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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]

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]

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]

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]

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]

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]

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]

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]

In this work, the perturbed restricted three–body problem is investigated numerically. The problem is applied to three real systems: Saturn–Hyperion, Saturn–Titan, and Earth–Moon, for analyzing the stability of first order resonant periodic orbits. In particular, the nature of periodic orbits is studied for all three systems, where their masses ratios represent small, moderate and large values. Using different types of numerical techniques, we have identified how the parameter of mass ratio, the Jacobi constant, and the oblateness coefficient affect the geometrical properties, and the periodic solutions of system. • Analysis of real systems in the structure of Saturn–Hyperion, Saturn–Titan and Earth–Moon systems. • Stability analysis of the first order resonant periodic orbits in the perturbed restricted three-body problem. • Effect of mass ratio, Jacobi constant and oblateness coefficient on geometrical parameters of the periodic orbits. • Identify the initial positions of periodic orbits using Poincaré surface of sections. • Identify the separation islands of zero size in Poincaré surface of sections. [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]

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]

Magnetic field lines are the trajectories of a 1² degree of freedom Hamiltonians. Plasmas in tokamaks are confined in regions where the magnetic field lines form closed toroidal surfaces. These surfaces are bounded by a separatrix, and outside the separatrix the magnetic field lines and the plasma flow to special regions of the walls called divertors. Both the confinement of the plasma and the feasibility of divertors are sensitive to the behavior of the magnetic field lines near the separatrix in the presence of non-axisymmetric magnetic perturbations. Separatrix manifold forms homoclinic tangle to preserve the symplectic invariant and topological neighborhood as the manifold evolves in canonical time. A scheme is developed based on these two invariants to calculate homoclinic tangle for actual tokamak equilibria that are subjected to non-axisymmetric perturbations. This scheme is used to study homoclinic tangles of a specific tokamak configuration. How non-ideal effects fill in the lobes formed by homoclinic tangles is demonstrated using a radial expansion operator to simulate plasma diffusion. It is found that for sufficiently rapid plasma diffusion the effects of the tangle on plasma footprint on collector plate are washed out. [ABSTRACT FROM AUTHOR]

Abstract: In its standard formulation, quantum mechanics presents a very serious inconvenience: given a quantum system, there is no possibility at all to unambiguously (causally) connect a particular feature of its final state with some specific section of its initial state. This constitutes a practical limitation, for example, in numerical analyses of quantum systems, which often make necessary the use of some extra assistance from classical methodologies. Here it is shown how the Bohmian formulation of quantum mechanics removes the ambiguity of quantum mechanics, providing a consistent and clear answer to such a question without abandoning the quantum framework. More specifically, this formulation allows to define probability tubes, along which the enclosed probability keeps constant in time all the way through as the system evolves in configuration space. These tubes have the interesting property that once their boundary is defined at a given time, they are uniquely defined at any time. As a consequence, it is possible to determine final restricted (or partial) probabilities directly from localized sets of (Bohmian) initial conditions on the system initial state. Here, these facts are illustrated by means of two simple yet physically insightful numerical examples: tunneling transmission and grating diffraction. [Copyright &y& Elsevier]

Abstract: An age structured model is considered in order to analyze the growth of a two sex population with a fixed age-specific sex ratio. The model is intended to give an insight into the dynamics of a population where the mating process takes place at random and the proportion between females and males is not influenced by environmental or social factors, but only depends on a differential mortality or on a possible transition from one sex to the other (e.g. in sequential hermaphrodite species). First a basic model, asymptotically linear, is considered and its ergodicity is studied. Survival thresholds and their dependence on the sex ratio are then analyzed, in connection with the optimal sex ratio to guarantee survival. A further model including logistic effect is also considered and discussed in connection with existence and stability of steady states. [Copyright &y& Elsevier]

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]

