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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]

Background. The study of dynamical systems that are invariant with respect to different groups of transformations is important both for the theory of differential equations and for its applications. Local bifurcations in generic two-parameter families of dynamical systems defined by vector fields invariant under the involution of a plane having a line of fixed points were described by H. Zholondek. The purpose of this research is to study some nonlocal bifurcations in such families. Materials and methods. The method of point mappings and other methods of the qualitative theory of differential equations are applied. Results. We consider a generic two-parameter family of planar vector fields with symmetry about the x-axis. It is assumed that at a zero value of the parameter, the field has a rough saddle, a weak saddle lying on the x axis, and two symmetrical contours formed by the separatrices of these saddles. A bifurcation diagram is obtained – a partition of the neighborhood of zero on the parameter plane by types of phase portraits in the neighborhood of a polycycle composed of these contours. In particular, we show that one stable rough limit cycle can be born from each contour. Conclusions. One of the possible scenarios for the occurrence of stable periodic oscillations when changing the parameters of a dynamical system with involutive symmetry is described.

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 magnetic separatrix surface is designed to provide the final and critical confinement to the hot stationary-operation core plasma in modern tokamak reactors in the absence of an external magnetic perturbation (MP) or transient magneto-hydrodynamic perturbation, while diverting the exhaust heat to divertor plates. All the stationary operational boundary plasma studies and reactor designs have been performed under this assumption. However, there has been a long-standing suspicion that a stationary-operation tokamak plasma even without external MPs or edge localized modes (ELMs) activities may not have a stable closed separatrix surface, especially near the magnetic X-point. Here, the first gyrokinetic numerical observation is reported that the divertor separatrix surface, due to homoclinic tangles caused by intrinsic electromagnetic turbulence, is not a stable closed surface in a stationary operation phase even without MPs or ELMs. Unlike the MP- or ELM-driven homoclinic tangles that could cause deleterious effects to core confinement or divertor plates, it is found that the micro-turbulence driven homoclinic tangles could connect the divertor plasma to the pedestal plasma in a constructive way by broadening the divertor heat-exhaust footprint and weakening the pedestal slope to the ELM-safe direction. Micro-turbulent homoclinic tangles can open a new research direction in understanding and controlling these two most troublesome and non-locally connected edge-plasma issues in a tokamak fusion reactor.

Predictive and real-time inference capability for the upstream separatrix electron density, ne, sep, is essential for design and control of core-edge integrated plasma scenarios. In this study, both supervised and semi-supervised machine learning algorithms are explored to establish direct mapping as well as indirect compressed representation of the pedestal profiles for predictions and inference of ne, sep. Based on the EUROfusion pedestal database for JET (Frassinetti et al., 2021), a tabular dataset was created, consisting of machine parameters, fraction of ELM cycle, high resolution Thomson scattering profiles of electron density and temperature, and ne, sep for 608 JET shots. Using the tabular dataset, the direct mapping approach provides a mapping of machine parameters and ELM percentage to ne, sep. Through representation learning, a compressed representation of the experimental pedestal electron density and temperature profiles is established. By conditioning the representation with machine control parameters, a probabilistic generative predictive model is established. For prediction, the machine parameters can be used to establish a conditional distribution of the compressed pedestal profiles, and the decoder that is trained as part of the algorithm can be used to decode the compressed representation back to full pedestal profiles. Although, in this work, a proof-of-principle for predicting and inferring ne, sep is given, such a representation learning can be used also for many other applications as the full pedestal profile is predicted. An implementation of this work can be found at https://github.com/fusionby2030/psi_2022.

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.

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]

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]

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]

At the present time, Magnetic Confinement Fusion (MCF) is considered as a way to produce energy. In this work, one of the Magnetohydrodynamic (MHD) limitations has been discussed. As closed magnetic surfaces, the Magnetic Islands (MIs) which are generated due to pressure effects, need to be surrounded by a separatrix which separate them from the other parts. External Magnetic Fields (EMFs), the safety factor (q) and the pressure profiles would be used to take the MIs under control. This could be achieved through an exterior medium, exclusively the Electron Cyclotron Heating (ECH) as well as the Current Drive (ECCD). Study of the magnetic flux surfaces and the effect of magnetic perturbation on tokamak plasmas, inform us about the formation of the MIs and their locations. In this work, together with the comprehensive review of the MIs and their importance, the conventional methods for improving the magnetic confinement has been introduced and discussed. In this regards, the Hot Limiter Biasing (HLB) method and the Resonant Helical Field (RHF) which is produced by external Helical Coils (HCs) were introduced and used. Then, the plasma current (), the Loop voltage, and the MO were obtained for different states. Finally, the Magnetic Islands Width (W) and their Growth Rate (GR) were calculated and compared with the experimental results.

