BIFURCATION OF LIMIT CYCLE FOR QUADRATIC SYSTEM WITH INVARIANT PARABOLA

In a previous paper, we have proved that a planar quadratic system with invariant parabola Г has at most one limit cycle. In this paper, we use geometric characteristics to give necessary and sufficient conditions under which a PQSГ with three non-degenerate singular points can be transformed into two different definite forms. In this way, we obtain ail the bifurcations of such a system.     

Phase transitions of McKean–Vlasov processes in double-wells landscape

The aim of this work is to establish the results for a particular class of inhomogeneous processes, the McKean–Vlasov diffusions. Such diffusions correspond to the hydrodynamical limit of an interacting particle system. In convex landscapes, existence and uniqueness of the invariant probability is a well-known result. However, previous results state the nonuniqueness of the invariant probabilities under nonconvexity assumptions. Here, we prove that there exists a phase transition. Below a critical value, there are exactly three invariant probabilities and above another critical value, there is exactly one. Under simple assumptions, these critical values coincide and it is characterized by a simple implicit equation. We also investigate other cases in which phase transitions occur. Finally, we provide numerical estimations of the critical values.     

Uniqueness of limit cycles bounded by two invariant parabolas

In this paper we consider a class of cubic polynomial systems with two invariant parabolas and prove in the parameter space the existence of neighborhoods such that in one the system has a unique limit cycle and in the other the system has at most three limit cycles, bounded by the invariant parabolas.     

一类单中心Hamilton系统在三次扰动下的Poincare分岔

使用一阶Mel‘nikov函数讨论了一类具有以抛物线与直线为边界的周期环域的单中心二次Hamilton系统的三次扰动下的Poincare分岔,得到其Poincare分岔最多可以产生两个极限环。     

Liénard systems for quadratic systems with invariant algebraic curves

We study the character of the friction function f(x) and the restoring force g(x) in the Liénard system to which a quadratic system with an invariant second-order algebraic curve (an ellipse that is a limit cycle, a hyperbola defining two separatrix cycles, or a parabola) or fourth-order algebraic curve with an oval being a limit cycle can be reduced. Invariant curves are constructed for quadratic systems in a five-parameter canonical family, which can readily be reduced to Liénard systems.     

具有二重抛物线解的二次系统

本文给出了具有二重抛物线解的二次系统的一般形状，并与具有并重抛物线解的二次系统相比较，证明了具有二重抛物线解的二次系统也有存在极限环的可能的，而且也是唯一的，但是二重抛物线解却是不可能成为二次系统的分界线不的。     

Limit cycles and invariant cylinders for a class of continuous and discontinuous vector field in dimention 2n

The subject of this paper concerns with the bifurcation of limit cycles and invariant cylinders from a global center of a linear differential system in dimension 2n perturbed inside a class of continuous and discontinuous piecewise linear differential systems. Our main results show that at most one limit cycle and at most one invariant cylinder can bifurcate using the expansion of the displacement function up to first order with respect to a small parameter. This upper bound is reached. For proving these results we use the averaging theory in a form where the differentiability of the system is not needed.     

The Poincaré Bifurcation of a Class of Planar Hamiltonian Systems

In this paper, we discuss the Poincare bifurcation of a class of Hamiltonian systems having a region consisting of periodic cycles bounded by a parabola and a straight line. We prove that the system can generate at most two limit cycles and may generate two limit cycles after a small cubic polynomial perturbation.     

Long-time averaging for integrable Hamiltonian dynamics

Summary. Given a Hamiltonian dynamical system, we address the question of computing the limit of the time-average of an observable. For a completely integrable system, it is known that ergodicity can be characterized by a diophantine condition on its frequencies and that this limit coincides with the space-average over an invariant manifold. In this paper, we show that we can improve the rate of convergence upon using a filter function in the time-averages. We then show that this convergence persists when a symplectic numerical scheme is applied to the system, up to the order of the integrator.     

The Paley–Wiener Theorem and Limits of Symmetric Spaces

We extend the Paley–Wiener theorem for Riemannian symmetric spaces to an important class of infinite-dimensional symmetric spaces. For this we define a notion of propagation of symmetric spaces and examine the direct (injective) limit symmetric spaces defined by propagation. This relies on some of our earlier work on invariant differential operators and the action of Weyl group invariant polynomials under restriction.     

On limit cycles of systems with a partial integral

For a certain class of two-dimensional autonomous systems of differential equations with an invariant curve that contains ovals, we indicate necessary and sufficient conditions for these ovals to be limit cycles of phase trajectories.     