Abstract: We consider a discretionary stopping problem that arises in the context of pricing a class of perpetual American-type call options, which include the perpetual American, Russian and lookback-American call options as special cases. We solve this genuinely two-dimensional optimal stopping problem by means of an explicit construction of its value function. In particular, we fully characterise the free-boundary that provides the optimal strategy, and which involves the analysis of a highly nonlinear ordinary differential equation (ODE). In accordance with other optimal stopping problems involving a running maximum process that have been studied in the literature, it turns out that the associated variational inequality has an uncountable set of solutions that satisfy the so-called principle of smooth fit. [Copyright &y& Elsevier]

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]

Abstract: This paper describes a new numerical method to compute the separatrix of the basins of attraction of coexisting attractors in a forced friction oscillator. Numerical results show that its intersection with a Poincaré section is a non-smooth curve. [Copyright &y& Elsevier]

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]

Abstract: The magnetic footprint on collector plates in a tokamak divertor is heavily constrained by the field lines obeying the equations of degree of freedom Hamiltonian mechanics. In a tokamak with broken axisymmetry, the last toroidal surface on which all field lines are confined encloses less toroidal flux than the ideal axisymmetric separatrix. The location of this surface is determined by the amplitude of the perturbations that resonate with the safety factor, q. Near a separatrix, such as that of a tokamak divertor, q has a logarithmic singularity as a function of the enclosed toroidal flux. The resonant perturbations produce islands in the field line trajectories that must overlap to fundamentally change the properties of the field line trajectories by making the field lines ergodically cover a volume. The qualitative features of the strike points of the file lines are determined by the mode numbers of the resonant perturbations. With low mode numbers a few large islands control the break up of the magnetic surfaces and the strike points lie within far sharper lines than when high mode numbers cause the break up. These features are studied using area preserving maps and the results are applied to the DIII-D divertor as an example. [Copyright &y& Elsevier]

Given a dynamics in configuration or phase-space, it is often important to map the barriers, the separatrices emanating from them, and the current distributions of the reaction paths. We describe a strategy to do this efficiently. [ABSTRACT FROM AUTHOR]

Abstract: We investigate entire radial solutions of the semilinear biharmonic equation in , , being a parameter. We show that singular radial solutions of the corresponding Dirichlet problem in the unit ball cannot be extended as solutions of the equation to the whole of . In particular, they cannot be expanded as power series in the natural variable . Next, we prove the existence of infinitely many entire regular radial solutions. They all diverge to −∞ as and we specify their asymptotic behaviour. As in the case with power-type nonlinearities [F. Gazzola, H.-Ch. Grunau, Radial entire solutions for supercritical biharmonic equations, Math. Ann. 334 (2006) 905–936], the entire singular solution plays the role of a separatrix in the bifurcation picture. Finally, a technique for the computer assisted study of a broad class of equations is developed. It is applied to obtain a computer assisted proof of the underlying dynamical behaviour for the bifurcation diagram of a corresponding autonomous system of ODEs, in the case . [Copyright &y& Elsevier]

Abstract: The hydrodynamic models that have recently been developed to investigate the nature of flow around coherent, rigid inclusions in simple shear reveal two contrasting patterns with eye-shaped and bow-tie shaped separatrix, even though all the models are based on Navier–Stokes law. In order to find the cause of this variance, this paper reviews the existing models in the light of different boundary conditions imposed on individual models. Scrutiny of the models reveals that inclusion–matrix systems, when considered infinitely extended in space, develop eye-shaped flows. However, those with finite dimensions essentially display bow-tie shaped flows. Using a finite element method (FEM), we advance the study to show the additional effects of model/inclusion dimension ratio (D R) and model aspect ratio (A R) under different boundary conditions. In the flow with bow-tie shaped separatrix, the regions of back flow define a nearly semi-circular geometry when D R is low (&#60;2). These regions assume a semi-elliptical shape with increasing D R. The distance of stagnation points from the inclusion is found to increase non-linearly with D R. Model results suggest that transformation of a flow with eye-shaped separatrix to that with bow-tie shaped separatrix can occur due to increasing A R under a specific boundary condition. Applying FEM results in geological situations thus requires the appropriate choice of dimensional parameters of the model as well as the kinematic conditions imposed at the model boundaries. [Copyright &y& Elsevier]

