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The problem of constructing n-linear (n ≥ 2) plane vector fields with an isolated critical point and given separatrices of prescribed types is considered. Such constructions are based on the use of vector algebra, the qualitative theory of second-order dynamic systems and classical methods for investigating their critical points. This problem is essentially an inverse problem of the qualitative theory of ordinary differential equations, and its solution can be used to synthesize mathematical models of controlled dynamical systems of various physical nature. [ABSTRACT FROM AUTHOR]

In this paper, we discuss a wide family of dynamical systems whose characteristic feature is a polynomial right-hand side containing coprime forms of the phase variables of the system. One of the equations of the system contains a third-degree polynomial (cubic form), the other equation contains a quadratic form. We consider the problem of constructing all possible phase portraits in the Poincaré disk for systems from the family considered and establish criteria for the implementation of each portrait that are close to coefficient criteria. This problem is solved by using the central and orthogonal Poincaré methods of sequential mappings and a number of other methods developed by the authors for the purposes of this study. We obtained rigorous qualitative and quantitative results. More than 250 topologically distinct phase portraits of various systems were constructed. The absence of limit cycles of systems of this family is proved. Methods developed can be useful for the further study of systems with polynomial right-hand sides of other forms. [ABSTRACT FROM AUTHOR]

We fill a gap concerning a dynamical description for a switching mosquito population suppression model proposed in Yu and Li (J Differ Equ 269:6193–6215, 2020), where a constant amount c of sterile mosquitoes is released after a waiting period T larger than the sexual lifespan T ¯ of the released male mosquitoes. The release amount thresholds g ∗ , c ∗ with g ∗ < c ∗ and the waiting period threshold T ∗ were found, and it was proved that the origin is locally asymptotically stable in D = { (c , T) : g ∗ < c < c ∗ , T < T ∗ } . However, the periodic solutions as well as the global asymptotical stability of the origin remains unknown. By ingeniously finding a useful separatrix L which can divide D into two sub-regions D 1 and D 2 , we show that the origin is globally asymptotically stable in D 1 , and the model admits exactly two periodic solutions in D 2 , with one stable, and the other unstable, and a unique periodic solution on L, which is semi-stable, respectively. Numerical examples to illustrate our theoretical results are also provided. [ABSTRACT FROM AUTHOR]

When fluid motion is induced through a porous medium, the distance travelled depends on the energy exerted and overall resistance to flow. Ostensibly intuitive, this concept of reach has proven elusive not only in delineation, but even mathematical definition. In wells surrounded by a homogeneous medium it is a circle and thus characterised by a single number referred to as the radius of influence. The simple definition thereof as the locus of zero radial velocity yields an infinite or severely overestimated result, and tempts custom modifications to reconcile the estimates with reality. In environmental applications such as aquifer sparging wells, hydraulic and petroleum wells, natural gas and landfill gas wells, the medium is often highly heterogeneous. The analysis presented herein shows that the zone of collection or dispersal changes shape dramatically based on the interrelation of resistance to flow posed by subdomains of distinct permeabilities. In contrast to the isotropic case, the mere presence of heterogeneity suggests a natural and unambiguous definition of this zone of interest as a set of separatrices connecting stagnation points in the flow field, and results in realistic estimates. The variability of its shape is remarkable and relates to the structural diversity of the parameter space underpinning the flow field. The findings help explain the longstanding problem of a reliable delimitation of these wells’ reach in practice. [ABSTRACT FROM AUTHOR]

The Lotka–Volterra model of competition has been studied by numerical simulations using the Runge–Kutta–Fehlberg algorithm. The stable fixed points, unstable fixed point, saddle node, basins of attraction, and the separatices are found. The transient behaviors associated with reaching the stable fixed point are studied systematically. It is observed that the time of reaching the stable fixed point in any one of the basins of attraction depends strongly on the initial distance from the separatrix. As the initial point approached the separatrix, this time was found to diverge logarithmically. The divergence of the time, required to reach the stable fixed point, indicates the critical slowing down near the critical point in equilibrium phase transition. A metastable behavior was also observed near the saddle fixed point before reaching the stable fixed point. [ABSTRACT FROM AUTHOR]

In this paper, blow-up solutions of autonomous ordinary differential equations (ODEs) which are unstable under perturbations of initial points, referred to as saddle-type blow-up solutions, are studied. Combining dynamical systems machinery (e.g., compactifications, timescale desingularizations of vector fields) with tools from computer-assisted proofs (e.g., rigorous integrators, the parameterization method for invariant manifolds), these blow-up solutions are obtained as trajectories on local stable manifolds of hyperbolic saddle equilibria at infinity. With the help of computer-assisted proofs, global trajectories on stable manifolds, inducing blow-up solutions, provide a global picture organized by global-in-time solutions and blow-up solutions simultaneously. Using the proposed methodology, intrinsic features of saddle-type blow-ups are observed: locally smooth dependence of blow-up times on initial points, level set distribution of blow-up times and decomposition of the phase space playing a role as separatrixes among solutions, where the magnitude of initial points near those blow-ups does not matter for asymptotic behavior. Finally, singular behavior of blow-up times on initial points belonging to different family of blow-up solutions is addressed. [ABSTRACT FROM AUTHOR]

