On Hopf bifurcations of piecewise planar Hamiltonian systems

In this paper we study the number of limit cycles appearing in Hopf bifurcations of piecewise planar Hamiltonian systems. For the case that the Hamiltonian function is a piecewise polynomials of a general form we obtain lower and upper bounds of the number of limit cycles near the origin respectively. For some systems of special form we obtain the Hopf cyclicity.     

Degenerate bifurcations of nontwisted heterodimensional cycles with codimension 3

Bifurcations of heterodimensional cycles with highly degenerate conditions are studied by establishing a suitable local coordinate system in three-dimensional vector fields. The existence, coexistence and noncoexistence of the periodic orbit, homoclinic loop, heteroclinic loop and double periodic orbit are obtained under some generic hypotheses. The bifurcation surfaces and the existence regions are located; the number of the bifurcation surfaces exhibits variety and complexity of the bifurcation of degenerate heterodimensional cycles. The corresponding bifurcation graph is also drawn.     

Stability of certain planar unbounded polycycles

We study the stability of a type of unbounded polycycles which appear in some planar differential equations. Each of these polycycles has hyperbolic corners, but the product of the hyperbolicity ratios of all its corners does not decide its stability. We obtain an explicit convergent integral whose sign gives the stability of the polycycle.     

Local cyclicity in low degree planar piecewise polynomial vector fields

In this work, we are interested in isolated crossing periodic orbits in planar piecewise polynomial vector fields defined in two zones separated by a straight line. In particular, in the number of limit cycles of small amplitude. They are all nested and surrounding one equilibrium point or a sliding segment. We provide lower bounds for the local cyclicity for planar piecewise polynomial systems, $M_p^c\left(n\right),$ with degrees 2, 3, $4,$ and 5. More concretely, $M_p^c\left(2\right)\ge 13,$ $M_p^c\left(3\right)\ge 26,$ $M_p^c\left(4\right)\ge 40,$ and $M_p^c\left(5\right)\ge 58.$ The computations use parallelization algorithms.     

Basin of attraction of a cusp–fold singularity in 3D piecewise smooth vector fields

Generic one-parameter families of piecewise smooth vector fields on R³${\mathbb{R}}^3$ presenting the so-called cusp–fold singularity are studied. The bifurcation diagrams are exhibited and the asymptotic and structural stabilities are discussed.     

On a class of new and practical performance indexes for approximation of fold bifurcations of nonlinear power flow equations

Efficient measurement of the performance index (the distance of a loading parameter from the voltage collapse point) is one of the key problems in power system operations and planning and such an index indicates the severity of a power system with regard to voltage collapse. There exist many interesting methods and ideas to compute this index. However, some successful methods are not yet mathematically justified while other mathematically sound methods are often proposed directly based on the bifurcation theory and they require the initial stationary state to be too close to the unknown turning point to make the underlying methods practical.This paper first gives a survey of several popular methods for estimating the fold bifurcation point including the continuation methods, bifurcation methods and the test function methods (Seydel's direct solution methods, the tangent vector methods and the reduced Jacobian method) and discuss their relative advantages and problems. Test functions are usually based on scaling of the determinant of the Jacobian matrix and it is generally not clear how to determine the behaviour of such functions. As the underlying nonlinear equations are of a particular type, this allows us to do a new analysis of the determinants of the Jacobian and its submatrices in this paper. Following the analysis, we demonstrate how to construct a class of test functions with a predictable analytical behaviour so that a suitable index can be produced. Finally, examples of two test functions from this class are proposed. For several standard IEEE test systems, promising numerical results have been achieved.     

Pitchfork bifurcations of invariant manifolds

A pitchfork bifurcation of an (m−1)-dimensional invariant submanifold of a dynamical system in Rm is defined analogous to that in R. Sufficient conditions for such a bifurcation to occur are stated and existence of the bifurcated manifolds is proved under the stated hypotheses. For discrete dynamical systems, the existence of locally attracting manifolds M₊ and M₋, after the bifurcation has taken place is proved by constructing a diffeomorphism of the unstable manifold M. Techniques used for proving the theorem involve differential topology and analysis. The theorem is illustrated by means of a canonical example.     

Homoclinic bifurcations of endomorphisms: The codimension one case

In this note, we consider the dynamic that appears when we unfold a quadratic degenerate homoclinic point of a generic one-parameter family of endomorphisms f_μ. This is done through a rescaling technique. Among other facts, it follows from our theorem the abundance of strange expanding sets.     