Phase portraits of integrable quadratic systems with an invariant parabola and an invariant straight line

We classify the phase portraits of the quadratic polynomial differential systems having an invariant parabola, an invariant straight line, and a Darboux first integral produced by these two invariant curves.     

Limit periodic travelling wave solution of a model for biological invasions

In this paper, we consider a class of biological invasion model with density-dependent migrations and Allee effect, which is reduced to one ordinary differential form via the travelling wave solution ansatz. For the corresponding planar system, we firstly obtain the first several weak focal values of its one equilibrium by computing the singular point quantities, then determine the existence of one stable limit cycle from its Hopf bifurcation. Thus a special periodic travelling wave solution which is isolate as a limit is obtained, and it corresponds to the particular real patterns of spread during biological invasions, which is an interesting discovery.     

Invariant surfaces of two-dimensional periodic systems with bifurcating rest points in the first approximation

Two-dimensional, time-periodic systems of differential equations with a small positive parameter whose first-approximation systems are conservative, depend on the parameter, and have one, two, or three rest points are considered. Explicit conditions on the coefficients under which the initial system has one or several two-dimensional invariant surfaces homeomorphic to the torus for all sufficiently small parameter values are obtained, and formulas for these surfaces are presented. As an example, a class of systems with three two-periodic invariant surfaces is constructed. __________ Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 38, Suzdal Conference-2004, Part 3, 2006.     

渐近时间周期系统

本文证明了渐近时间周期系统的Ｐｏｉｎｃａｒｅ映射的极限集是极限系统的Ｐｏｉｎｃａｒｅ映射的不变集。     

Quantities at infinity in translational polynomial vector fields

In this paper, we study quantities at infinity and the appearance of limit cycles from the equator in polynomial vector fields with no singular points at infinity. We start by proving the algebraic equivalence of the corresponding quantities at infinity (also focal values at infinity) for the system and its translational system, then we obtain that the maximum number of limit cycles that can appear at infinity is invariant for the systems by translational transformation. Finally, we compute the singular point quantities of a class of cubic polynomial system and its translational system, reach with relative ease expressions of the first five quantities at infinity of the two systems, then we prove that the two cubic vector fields perturbed identically can have five limit cycles simultaneously in the neighborhood of infinity and construct two systems that allow the appearance of five limit cycles respectively. The positions of these limit cycles can be pointed out exactly without constructing Poincaré cycle fields. The technique employed in this work is essentially different from more usual ones, The calculation can be readily done with using computer symbol operation system such as Mathematics.     

Invariant surfaces of standard two-dimensional systems with conservative first approximation of the third order

We simultaneously study two classes of two-dimensional time-periodic systems of differential equations with a small positive parameter, namely, systems with “slow” or “fast” time whose first-approximation systems are autonomous and conservative and do not contain terms of order higher than three. Thus, the corresponding unperturbed systems have one, two, or three rest points.For the perturbations, we indicate explicit conditions, independent of the small parameter, under which every original system of either class with coefficients three times continuously differentiable with respect to the phase variables and the parameter in a neighborhood of zero has finitely many two-dimensional invariant surfaces homeomorphic to tori for all sufficiently small parameter values. We also give formulas for these surfaces.     

具三次曲线解的二次系统至多有一个极限环

本文研究具有三次曲线解x^3-x^2-y^2=0的二次系统,证明此类二次系统最多只有一个极限环,进而证明了具有三次的曲线解的二次系统至多有一个极限环。     

On the structure of invariant attracting sets of systems of finite-difference equations generating Hénon-Lozi mappings

For Hénon-Lozi mappings F, we find sufficient conditions under which on the plane there exists a domain U such that its closure is mapped by F strictly inside U. This ensures the existence of a compact invariant set in U. We prove the existence of an open set of parameter values for which this invariant set contains a zero-dimensional locally maximal topologically transitive Markov set such that the restriction of the mapping to this set is topologically conjugate to the shift automorphism in the space of sequences of two symbols. We show that if this Markov set is hyperbolic, then the above-mentioned compact invariant set coincides with the closure of the unstable manifold of F at a fixed point lying in that set and is a topologically indecomposable one-dimensional continuum. We present the parameter values for which these results hold for the Hénon mapping. We thereby prove the existence of a parameter range in which the invariant set of the Hénon mapping is a one-dimensional topologically indecomposable Brauer-Janiszewski continuum that contains a zero-dimensional locally maximal set and lies in the attraction domain of itself.