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]

WEST database analysis shows a correlation of the recycled neutral source around the separatrix with core performances. This observation questions the causality chain between particle source and turbulent transport up to the core in L-mode, high recycling plasmas, an unavoidable phase of all scenarios. The best core performances correlate with the lowest values of the density at the separatrix, ${n_{{\text{sep}}}}$ , similarly to ASDEX Upgrade (AUG) tokamak and Joint European Torus (JET) tokamak in H-mode (Verdoolaege et al 2021 Nucl. Fusion 61 076006). Reflectometry in the midplane provides ${n_{{\text{sep}}}}$ , while the temperature at the separatrix, ${T_{{\text{sep}}}}$ is inferred by the ‘two-point model’ using Langmuir probe data on divertor targets. Lower separatrix resistivity does not correlate with better core performances, unlike H-mode observations (Eich et al 2020 Nucl. Fusion 60 056016). As expected in the presence of an efficient neutral source due to recycling fluxes, ${n_{{\text{sep}}}}$ correlates with the D recycled particle flux at the divertor measured by visible spectroscopy. Coherently, at a given controlled central line integrated density $\bar n$ , lower ${n_{{\text{sep}}}}$ correlates with a larger density gradient around the separatrix as well as a larger global density peaking, $\bar n/\langle n\rangle $ , measured by interferometry. The latter correlates as well with lower collisionality in the core, similarly to JET and AUG H-modes (Angioni et al 2007 Nucl. Fusion 47 1326). The correlations reported allow phrasing the subsequent causality question: what is the interplay chain between low neutral recycling at the divertor plates, low density at the separatrix, high density peaking at the separatrix, high global density peaking, higher central temperature and better core energy confinement quality? Understanding the causality chain is essential to prepare ITER operation and design DEMO scenarios where the ratio of the divertor leg to the ionization length will be larger and where the pumped flux with respect to the plasma volume will be lower than presently operating tokamaks.

The quasi-continuous exhaust (QCE) regime, formerly known as either type-II ELM or small ELM regime is studied in ASDEX Upgrade. The regime is a natural type-I ELM-free H-mode. The operational space of QCE discharges in ASDEX Upgrade with respect to their separatrix conditions and their power exhaust capabilities are presented. A significant broadening of the power fall-off length is observed, correlating to an increased separatrix density and pressure. Moreover, the possible reactor relevance of this regime is demonstrated by expanding the operational space to low edge safety factor and demonstrating the benign tungsten impurity behaviour. A discharge without any type-I ELM from start to end reaching a partially detached divertor at high normalised energy confinement time is presented.

A myriad of physical phenomena, such as fluid flows, magnetic fields, and population dynamics are described by vector fields. More often than not, vector fields are complex and their analysis is challenging.Vector field topology is a powerful analysis technique that consists in identifying the most essential structure of a vector field. Its topological features include critical points and separatrices, which segment the domain into regions of coherent flow behavior, provide a sparse and semantically meaningful representation of the underlying data.However, a broad adoption of this formidable technique has been hampered by the lack of open source software implementing it. The Visualization Toolkit (VTK) now contains the filter vtkVectorFieldTopology that extracts the topological skeleton of 2D and 3D vector fields. This paper describes our implementation and demonstrates its broad applicability with two real-world examples from hydrology and space physics.

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]

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]

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]

Abstract A well-known result by L. Markus [6] , later extended by D.A. Neumann [7] , states that two continuous flows on a surface are equivalent if and only if there is a surface homeomorphism preserving orbits and time directions of their separatrix configurations. In this paper we present several examples showing that, as originally formulated, the Markus–Neumann theorem needs not work. Besides, we point out the gap in its proof and show how to restate it in a correct (and slightly more general) way. [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]

We study a 2-species Lotka–Volterra type differential system, modeling competition between two species and having a coexistence equilibrium in the first quadrant. In case that this equilibrium is of saddle type, its stable manifold divides the first quadrant into two zones. Then, depending on the zone where the initial condition lies, one of the species will extinct and the other will go to an equilibrium. Using this separatrix we introduce a measure to discern which species has more chance of surviving. This measure is given by a non-negative real number κ , that we will call persistence ratio , that only depends on the parameters of the system. In some cases, we can give simple explicit expressions for κ . When this is not possible, we use several dynamical tools to obtain effective approximations of it. [ABSTRACT FROM AUTHOR]

A large variety of antipredator defenses are exhibited by plants, animals and microbes in nature. A deep understanding of the dynamic consequences of prey responses to predation risk is essential for building a comprehensive theory of food webs. Here we present a simple classification of prey defenses based on the sensitivity of prey immunity to predation respect to abundances of prey and predators. Only three out of six defense types have been analytically studied in the context of predator–prey dynamics, which reveals a serious gap in our current knowledge of ecological interactions. In this study we present a mathematical analysis on a widely occurring type of prey defense whose behavior has not been established in exact terms. The study model considers prey whose average immunity to predators is enhanced by predator abundance. This case, known as inducible defenses, has been reported for a wide array of species. Our results reveal a rich dynamic behavior, in which the predator-prey system exhibits either one, two or three positive equilibrium points, with up to two attractors. Thus, inducible defenses constitute a mechanism that could drive alternative stable states even in very simple food web models. [ABSTRACT FROM AUTHOR]