Many vibrating systems, over some ranges of parameter values, exhibit a single unstable mode. Adding a small resonant secondary system to the unstable system is a well-known stabilization strategy. Here we show that even a nonresonant secondary system, if equipped with a limit cycle of its own, can stabilize the unstable mode of the primary system. The primary system is modeled here as a linear spring block system with negative damping. The secondary system is a van der Pol oscillator. Smallness of the latter's parameters allows use of the method of multiple scales. The resulting slow amplitude equations decouple from the phases and a two-dimensional system is obtained. The secondary system's amplitude evolves faster than that of the primary system, which simplifies analysis. A parameter-dependent transformation casts the system in a canonical form with a single free parameter c 1 > 0 in addition to the small perturbation parameter. The canonical phase portrait involves two key straight lines. When c 1 < 4 these lines intersect and a separatrix passes through that intersection. Solutions on one side of the separatrix show quenching of the primary instability with limit cycle oscillation of the secondary system. Solutions on the other side of the separatrix show significant oscillations of the primary system at its natural frequency, with the secondary limit cycle being quenched. When c 1 > 4 , stabilization fails for all initial conditions. In summary, for the case of a negatively damped oscillator interacting with a small nonresonant secondary limit cycle oscillator, we show stabilization, provide a pair of canonical equations with one free parameter, and present a complete qualitative characterization of the dynamics. [ABSTRACT FROM AUTHOR]

The theta aurora is reproduced by global simulation. First, we construct a solution for the stationary northward interplanetary magnetic field (IMF) forming the separatrices, the separators, the nulls, and the stemlines. From the drawing of last‐closed field lines, the overall structure under this condition is summarized as the northern lobe is generated by a separatrix emanating from the southern null. In this paper, all variations are antisymmetric in the southern hemisphere. Afterward, the IMF By is switched to reproduce the theta aurora. The ionospheric theta aurora is reproduced as closed magnetic field regions. The polar cap is divided to old and new parts, by the theta bar. In the magnetosphere, two dayside nulls occur corresponding to the new IMF and two nulls corresponding to the old IMF retreat tailward. The four nulls form a structure connected by four separators, constructing the magnetospheric topology corresponding to the theta aurora. In this topology, old and new nulls in the southern hemisphere generate old and new lobes in the northern hemisphere. Each lobe is projected onto northern old and new polar caps. The origin of the theta bar is the stagnating closed magnetic field region that occurs between old and new lobes. Separator reconnection occurs between the old lobe in the southern and the new lobe in the northern hemispheres, reducing southern old polar caps. This is the cause of the movement of the theta bar. The theta aurora is the phenomenon that demonstrates the existence of the null‐separator structure. Key Points: The theta aurora is an image of the magnetospheric null‐separator structure projected to the virtual mirror in the ionosphereNorthern new and old lobes are respectively generated from new dayside and old retreating nulls in the southern hemisphereThe theta bar is the projection of stagnating closed magnetic field regions accumulating at the interface between old and new lobes [ABSTRACT FROM AUTHOR]

A Leslie-Gower type predator-prey model including group defense formation is analyzed. This phenomenon, described by a non-monotonic function originates interesting dynamics; positiveness, boundedness, permanence of solutions, and existence of up to three positive equilibria are established. The solutions are highly sensitive to initial conditions since there exists a separatrix curve dividing their behavior. Two near trajectories can have far omega-limit sets. The weakness of a singularity is established showing two limit cycles can exist. Numerical simulations endorse the analytical outcomes. [ABSTRACT FROM AUTHOR]

We construct new substantive examples of nonautonomous vector fields on a -dimensional sphere having simple dynamics but nontrivial topology. The construction is based on two ideas : the theory of diffeomorphisms with wild separatrix embedding and the construction of a nonautonomous suspension over a diffeomorphism. As a result, we obtain periodic, almost periodic, or even nonrecurrent vector fields that have a finite number of special integral curves possessing exponential dichotomy on such that among them there is one saddle integral curve (with a dichotomy type) with a wildly embedded -dimensional unstable separatrix and a wildly embedded -dimensional stable manifold. All other integral curves tend to these special integral curves as . We also construct other vector fields having special saddle integral curves with the tamely embedded -dimensional unstable separatrices forming mildly wild frames in the sense of Debrunner–Fox. In the case of periodic vector fields, the corresponding specific integral curves are periodic with the period of the vector field, and are almost periodic in the case of an almost periodic vector field. [ABSTRACT FROM AUTHOR]

