Rotated Vector Fields Research Articles

The constant angle problem for mean curvature flow inside rotational tori

We flow a hypersurface in Euclidean space by mean curvature flow with a Neumann boundary condition, where the boundary manifold is any torus of revolution. If we impose the conditions that the initial manifold is compatible and does not contain the rotational vector field in its tangent space, then mean curvature flow exists for all time and converges to a flat cross-section as $t \rightarrow \infty$.

Periodic Solutions and Homoclinic Bifurcations of Two Predator-Prey Systems with Nonmonotonic Functional Response and Impulsive Harvesting

Two predator-prey models with nonmonotonic functional response and state-dependent impulsive harvesting are formulated and analyzed. By using the geometry theory of semicontinuous dynamic system, we obtain the existence, uniqueness, and stability of the periodic solution and analyse the dynamic phenomenon of homoclinic bifurcation of the first system by choosing the harvesting rateβas control parameter. Besides, we also study the homoclinic bifurcation of the second system about parameterδon the basis of the theory of rotated vector field. Finally, numerical simulations are presented to illustrate the results.

Bifurcation diagram and stability for a one-parameter family of planar vector fields

We consider the one-parameter family of planar quintic systems, x˙=y3−x3, y˙=−x+my5, introduced by A. Bacciotti in 1985. It is known that it has at most one limit cycle and that it can exist only when the parameter m is in (0.36,0.6). In this paper, using the Bendixson–Dulac theorem, we give a new unified proof of all the previous results. We shrink this interval to (0.547,0.6) and we prove the hyperbolicity of the limit cycle. Furthermore, we consider the question of the existence of polycycles. The main interest and difficulty for studying this family is that it is not a semi-complete family of rotated vector fields. When the system has a limit cycle, we also determine explicit lower bounds of the basin of attraction of the origin. Finally, we answer an open question about the change of stability of the origin for an extension of the above systems.

HETEROCLINIC BIFURCATIONS OF A PREY–PREDATOR FISHERY MODEL WITH IMPULSIVE HARVESTING

In this paper, we consider a prey–predator fishery model with Allee effect and state-dependent impulsive harvesting. First, we investigate the existence of order-1 heteroclinic cycle. Second, choosing p as a control parameter, we obtain the sufficient conditions for the existence and uniqueness of order-1 periodic solution of system (2.3) by using the geometry theory of semi-continuous dynamic systems. Finally, on the basis of the theory of rotated vector fields, heteroclinic bifurcation to perturbed system of system (2.3) is also studied. The methods used in this paper are novel to prove the existence of order-1 heteroclinic cycle and heteroclinic bifurcations.

A method of Lie-symmetry GL(n,[formula omitted]) for solving non-linear dynamical systems

The group-preserving scheme (GPS) developed in Liu [1] for solving a non-linear dynamical system x˙=f(x,t) was based on the Lie-symmetry SOo(n,1). Here we derive a more fundamental GL(n,R) dynamics for x, and develop a relevant Lie-group scheme based on the Lie-symmetry GL(n,R), which is a Lie-group set large enough to cope all non-linear ordinary differential equations (ODEs). We find that the first-order explicit scheme based on GL(n,R) is equivalent to the GPS. Moreover, when one uses an implicit scheme based on GL(n,R), it converges very fast at each time marching step and the accuracy is raised several orders than the explicit method. For the dynamical system endowed with a rotational vector field we also develop an implicit SO(n) Lie-group integration method. Several numerical examples are examined, showing that the GL(n,R) and SO(n) Lie-group schemes have superior efficiency, accuracy and stability.

Energy Estimates and Gravitational Collapse

We study the cancelations in the energy estimates for Einstein vacuum equations in order to prove the formation of black holes along evolutions. The novelty of the paper is that, we completely avoid using rotation vector fields to establish the global existence theorem of the solution. More precisely, we use only canonical null directions as commutators to derive energy estimates at the level of one derivatives of null curvature components.

