antialigned in 2000. Numerical integrations reproduce both the longitude drifts and the jumps. The latter have been attributed to the greater strength of interactions near apse antialignment (every 6.2 yr), and it has been assumed that this drift-jump behavior will continue indefinitely. We re-examine the dynamics of the Prometheus–Pandora system by analogy with that of a nearly adiabatic, parametric pendulum. In terms of this analogy, the current value of the action of the satellite system is close to its maximum in the chaotic zone. Consequently, at present, the two separatrix crossings per precessional cycle occur close to apse antialignment. In this state libration only occurs when the potential's amplitude is nearly maximal, and the “jumps” in mean motion arise during the short intervals of libration that separate long stretches of circulation. Because chaotic systems explore the entire region of phase space available to them, we expect that at other times the Prometheus–Pandora system would be found in states of medium or low action. In a low action state it would spend most of the time in libration, and separatrix crossings would occur near apse alignment. We predict that transitions between these different states can happen in as little as a decade. Therefore, it is incorrect to assume that sudden changes in the orbits only happen near apse antialignment.

Phase change between separatrix crossings in slow–fast Hamiltonian systems

We consider a Hamiltonian system with slow and fast motions, one degree of freedom corresponding to fast motion, and the other degrees of freedom corresponding to slow motion. Suppose that at frozen values of the slow variables there is a non-degenerate saddle point and a separatrix on the phase plane of the fast variables. In the process of variation of the slow variables, the projection of a phase trajectory onto the phase plane of the fast variables may repeatedly cross the separatrix. These crossings are described by the crossing parameter called the pseudo-phase. We obtain an asymptotic formula for the pseudo-phase dependence on the initial conditions, and calculate the change of the pseudo-phase between two subsequent separatrix crossings.

Resonances and Particle Stochastization in Nonhomogeneous Electromagnetic Fields

In the present paper we investigate the resonant interaction between monochromatic electromagnetic waves and charged particles in configurations with magnetic field reversals (e.g., in the earth magnetotail). The smallness of certain physical parameters allows us to solve this problem using perturbation theory, reducing the problem of resonant wave–particle interaction to the analysis of slow passages of a particle through a resonance. We discuss in detail two of the most important resonant phenomena: capture into resonance and scattering on resonance. We show that these processes result in destruction of the adiabatic invariants and chaotization of particles; they also may lead to significant (almost free) acceleration of particles and may govern transport in the phase space. We calculate the characteristic times of mixing due to resonant effects and separatrix crossings, and discuss the relative importance of these phenomena.

Origin of chaos in the Prometheus–Pandora system

We demonstrate that the chaotic orbits of Prometheus and Pandora are due to interactions associated with the 121:118 mean motion resonance. Differential precession splits this resonance into a quartet of components equally spaced in frequency. Libration widths of the individual components exceed the splitting, resulting in resonance overlap which causes the chaos. Mean motions of Prometheus and Pandora wander chaotically in zones of width 1.8 and 3.1 deg yr −1, respectively. A model with 1.5 degrees of freedom captures the essential features of the chaotic dynamics. We use it to show that the Lyapunov exponent of 0.3 yr −1 arises because the critical argument of the dominant member of the resonant quartet makes approximately two separatrix crossings every 6.2 year precessional cycle.

Geometric and statistical properties induced by separatrix crossings in volume-preserving systems

We consider a three-dimensional volume-preserving system inside the unit sphere. The system is a small perturbation of an integrable one. Almost all phase trajectories of the integrable system are closed curves. Under an arbitrarily small non-zero perturbation all the interior of the sphere, up to a residue of small measure, is apparently a domain of chaotic motion. The phenomenon is described as a result of jumps in an adiabatic invariant of the system occurring when a phase trajectory crosses the two-dimensional separatrix of the unperturbed (integrable) system. The dynamical properties can be understood as a consequence of splitting of invariant manifolds under the perturbation. A two-dimensional return map generated by the system possesses strong stretching properties. This makes the dynamics of the system close to hyperbolic. Numerical investigation of the statistical properties of the system demonstrates a good agreement with theoretical predictions made under the assumption that the system has a large ergodic component whose measure tends to the measure of the sphere's interior as the perturbation tends to zero. However, the system is not ergodic, and stable periodic solutions are found, surrounded by stability islands of measure exponentially small with the perturbation.

Aspiration-Based and Reciprocity-Based Rules in Learning Dynamics for Symmetric Normal-Form Games

Psychologically based rules are important in human behavior and have the potential of explaining equilibrium selection and separatrix crossings to a payoff dominant equilibrium in coordination games. We show how a rule learning theory can easily accommodate behavioral rules such as aspiration-based experimentation and reciprocity-based cooperation and how to test for the significance of additional rules. We confront this enhanced rule learning model with experimental data on games with multiple equilibria and separatrix-crossing behavior. Maximum likelihood results do not support aspiration-based experimentation or anticipated reciprocity as significant explanatory factors, but do support a small propensity for non-aspiration-based experimentation by random belief and non-reciprocity-based cooperation.