In this paper, by performing two‐dimensional (2‐D) particle‐in‐cell simulations of collisionless magnetic reconnection in a Harris current sheet, we analyze the formation of electron inflow along the separatrices toward the X‐line from the perspective of fluid. The results show that both the parallel electric field and mirror force can drive the electron inflow to stream toward the X‐line. The speed of the electron inflow can reach about 4VA(where VA is the Alfvén speed based on the upstream asymptotic magnetic field magnitude and peak density in the current sheet). Although the contribution of the parallel electric field is much larger than that of the mirror force, the mirror force cannot be ignored. When the electron inflow streams toward the X‐line, it is heated in the parallel direction. The resulted gradient of the parallel electron pressure leads to the attenuation of the electron inflow. We also investigate the effects of the density and temperature of the background plasma on the formation of the electron inflow. With the decrease of the background plasma density, the contribution of the parallel electric field becomes larger while that of the mirror force almost remains unchanged. With the decrease of the background plasma temperature, the contribution of the parallel electric field becomes larger while that of the mirror force becomes smaller. Plain Language Summary: Magnetic reconnection is a universal process in which the topology of the magnetic field changes, accompanied by plasma heating and particle acceleration. During reconnection, electrons moving along the separatrices toward the X‐line form the electron inflow. The electron inflow has been confirmed in simulations and observations, but its generation mechanism remains unclear. This paper focuses on the dynamics of inflow electrons along the separatrices during magnetic reconnection, and quantitatively analyzes the formation of electron inflow. We found that the formation of electron inflow is contributed by three effects: accelerated by the parallel electric field, accelerated by the magnetic mirror force, and decelerated by the gradient of parallel electron pressure. The acceleration contribution of the parallel electric field is dominant, but that of mirror force cannot be ignored, especially when the background plasma density and temperature are sufficiently high. Key Points: Electron inflow is accelerated by parallel electric field and mirror force and decelerated by the gradient of parallel electron pressureParallel electric field plays a major role in the formation of electron inflow, while the contribution of mirror force cannot be ignoredThe contributions of parallel electric field and mirror force change with the background plasma density and temperature [ABSTRACT FROM AUTHOR]

The problem of inhibition of horseshoe chaos in a nonlinear hysteretic systems using negative stiffness is investigated in this paper. The Bouc–Wen model is used to describe the force produced by both the purely hysteretic and linear elastic springs. The analytical investigation of the Hamiltonian shows that the appearance of separatrix in the system is directly related to the parameters of the hysteretic forces. This means that the transverse intersection between the perturbed and unperturbed separatrix can be controlled according to the shape parameters of the hysteretic model. [ABSTRACT FROM AUTHOR]

Let F be a germ of holomorphic foliation defined in a neighborhood of the origin of C° that has a germ of irreducible holomorphic invariant curve γ. We provide a lower bound for the vanishing multiplicity of F at the origin in terms of the equisingularity class of γ. Moreover, we show that such a lower bound is sharp. Finally, we characterize the types of dicritical singularities for which the multiplicity of F can be bounded in terms of that of γ, and provide an explicit bound in this case. [ABSTRACT FROM AUTHOR]

We consider a C 1 vector field X defined on an open subset U of the plane with compact closure. If X has no singular points and if U is simply connected, a weak version of the Poincaré–Bendixson theorem says that the limit sets of X in U are empty but that one can define non empty extended limit sets contained in the boundary of U. We give an elementary proof of this result, independent of the classical Poincaré–Bendixson theorem. A trapping triangle T based at p, for a C 1 vector field X defined on an open subset U of the plane, is a topological triangle with a corner at a point p located on the boundary ∂ U and a good control of the tranversality of X along the sides. The principal application of the weak Poincaré–Bendixson theorem is that a trapping triangle at p contains a separatrix converging toward the point p. This does not depend on the properties of X along ∂ U . For instance, X could be non differentiable at p, as in the example presented in the last section. [ABSTRACT FROM AUTHOR]