HOMOCLINIC BIFURCATION IN SEMI-CONTINUOUS DYNAMIC SYSTEMS

In this paper, a class of homoclinic bifurcations in semi-continuous dynamic systems are investigated. On the basis of rotated vector fields theory, existence of order-1 periodic solution and the rotated vector fields of the semi-continuous dynamic system are discussed. Furthermore, homoclinic cycles and homoclinic bifurcations are described. Finally, an example is provided to show the validity of our theoretical results.

The limit cycles of a general Kolmogorov system

In this paper, the general Kolmogorov system {dxdt=ϕ(x)f(x,y),dydt=ρ(y)g(x,y) is studied. By using the theory of generalized rotated vector fields and the Poincare–Bendixson annular region theorem, we prove the sufficient conditions for the existence and uniqueness of limit cycles in this system. And our results extend almost all the related existing studies on the Kolmogorov and predator–prey systems.

Γ-Convergence of 2D Ginzburg-Landau functionals with vortex concentration along curves

We study the variational convergence of a family of twodimensional Ginzburg-Landau functionals arising in the study of superfluidity or thin-film superconductivity as the Ginzburg-Landau parameter ε tends to 0. In this regime and for large enough applied rotations (for superfluids) or magnetic fields (for superconductors), the minimizers acquire quantized point singularities (vortices). We focus on situations in which an unbounded number of vortices accumulate along a prescribed Jordan curve or a simple arc in the domain. This is known to occur in a circular annulus under uniform rotation, or in a simply connected domain with an appropriately chosen rotational vector field. We prove that if suitably normalized, the energy functionals Γ-converge to a classical energy from potential theory. Applied to global minimizers, our results describe the limiting distribution of vortices along the curve in terms of Green equilibrium measures.

Uniqueness theorem for black hole space-times with multiple disconnected horizons

We show uniqueness of stationary and asymptotically flat black hole spacetimes with multiple disconnected horizons and with two rotational Killing vector fields in the context of five-dimensional minimal supergravity (Einstein-Maxwell-Chern-Simons gravity). The novelty in this work is the introduction in the uniqueness theorem of intrinsic local charges measured near each horizon as well as the measurement of local fluxes besides the asymptotic charges that characterize a particular solution. A systematic method of defining the boundary conditions on the fields that specify a black hole space-time is given based on the study of its rod structure (domain structure). Also, an analysis of known solutions with disconnected horizons is carried out as an example of an application of this theorem. ”But the perfect scientist is also a gardener: he believes that beauty is knowledge.” Gonçalo M. Tavares in Brief Notes on Science

Energy splitting for solutions of multi-dimensional isotropic symmetric hyperbolic systems

This note, submitted in honor of Walter Strauss’ 70th birthday, offers a new look at the elegant paper by Costa and Strauss entitled Energy Splitting, [3]. In it, they studied finite energy solutions to linear, constant coefficient, isotropic, symmetric hyperbolic systems. They showed that solutions separate, in L2, into a superposition of outgoing plane waves, using a decomposition based on the Radon transform. Our approach to energy splitting is based on the local energy decay result obtained by the author with B. Thomases in [9], for rotationally and scaling invariant isotropic systems. Inspired by Costa and Strauss, here we replace rotational invariance by the isospectral condition of [3]. This involves a nonlocal modification of the rotational vector fields. However, the final result is entirely local. Solutions are decomposed locally according to the spectral projections of the symbol, and individual wave families concentrate near characteristic cones. After a few preliminaries, we state the main result and compare it with [3]. The proof is entirely self-contained. To conclude, we illustrate with three examples. Energy splitting is closely related to the phenomenon of energy equipartition. In the context of symmetric hyperbolic systems with constant coefficients, there have been numerous works on these topics, among them [2, 4, 10, 11].

The analysis on the single particle model of CDW

Grüner put forward a single particle model of charge-density wave, which is a typical nonlinear differential equation, and also a mathematical model of pendulum. This Letter analyzes the solution of equation by the rotated vector fields theory, providing the relation between the applied field E and the periodic solution, and the conclusion that the critical value of E for the periodic solution is fixed in the over-damped situation. With these conclusions, it derives the formulae of nonlinear conductivity, narrow-band noise, which are consistent with the empirical ones given by Fleming.