We model the longitudinal librations of Hyperion with a second order nonlinear differential equation in which the terms of order e 2 or higher, where e is the eccentricity of the satellite's elliptical orbit, are neglected. With a suitable change of variable we transform the differential equation into an equivalent one that describes the motion of a forced pendulum with a vertically oscillating pivot. We prove that the pendulum has uncountably many chaotic orbits. Hence, the proposed model indicates, although it does not prove, that Hyperion's longitudinal librations are chaotic. [ABSTRACT FROM AUTHOR]

The main purpose of this investigation is to show that a pendulum, whose pivot oscillates vertically in a periodic fashion, has uncountably many chaotic orbits. The attribute chaotic is given according to the criterion we now describe. First, we associate to any orbit a finite or infinite sequence as follows. We write 1 or -1 every time the pendulum crosses the position of unstable equilibrium with positive (counterclockwise) or negative (clockwise) velocity, respectively. We write 0 whenever we find a pair of consecutive zero's of the velocity separated only by a crossing of the stable equilibrium, and with the understanding that different pairs cannot share a common time of zero velocity. Finally, the symbol ω, that is used only as the ending symbol of a finite sequence, indicates that the orbit tends asymptotically to the position of unstable equilibrium. Every infinite sequence of the three symbols {1,-1, 0} represents a real number of the interval [0, 1] written in base 3 when -1 is replaced with 2. An orbit is considered chaotic whenever the associated sequence of the three symbols {1, 2, 0} is an irrational number of [0, 1]. Our main goal is to show that there are uncountably many orbits of this type. [ABSTRACT FROM AUTHOR]

A resonant slow-extraction method from synchrotron rings has been successfully developed in order to deliver a beam for a period of over several seconds. In the resonant slow-extraction method, an optimization of the parameters is necessary in order to keep the beam size and position constant during the extraction for sufficient use of the extracted beam. For this purpose, a simple method to measure the outgoing separatrix was proposed and verified at the HIMAC synchrotron. In this method, two tantalum rods each with diameter of 1 mm are used as internal probes located near the extraction channel. By observing the extracted beam under the rods inserted into the ring, the outgoing separatrix can be measured. This technique will significantly contribute to the operation of a synchrotron employing resonant slow-extraction. [Copyright &y& Elsevier]

The structure of edge magnetic field lines of the l =2 Yamator (the region outside closed magnetic surfaces) is numerically analysed. Yamator is a new stellarator-type magnetic system, where a significant magnetic well can be formed at moderate values of basic magnetic surface characteristics. In comparison with a straight classical stellarator or a torsatron, this system has a double number of separatrix ribs (or so-called "X-points"). The present investigations give a general idea about the behavior of the edge magnetic field lines in the l =2 Yamator and give grounds to discuss a feasible concept of the divertor. [ABSTRACT FROM AUTHOR]

A novel study of midfrequency oscillation and stability in permanent-magnet stepping motors is presented with an attempt to give a comprehensive description for the midfrequency oscillation and the corresponding failure. It is shown that nonlinearity plays a crucial role in midfrequency oscillation, and the relationship between local stability and midfrequency oscillation is clarified. A sufficient and necessary condition of the local stability is given, which is more applicable than the existing results, and can be used to estimate the frequency range where the oscillation occurs. The separatrices are calculated and analyzed based on a fourth-order state-space model of stepping motors. The stability at high frequencies and failure are evaluated in terms of the calculated separatrix. A novel physical quantity, which is easily calculated and analyzed, is used to evaluate the high-frequency stability. The method and results developed in the paper are useful in understanding the complex oscillation and failure phenomena. © 1999 Scripta Technica, Electr Eng Jpn, 129(3): 59–68, 1999 [ABSTRACT FROM AUTHOR]

We simulated electron motion in confronting divergent magnetic fields (CDMFs) using a Monte Carlo method. Fundamental plasma structure under a filtering effect of the separatrix of CDMFs was depicted. Hot spots of energy deposition into electrons were observed not only near the RF antenna but also near the equistrength surface of \(2B_{\rm ECR}\) , where \(B_{\rm ECR}\) is the RF-resonant magnetic field strength. [ABSTRACT FROM PUBLISHER]