Waves driven by electron beams are frequently observed within the narrow separatrices of magnetic reconnection. Strong plasma instabilities can grow from mixing between the reconnection inflow and outflow, resulting in electron heating which ultimately influences the reconnection process. Observations of a separatrix encounter by the Magnetospheric MultiScale mission on July 11, 2017 feature an anti‐parallel electron beam accelerated in a sub‐ion‐scale layer containing high amplitude (>50 mV/m) electrostatic waves. In two consecutive separatrix crossings, one exhibits faster electron beams, greater magnetic shear, higher amplitude waves, and more electron heating than the other. The observations suggest that variable reconnection outflow pressure is associated with these changes. Coincident with lower hybrid waves are Buneman and beam‐mode instabilities, likely responsible for parallel heating of electrons. Dissipation (J⋅E) within the mixing layer is found to be negative where wave activity is strongest, indicating a slowing of electrons and growth of electromagnetic fields likely related to local thinning of the separatrix. Calculation of momentum balance via the generalized Ohm's law indicates an overall imbalance of >40 mV/m in the more extreme separatrix crossing, requiring a significant dissipative term or error in the electron inertial term to make up the remainder. Finally, an analysis of quasi‐viscous heating shows that active heating of electrons is more prevalent in the presence of a high speed electron inflow and increased magnetic shear. Plain Language Summary: Magnetic reconnection is a process which converts magnetic energy into kinetic energy of charged particles. Particles flowing into a region of reconnection are separated from those in the outflow by a narrow region called the separatrix. We look at a reconnection event observed by the Magnetospheric Multiscale mission where the separatrix appears to be compressed over time. When in the compressed state, violent wave activity is observed which enables mixing between inflowing and outflowing plasma. These waves heat electrons in the inflow, partially counteracting the thinning of the separatrix layer. Detailed analysis reveals evidence for other small scale, non‐ideal processes which may be responsible for acceleration of the inflow electrons as well. Since the inflow feeds in to the reconnection site, electron heating and acceleration by compression may act as a feedback mechanism for the reconnection process. Key Points: A rapidly evolving separatrix exhibits thinning, increased shear, and enhanced electron beams associated with changes in the ion outflowEvaluation of force balance shows contributions from time variation, electron inertia, or the resistive term may be significantQuasi‐viscous effects qualitatively agree with predictions and are largest in the region of increased shear and enhanced inflow [ABSTRACT FROM AUTHOR]

An ideal triangulation of a singular flat surface is a geodesic triangulation such that its vertex set is equal to the set of singular points of the surface. Using the fact that each pair of points in a surface has a finite number of geodesics having length ≤ L connecting them, where L is any positive number, we prove that each singular flat surface has an ideal triangulation provided that the surface has singular points when it has no boundary components, or each of its boundary components has a singular point. Also, we prove that such a surface contains a finite number of geodesics which connect its singular points so that when we cut the surface through these arcs we get a flat disc with a nonsingular interior. [ABSTRACT FROM AUTHOR]

Invariant manifolds are important sets arising in the stability theory of dynamical systems. In this article, we take a brief review of invariant sets. We provide some results regarding the existence of invariant lines and parabolas in planar polynomial systems. We provide the conditions for the invariance of linear subspaces in fractional-order systems. Further, we provide an important result showing the nonexistence of invariant manifolds (other than linear subspaces) in fractional-order systems. [ABSTRACT FROM AUTHOR]

Midplane separatrix density is a crucial parameter in tokamaks since it strongly impacts divertor conditions. Scaling midplane separatrix density, ne, SEP, and pedestal density, ne, PED, as function of engineering parameters such as auxiliary heating Pinjected, toroidal magnetic field BT, and plasma poloidal current Ip are relevant to observe the effect of tuning these parameters on, for example, quality of confinement and divertor regime governed by ne, PED and ne, SEP, respectively. Thus, a dataset of JET H‐mode pulses performed with Iter like wall (ILW) has been analysed. Midplane density data are collected from an HRTS (high‐resolution Thomson scattering) diagnostic and ne, SEP is determined using the power balance method. Parallel heat flux model is chosen using transport code SOLEDGE2D (S2D) applying power balance method over a simulated ne, SEP and Te, SEP profiles to obtain separatrix positions. The parameters are averaged over time windows with order of (85–185 ms) and the magnetic configuration has been fixed to avoid divertor geometrical effect on ne, SEP determination, configuration chosen is corner–corner. A ratio between separatrix density and pedestal density at outer midplane ranges between 0.3 and 0.7 on the data set. A scaling law of ne, SEP/ne, PED is obtained as function of Pinjected, BT, and IP. [ABSTRACT FROM AUTHOR]

Interactions between magnetic reconnection inflows and outflows can result in a violent mixing process. In Magnetospheric MultiScale observations of asymmetric, low guide‐field reconnection, highly sheared electron flow paired with sharp density and temperature gradients have been found in association with bursts of strong (≥100 mV/m) electric fields parallel to the ambient magnetic field. It is likely that large spikes in E‖ are part of a dynamic, small‐scale structure which results from mixing between plasmas. In this study, a 1‐D Vlasov simulation with parameters directly comparable to the observed plasma environment and interaction timescale is used to demonstrate that mixing at a sharp boundary between magnetospheric and magnetosheath electrons is qualitatively consistent with measured particle distributions and signatures in E‖. Properties of mixing structures such as net electric potential are estimated and found capable of accelerating electron beams toward the electron diffusion region but are not necessarily sufficient to generate the strongest observed jets. Obliquely propagating lower hybrid drift waves are also present and likely provide most of the energy for acceleration. Drift waves may be responsible for cross‐field transport required to begin the mixing process. We conclude that parallel mixing primarily acts to mediate plasma boundaries, thermalizing electron beams contributing to the high anisotropy (Te‖>Te⊥) electron distributions found in the dayside reconnection magnetospheric inflow region. Key Points: Violent mixing between plasmas is observed within an electron‐scale boundary along the dayside magnetosphere reconnection separatrixConsecutive negative spikes in parallel electric field can be explained by a single oscillating double‐layer‐like structureElectrostatic mixing signatures likely contribute primarily to thermalization and heating rather than linear acceleration toward the X line [ABSTRACT FROM AUTHOR]