COLOR BOSONIZATION, CHIRAL PARAMETRIZATION OF GLUONIC FIELD AND QCD EFFECTIVE ACTION

We develop a color bosonization approach to treat QCD gauge field ("gluons") at low energies in order to derive an effective color action of QCD taking into account the quark chiral anomaly in the case of SU(2) color. We have found that there exists such a region in the chiral sector of color space, where a gauge field coincides with chirally rotated vector field, while an induced axial vector field disappears. In this region, the unit color vector of chiral field plays a defining role, and a gauge field is parametrized in terms of chiral parameters, so that no additional degrees of freedom are introduced by the chiral field. A QCD gauge field decomposition in color bosonization is a sum of a chirally rotated gauge field and an induced axial-vector field expressed in terms of gluonic variables. An induced axial-vector field defines the chiral color anomaly and an effective color action of QCD. This action admits existence of a gauge invariant d = 2 condensate of induced axial-vector field and mass.

Wintner-Perko termination principle, parameters rotating a field, and limit-cycle problem

In this paper, limit cycles of polynomial dynamical systems are studied. For the global analysis of bifurcations of limit cycles, we use the Wintner–Perko termination principle. Monotone families of limit cycles and rotated vector fields and limit-cycle problems for quadratic systems are also discussed.

Bifurcation of limit cycles and separatrix loops in singular Lienard systems

We give sufficient conditions for the existence of one or two limit cycles of singular Lienard systems through the construction of a Poincare–Bendixson domain. With the help of the theory of rotated vector fields,we develop a method to compute bifurcation value at Saddle-node bifurcation of limit cycles and homoclinic or symmetric heteroclinic bifurcations. We also present application examples and prove the existence of duck cycles.

Introduction to isolated horizons in numerical relativity

We present a coordinate-independent method for extracting the mass ${(M}_{\ensuremath{\Delta}})$ and angular momentum ${(J}_{\ensuremath{\Delta}})$ of a black hole in numerical simulations. This method, based on the isolated horizon framework, is applicable both at late times when the black hole has reached equilibrium, and at early times when the black holes are widely separated. Assuming that the spatial hypersurfaces used in a given numerical simulation are such that apparent horizons exist and have been located on these hypersurfaces, we show how ${J}_{\ensuremath{\Delta}}$ and ${M}_{\ensuremath{\Delta}}$ can be determined in terms of only those quantities which are intrinsic to the apparent horizon. We also present a numerical method for finding the rotational symmetry vector field (required to calculate ${J}_{\ensuremath{\Delta}})$ on the horizon.

An analogue rotated vector field of polynomial system

A class of polynomial system was structured, which depends on a parameter δ. When δ monotonous changes, more than one neighbouring limit cycles located in the vector fiel of this polynomial system can expand (or reduce) together with the δ. But the expansion (or reduction) of these limit cycles is not surely monotonous. This vector field is like the rotated vector field. So these limit cycles of the polynomial system are called to constitute an “analogue rotated vector field” with δ. They may become an effective tool to study the bifurcation of multiple limit cycle or fine separatrix cycle.

Homoclinic orbits and Lie rotated vector fields

Based on the definition of Lie rotated vector fields in the plane, this paper gives the property of homoclinic orbit as parameter is changed and the singular points are fixed on Lie rotated vector fields. It gives the conditions of yielding limit cycles as well.

Singular points and Lie rotated vector fields

This paper gives the definition of Lie rotated vector fields in the plane and the conditions of movement of singular points on Lie rotated vector fields with variable parameters.

Bifurcation and Distribution of Limit Cycles for Quadratic Differential Systems of Type II

In this paper, we first solve an open problem of quadratic differential systems of type (II)l=0in the Chinese classification. Then by making use of the theory of rotated vector fields we investigate the bifurcation phenomena and all the possible distributions of the limit cycles for quadratic systems of type II. Moreover, under a conjecture of Ye Yanqian we prove that the (2,2) limit cycle distribution is impossible.