Long-time discrete particle effects versus kinetic theory in the self-consistent single-wave model

The influence of the finite number N of particles coupled to a monochromatic wave in a collisionless plasma is investigated. For growth as well as damping of the wave, discrete particle numerical simulations show an N-dependent long time behavior resulting from the dynamics of individual particles. This behavior differs from the one due to the numerical errors incurred by Vlasov approaches. Trapping oscillations are crucial to long time dynamics, as the wave oscillations are controlled by the particle distribution inhomogeneities and the pulsating separatrix crossings drive the relaxation towards thermal equilibrium.

Change of the adiabatic invariant at a separatrix in avolume-preserving 3D system

A 3D volume-preserving system is considered. The system differs by a small perturbation from an integrable one. In the phase space of the unperturbed system there are regions filled with closed phase trajectories, where the system has two independent first integrals. These regions are separated by a 2D separatrix passing through non-degenerate singular points. Far from the separatrix, the perturbed system has an adiabatic invariant. When a perturbed phase trajectory crosses the two-dimensional separatrix of the unperturbed system, this adiabatic invariant undergoes a quasi-random jump. The formula for this jump is obtained. If the geometry of the system allows for multiple separatrix crossings, the destruction of adiabatic invariance is possible, leading to chaotic behaviour in the system. An example of such a system is given.

E×B guiding-centre motion in three electrostatic waves: statistics of separatrix crossings

E×B guiding-centre (GC) motion in a special configuration of three low-frequency electrostatic waves can be considered as a paradigmatic Hamiltonian system for studying adiabatic motion and separatrix crossings. A peculiarity of this system is that a single initial condition gives rise to two stroboscopic phase-space trajectories. According to the classical Hamiltonian theory, the proportion of points on the stroboscopic trajectories is a function of the time evolution of the surfaces enclosed by the separatrices in the phase space. This behaviour is qualitatively observed in test-particle numerical experiments. The ability of numerical integration methods like the ‘classical’ fourth-order Runge–Kutta integration scheme or a third-order symplectic integrator to reproduce the statistics is analysed.

Chaotic advection in a cubic Stokes flow

A stationary incompressible Stokes flow in a sphere is considered. The flow was introduced by Stone et al. (1991) as a flow inside a neutrally buoyant spherical drop immersed in a linear flow. The velocity field of the flow is a result of a small perturbation of an integrable velocity field with almost all streamlines closed. Under arbitrarily small pertubation a large domain of chaotic advection within the sphere arises. This phenomenon is explained by quasirandom changes in the adiabatic invariant of the flow, which occur as a streamline crosses the two-dimensional separatrix of the unperturbed flow. Phase portraits of the averaged system are constructed. An asymptotic formula for the change in the adiabatic invariant due to the separatrix crossing is derived. The process of diffusion of the adiabatic invariant due to multiple separatrix crossings is described.

Application of separatrix crossing theory to nondiffusion model of current sheet resonance

The resonance phenomenon in charged particle dynamics in the Earth's magnetotail is investigated in terms of quasiadiabatic theory. The resonant properties of particle motion are found to be related to non‐diffusion chaotic migration and separatrix crossings.

Length scale, quasiperiodicity, resonances, separatrix crossings, and chaos in the weakly relativistic Zakharov equations.

Nonlinear saturation of unstable solutions to the weakly relativistic, one-dimensional Zakharov equations is considered in this paper. In order to perform the analysis, two quantities are introduced. One of them, ${\mathrm{\ensuremath{\rho}}}_{\mathrm{*}}$, is proportional to the initial energy of the high-frequency field, and the other is the basic wave vector of the low-frequency perturbing mode k=2\ensuremath{\pi}/L, with L as the length scale. With these quantities it becomes possible to identify a number of regions on a ${\mathrm{\ensuremath{\rho}}}_{\mathrm{*}}$ versus k parametric plane. For very small values of ${\mathrm{\ensuremath{\rho}}}_{\mathrm{*}}$, steady-state solutions become unstable when k is also very small. In this case ion-acoustic dynamics is found to be unimportant and the system is numerically shown to be approximately integrable, even if k is below a critical value where the solutions are not simply periodic. For larger values of ${\mathrm{\ensuremath{\rho}}}_{\mathrm{*}}$ the unstable wave vectors also become larger and the ion-acoustic fluctuations turn into active dynamical modes of the system, driving a transition to chaos, which follows initial inverse pitchfork bifurcations. The transition includes resonant and quasiperiodic features; separatrix crossing phenomena are also found. The influence of relativistic terms on the chaotic dynamics is studied in the context of the Zakharov equations; it is shown that relativistic terms generally enhance the instabilities of the system, therefore anticipating the transition.

Collisionless diffusive fluxes of locally trapped ions in Tokamaks with rippled magnetic field

The effect of the adiabatic invariant diffusion caused by repeated separatrix crossings, when the ion emerges step by step from and is entrapped by the local magnetic well, on the transport of locally trapped ions in Tokamaks with a rippled magnetic field is considered. The corresponding alpha particle loss rate is comparable with that induced by the stochastic ripple diffusion of banana tips

Regular and chaotic aspects of charged particle motion in a magnetotail‐like field with a neutral line

The motion of charged particles is studied numerically and analytically using a test particle approach. Acceleration near the X‐type neutral line is described in terms of adiabatic invariants and separatrix crossings. The relation between regular and chaotic properties of the process is discussed for a realistic source in the plasma mantle.

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.