The separatrix region is the region between the separatrix and the reconnection jet. Due to the E×B drift and velocity filter effect in which high‐energy particles with high parallel speed can be seen prior to low‐energy particles along the field line, electrons are separated from ions. The electron dynamics in this region is of interest; however it has not been studied in detail, because of the insufficient resolution of plasma data. We present a slow separatrix crossing event observed by Magnetospheric Multiscale (MMS) satellite constellation on 1 January 2016, from the magnetosheath side with high‐resolution burst mode data. The electron edge and ion edge are clearly distinguished in the separatrix region. Two types of electron dispersion, one with a short duration (~0.3 s) and the other with a longer duration (~13 s) were detected between the electron and ion edges. The rapid dispersion (with small time scale) is mainly in the parallel direction, which might originate from a thin layer with non‐frozen‐in electrons close to the separatrix. The gradual (long time scale) dispersion is seen from parallel to perpendicular directions, which comes from the E×B drift of a curved D‐shape distribution of electrons. The width of the electron diffusion region on the magnetosheath side is estimated based on MMS observation. The observation also reveals an unexpected parallel electron beam outside of the electron edge. Wave‐particle interaction or parallel potential in the inflow region may be responsible for the generation of this electron population. Plain Language Summary: Magnetic reconnection is a process commonly occurred at the Sun, the dayside magnetopause, and in the magnetotail that changes magnetic topology, and converts electromagnetic energy into plasma kinetic and thermal energies. Due to the different parallel velocities of the particles and field line convection, electrons are separated from ions, and this creates the electron edge and ion edge in the separatrix region. These separations downstream of the X‐line widen the distance in the convection direction between the field lines involved in the reconnection. This, in turn, makes it easier for the satellite to observe the changes in the magnetic field topology and electronic dynamics during magnetic reconnection. However, high‐resolution satellite data are still required for the observation of the separatrix region in the downstream. The unprecedented high temporal and spatial resolution measurements of the Magnetospheric Multiscale (MMS) mission provide a unique opportunity to perform such studies. By analyzing the magnetic field and plasma data from MMS, this study has revealed the field line topology and electron dynamics between the electron and ion edges. Possible generation mechanisms based on the electron dynamics are also discussed in this paper. Key Points: Rapid dispersion is observed at the electron edge, indicating that electrons are decoupled from the magnetic field close to separatrixThe width of the electron diffusion region on the magnetosheath side is estimated using in situ observationsA parallel electron beam is observed outside of the separatrix in the magnetosheath for the first time [ABSTRACT FROM AUTHOR]

In mathematical modeling, it is often required the analysis of the vector field topology in order to predict the evolution of the variables involved. When a dynamical system is multistable, the trajectories approach different stable states, depending on the initial conditions. The aim of this work is the detection of the invariant manifolds of the saddle points to analyze the boundaries of the basins of attraction. Once that a sufficient number of separatrix points is found, a moving least squares meshfree method is involved to reconstruct the separatrix manifolds. Numerical results are presented to assess the method referring to tri‐stable models with complex attractors such as limit cycles or limit tori. [ABSTRACT FROM AUTHOR]

The behavior of systems with fast and slow time scales is organized by families of locally invariant slow manifolds. Recently, numerical methods have been developed for the approximation of attracting and repelling slow manifolds. However, the accurate computation of saddle slow manifolds, which are typical in higher dimensions, is still an active area of research. A saddle slow manifold has associated stable and unstable manifolds that contain both fast and slow dynamics, which makes them challenging to compute. We give a precise definition for the stable manifold of a saddle slow manifold and design an algorithm to compute it; our computational method is formulated as a two-point boundary value problem that is solved by pseudo-arclength continuation with Auto. We explain how this manifold acts as a separatrix and determines the number of spikes in the transient response generated by a stimulus with fixed amplitude and duration in two different models. [ABSTRACT FROM AUTHOR]