The qualitative analysis of a class of planar Filippov systems

We use the theory of differential inclusions, Filippov transformations and some appropriate Poincaré maps to discuss the special case of two-dimensional discontinuous piecewise linear differential systems with two zones. This analysis applies to uniqueness and non-uniqueness for the initial value problem, stability of stationary points, sliding motion solutions, number of closed trajectories, existence of heteroclinic trajectories connecting two saddle points forming a heteroclinic cycle and existence of the homoclinic trajectory     

Rank-1 codimension one singularities of positive quadratic differential forms

We complete the study of first-order structural stability at singular points of positive quadratic differencial forms on two manifolds. For this, we consider the generic 1-parameter bifurcation of a D₂₃-singular point. This situation consists in having, before the bifurcation, two locally stable singular points (one of type D₂ and the other of type D₃) which collapse at the D₂₃-singular point when the bifurcation parameter is reached, and afterwards disappear. In local (x,y)-coordinates, such a point appears at the origin of a planar differential equation of the form with (b²-ac)(x,y)?0, such that

(1)

the first jet of the map (a,b,c) at the origin is T₁(a,b,c)(0,0)=(y,0,-y) and

(2)

     

Bifurcation of periodic orbits in a class of planar Filippov systems

In this paper we discuss the perturbations of a general planar Filippov system with exactly one switching line. When the system has a limit cycle, we give a condition for its persistence; when the system has an annulus of periodic orbits, we give a condition under which limit cycles are bifurcated from the annulus. We also further discuss the stability and bifurcations of a nonhyperbolic limit cycle. When the system has an annulus of periodic orbits, we show via an example how the number of limit cycles bifurcated from the annulus is affected by the switching.     

Controllable Hopf bifurcations of codimension 1 and 2 in nonlinear control systems

The purpose of this work, given a nonlinear control system, is to design a four-parameter family of static state feedback such that the corresponding closed-loop control system exhibits controllable Hopf bifurcations of codimension 1 and 2. More precisely, the scalar law designed by us permits the control of the stability of the equilibrium points and the orientation and stability of the periodic orbits.     

与比率有关的捕食——被捕食生态模型

本文讨论了一类生态模型的有效性，种群不灭性，闭轨和同窗轨的存在性，平衡点的稳定性，并定义了向量场同胚映射。     

Normal toric ideals of low codimension

Every normal toric ideal of codimension two is minimally generated by a Gröbner basis with squarefree initial monomials. A polynomial time algorithm is presented for checking whether a toric ideal of fixed codimension is normal.     

The loop quantities and bifurcations of homoclinic loops

The stability and bifurcations of a homoclinic loop for planar vector fields are closely related to the limit cycles. For a homoclinic loop of a given planar vector field, a sequence of quantities, the homoclinic loop quantities were defined to study the stability and bifurcations of the loop. Among the sequence of the loop quantities, the first nonzero one determines the stability of the homoclinic loop. There are formulas for the first three and the fifth loop quantities. In this paper we will establish the formula for the fourth loop quantity for both the single and double homoclinic loops. As applications, we present examples of planar polynomial vector fields which can have five or twelve limit cycles respectively in the case of a single or double homoclinic loop by using the method of stability-switching.     

First integrals and bifurcations of a Lane-Emden equation of the second kind

First integrals admitted by an approximate Lane-Emden equation modelling a thermal explosion in a rectangular slab and cylindrical vessel are investigated. By imposing the boundary conditions on the first integrals we obtain a nonlinear relationship between the temperature at the center of the vessel and the temperature gradient at the wall of the vessel. For a rectangular slab the presence of a bifurcation indicates multivalued solutions for the temperature at the center of the vessel when the temperature gradient at the wall is fixed. For a cylindrical vessel we find a bifurcation indicating multivalued solutions for the temperature gradient at the walls of the vessel when the temperature at the center of the vessel is fixed.     

Codimension One Bifurcation of 2-Dimensional Tori Born from an Equilibrium Family in Systems with Cosymmetry

Mathematical Notes -     

Exact number of limit cycles for a family of rigid systems

For a given family of planar differential equations it is a very difficult problem to determine an upper bound for the number of its limit cycles. Even when this upper bound is one it is not always an easy problem to distinguish between the case of zero and one limit cycle. This note mainly deals with this second problem for a family of systems with a homogeneous nonlinear part. While the condition that allows us to separate the existence and the nonexistence of limit cycles can be described, it is very intricate.

     

Limit cycle bifurcations of some Liénard systems

In this paper we first give some general theorems on the limit cycle bifurcation for near-Hamiltonian systems near a double homoclinic loop or a center as a preliminary. Then we use these theorems to study some polynomial Liénard systems with perturbations and give new lower bounds for the maximal number of limit cycles of these systems.