Search

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]

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]

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]

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]

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]

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]

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]

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]

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 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]

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]

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]

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]

Magnetic reconnection is a main process converting the magnetic energy into thermal and kinetic energy in plasmas. It is one of the fundamental problems of crucial importance not only to space plasmas physics and space weather studies, such as the solar flare, coronal mass ejections and magnetospheric substorms, but also to the stability analysis in magnetically confined fusion. In general, except for cases with periodical boundary conditions, three-dimensional (3D) magnetic reconnection occurs on magnetic separatrices generated by magnetic nulls. Here we briefly introduce/review the theories and some recent satellite observations of 3D magnetic reconnection. Topics to be further studied are also discussed. [ABSTRACT FROM AUTHOR]

In this paper a topological invariant is introduced for the class of supertransitive flows on closed nonorientable surfaces M of negative Euler characteristic. We describe properties of this invariant and prove that it provides necessary conditions for the topological equivalence of flows belonging to the above-mentioned class of supertransitive flows. [ABSTRACT FROM AUTHOR]

Copyright of Boletim da Sociedade Brasileira de Matematica is the property of Springer Nature and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

This paper discusses the dynamics of bi-stable competition systems. It is assumed that in the initial state one species is numerically superior to the other species at every point of their habitat. If the two species do not migrate, then the superior species will wipe out the inferior one. It is shown, however, that the superior species may become extinct if the diffusion is taken into consideration. [ABSTRACT FROM AUTHOR]