On 20 September 2015, the Magnetospheric Multiscale (MMS) spacecraft crossed the dusk magnetopause after a compression of the magnetosphere. Enhanced densities and fluxes of both colder (≤10 eV) and hotter (>1 keV) magnetospheric and magnetosheath heavy ion species were observed reaching the magnetopause. The evolution of the velocity distributions for H+, He+, and O+ measured by the Hot Plasma Composition Analyzer on MMS during this magnetopause crossing is presented. In particular, this study focuses on the changes in the species' distribution functions as MMS passes from the magnetosphere through the electron edge of the low-latitude boundary layer (LLBL) separatrix and then into the LLBL. Two types of processes are suggested to play a role in the heating of colder magnetospheric ions across the LLBL separatrix in the region between the separatrix and the electron and ion edges of the LLBL. One mechanism leads to the formation and enhancement of ring distributions in this layer of the LLBL as the magnetospheric ions propagate across the separatrix. A second mechanism leading first to perpendicular heating and then to parallel heating of colder protons may arise from a possible two-stream instability as the magnetospheric ions first encounter the warmer magnetosheath electrons in the electron layer and then the warmer magnetosheath ions between the electron and ion edges of the LLBL separatrix. Perpendicular heating of He+ and O+ is seen more so in the main reconnection exhaust, due to nonadiabatic behavior of these ions as they are accelerated up to the bulk flow speed. [ABSTRACT FROM AUTHOR]

It has long been suggested that whistler waves play an active role during magnetic reconnection. However, all previous observations were based on case studies and could not give a detailed picture as to where the whistler waves occur in the reconnection region. In this paper, a statistical study by using the Cluster data is performed to investigate the spatial distribution and the occurrence rate of whistler waves in the magnetotail reconnection region. It is found that the occurrence rate of the whistler waves is large in the separatrix region (| Bx/ B0| > 0.4) and in the pileup region (| Bx/ B0| < 0.2, | θ| > 45°, where θ = arctan( Bz/ Bx)), but is very small in the vicinity of the X-line. These statistical results are well consistent with recent kinetic simulations and with previous observational case studies. [ABSTRACT FROM AUTHOR]

Trajectories of magnetic field lines are a 1½ degree of freedom Hamiltonian system. The unperturbed separatrix and the perturbed separatrix in a divertor tokamak are fundamentally different. Magnetic asymmetries cause the separatrix to form extremely complicated structures known as homoclinic tangles. After each toroidal circuit, the perturbed separatrix manifolds meet in a fixed poloidal plane and intersect to form homoclinic tangle in order to preserve the topological invariants. This tangle becomes extremely complicated as the magnetic field lines take more and more toroidal turns. This effect is most pronounced near the X-point. The homoclinic tangles of the DIII-D tokamak separatrix from the magnetic perturbation representing the peeling-ballooning modes are studied. The homoclinic tangles can have important implications for the edge physics in divertor tokamaks. [ABSTRACT FROM PUBLISHER]

We consider a new family of derivatives whose payoffs become strictly positive when the price of their underlying asset falls relative to its historical maximum. We derive the solution to the discretionary stopping problems arising in the context of pricing their perpetual American versions by means of an explicit construction of their value functions. In particular, we fully characterise the free-boundary functions that provide the optimal stopping times of these genuinely two-dimensional problems as the unique solutions to highly nonlinear first order ODEs that have the characteristics of a separatrix. The asymptotic growth of these free-boundary functions can take qualitatively different forms depending on parameter values, which is an interesting new feature. [ABSTRACT FROM AUTHOR]

The paper deals with rotation of gyroscope in Hess' conditions. Motion equations of a solid body are established on the base of Hamiltonian formalism. There are some analytical researches and computer experiments were made on the base of numeral study of phase portrait of equations, which describe gyroscope's motion. The movements of gyroscope, which is submitted to Hess' conditions in the null constant of integral of an area and a light weight of the body, are investigated more detailed. The motion equations and integrals are expressed in variables Andoyer-Deprit. The heteroclinic trajectories of the dynamical system are examined by means of the new canonical variables. [ABSTRACT FROM AUTHOR]

Transport mechanism is critical for understanding natural phenomena in the solar system and is beneficial to space mission design. In this study, transport mechanism in the bicircular four-body problem is numerically explored by using Lagrangian coherent structure (LCS), a tool widely used for identifying transport barriers in fluid flow. First, equations of motion of the bicircular problem are derived and five topology configurations of forbidden region are presented. Then, definition and computational method of LCS are introduced. Finally, properties of LCS which are useful for revealing transport mechanism in the four-body problem are numerically investigated. [ABSTRACT FROM AUTHOR]

As memory density continues to grow in modern systems, accurate analysis of static RAM (SRAM) stability is increasingly important to ensure high yields. Traditional static noise margin metrics fail to capture the dynamic characteristics of SRAM behavior, leading to expensive over design and disastrous under design. One of the central components of more accurate dynamic stability analysis is the separatrix; however, its straightforward extraction is extremely time-consuming, and efficient methods are either nonaccurate or extremely difficult to implement. In this paper, we propose a novel algorithm for fast separatrix tracing of any given SRAM topology, designed with industry standard transistor models in nanoscaled technologies. The proposed algorithm is applied to both standard 6T SRAM bitcells, as well as previously proposed alternative subthreshold bitcells, providing up to three orders-of-magnitude speedup, as compared with brute force methods. In addition, for the first time, statistical Monte Carlo separatrix distributions are plotted. [ABSTRACT FROM PUBLISHER]

Curves dividing the parameter plane into regions according to the presence or absence of homoclinic or heteroclinic tangle corresponding to the periodically perturbed saddle of the piecewise smooth oscillator are studied using Melnikov analysis. The analysis is not simplified by choosing the discontinuity plane at a convenient location. Separatrix of the unperturbed system is parametrized exactly in a piecewise manner. Switching times, i.e. parameter values at which the separatrix crosses the discontinuity plane, are obtained. Switching times split the Melnikov integral into various subintegrals which are evaluated either exactly using term-wise integration of the infinite series of the integrand or approximately using a finite-term series approximation of the integrand, the latter being computationally an extensive task. Integral evaluations though approximate, are purely analytical expressions in terms of special functions such as digamma and hypergeometric. Melnikov plots show that the boundary between three regions in the parameter plane differ qualitatively in case of parametric and external excitations, however; adding self-excitation to the external one does not much alter the boundary qualitatively and quantitatively. [ABSTRACT FROM AUTHOR]

The idea of adiabatic invariance is presented in the context of simple classical mechanical models. The adiabatic invariant jumps across the separatrix - an attempt has been made to bring out the basic ideas underlying the Melnikov-Arnold integral. This becomes important as soon as a perturbation to a regular, stable system makes it dynamically unstable. [ABSTRACT FROM AUTHOR]

Bifurcations in degenerate resonance zones for Hamitonian systems with 3/2 degrees of freedom close to nonlinear integrable ones and for symplectic maps of a cylinder are discussed. [ABSTRACT FROM AUTHOR]

A small periodic perturbation results in a complicated dynamics near separatrices and saddle points. A two-parameter family of asymptotic solutions staying near separatrices for a long time is constructed. Solutions from this family depend nonsmoothly on the perturbation parameter. An example is given in which the values of the perturbation parameter for this family of solutions are determined by a set with structure of the type of the Cantor set. [ABSTRACT FROM AUTHOR]

The pull-out frequency of a second-order phase lock loop (PLL) is an important parameter that quantifies the loop's ability to stay frequency locked under abrupt changes in the reference input frequency. In most cases, this must be determined numerically or approximated using asymptotic techniques, both of which require special knowledge, skills, and tools. An approximating formula is derived analytically for computing the pull-out frequency for a second-order Type II PLL that employs a sinusoidal characteristic phase detector. The pull-out frequency of such PLLs can be easily approximated to satisfactory accuracy with this formula using a modern scientific calculator. [ABSTRACT FROM AUTHOR]

The standard flare model, which was proposed based on observations and magnetohydrodynamic theory, can successfully explain many observational features of solar flares. However, this model is just a framework, with many details awaiting to be filled in, including how reconnection is triggered. In this paper, we address an unanswered question: where do flare ribbons stop? With the data analysis of the 2003 May 29 flare event, we tentatively confirmed our conjecture that flare ribbons finally stop at the intersection of separatrices (or quasi-separatrix layer in a general case) with the solar surface. Once verified, such a conjecture can be used to predict the final size and even the lifetime of solar flares. [ABSTRACT FROM AUTHOR]

The concept of a new form of solitary waves—super solitary waves—is proposed, specific for embracing one or several interior separatrices on their wave phase portraits. The super solitary waves of an ion-acoustic type exist, for example, in nonmagnetized plasma containing five species of charged particles. For such plasma, electrostatic potential for ion-acoustic super solitary waves is calculated. The super solitary waves can be easily identified among usual solitons, e.g., in differential circuits installed into the measuring channel. [ABSTRACT FROM PUBLISHER]

The concept of limiting phase trajectories (LPT) has been introduced by one of the authors to describe intensive energy exchange between weakly coupled oscillators or oscillatory chains. It turns out that LPT can be considered as an alternative to nonlinear normal modes (NNMs), which are characterized by conservation of energy. LPT (in the introduced coordinates) describes the vibroimpact-type process with saw-tooth amplitude and a discontinuous derivative. It was shown earlier that this concept could also be extended to systems with one degree of freedom (DoF). In this case energy exchange between the oscillator and the source of energy can occur. In this paper, we generalize the above results in several ways, namely: (1) a consideration of the asymmetry of elastic potential; (2) a detailed description of the origin of vibroimpact-type behavior and the transition from nonresonant nonstationary oscillations to resonant ones (3) a direct application of obtained results to transient vibrations in strongly asymmetric 2DoF systems. [ABSTRACT FROM AUTHOR]

We consider homoclinic trajectories under the mapping of a two-dimensional manifold onto itself, define various invariants of homoclinic trajectories, and establish relations between them. We estimate the number of homoclinic trajectories whose distinct invariants possess values within prescribed limits. [ABSTRACT FROM AUTHOR]

It is proved in Ann. Math. (2)115 (1982) 579-595 that, for any germ of holomorphic nondicritic vector field in (ℂ2, 0), there exists at least one separatrix (invariant analytic curve containing the origin). In Proc. Amer. Math. Soc.125 (1997) 2649-2650 a simple criterion was given to find, at each level of the blow-up, a singular point which leads to an analytical invariant curve. In this paper we prove shortly and strictly combinatorially, the existence of a separatrix, and show that for any germ of holomorphic nondicritic vector field in (ℂ2, 0), there exists at least one separatrix issuing from each connected component of the exceptional divisor of its nice blow-up with nodal corner points deleted. [ABSTRACT FROM AUTHOR]

Barotropic f-plane dipolar vortices were generated in a rotating fluid and a comparison was made with the so-called supersmooth f-plane solution which—in contrast to the classical Lamb–Chaplygin solution—is marked by an elliptical separatrix and a doubly continuously differentiable vorticity field. Dye-visualization and high-resolution particle-tracking techniques revealed that the observed dipole characteristics (separatrix aspect ratio, cross-sectional vorticity distribution and vorticity versus streamfunction relationship) are in close agreement with those of the supersmooth f-plane solution for the entire lifespan of the dipolar vortex. [ABSTRACT FROM AUTHOR]

In this article we examine 2 × 2 first-order systems of ordinary differential equations and show how to identify separatrices for phase plane portraits when the system has a saddle point critical value. We describe how to use a computer algebra system to generate trajectories from contour plots, when possible, and determine the equation of the separatrix. When this approach is not possible, we describe how to use numerical investigations to determine the separatrix. Generating a phase plane portrait is useful, for at a glance one can observe what initial values give rise to bounded solutions, periodic solutions and other important features. It also permits the instructor to concentrate on the qualitative aspects of the model under investigation rather than the calculational difficulties associated with finding solutions. A list of examples suitable for student study is given in an appendix. [ABSTRACT FROM AUTHOR]

We consider the class G4 of Morse—Smale diffeomorphisms on $$ \mathbb{S} $$3 with nonwandering set consisting of four fixed points (namely, one saddle, two sinks, and one source). According to Pixton, this class contains a diffeomorphism that does not have an energy function, i.e., a Lyapunov function whose set of critical points coincides with the set of periodic points of the diffeomorphism itself. We define a quasi-energy function for any Morse—Smale diffeomorphism as a Lyapunov function with the least number of critical points. Next, we single out the class G4,1 ⊂ G4 of diffeomorphisms inducing a special Heegaard splitting of genus 1 of the sphere $$ \mathbb{S} $$3. For each diffeomorphism in G4,1, we present a quasi-energy function with six critical points. [ABSTRACT FROM AUTHOR]

We consider the class G4 of Morse—Smale diffeomorphisms on $$ \mathbb{S} $$3 with nonwandering set consisting of four fixed points (namely, one saddle, two sinks, and one source). According to Pixton, this class contains a diffeomorphism that does not have an energy function, i.e., a Lyapunov function whose set of critical points coincides with the set of periodic points of the diffeomorphism itself. We define a quasi-energy function for any Morse—Smale diffeomorphism as a Lyapunov function with the least number of critical points. Next, we single out the class G4,1 ⊂ G4 of diffeomorphisms inducing a special Heegaard splitting of genus 1 of the sphere $$ \mathbb{S} $$3. For each diffeomorphism in G4,1, we present a quasi-energy function with six critical points. [ABSTRACT FROM AUTHOR]

We study the ray propagation path in a plane smoothly irregular waveguide. The following two modes of ray propagation are possible: with reflections and without reflections from the waveguide walls. In each of these modes, the problem has an adiabatic invariant. We obtain an asymptotic formula for the value of the adiabatic invariant jump as the propagation mode changes. [ABSTRACT FROM AUTHOR]

We study a Hamiltonian describing a pendulum coupled with several anisochronous oscillators, giving a simple construction of unstable KAM tori and their stable and unstable manifolds for analytic perturbations. We extend analytically the solutions of the equations of motion, order by order in the perturbation parameter, to a uniform neighborhood of the time axis. [ABSTRACT FROM AUTHOR]