Spectral Flow,Maslov Index and Bifurcation of Semi-Riemannian Geodesics

We give a functional analytical proof of the equalitybetween the Maslov index of a semi-Riemannian geodesicand the spectral flow of the path of self-adjointFredholm operators obtained from the index form. This fact, together with recent results on the bifurcation for critical points of strongly indefinite functionals imply that each nondegenerate and nonnull conjugate (or P-focal)point along a semi-Riemannian geodesic is a bifurcation point.In particular, the semi-Riemannian exponential map is notinjective in any neighborhood of a nondegenerate conjugate point,extending a classical Riemannian result originally due to Morse and Littauer.     

简单高阶奇异性的数值模拟

本文对高阶奇异性的分歧数值计算问题进行了研究，构造了计算简单高阶分歧点的扩充系统，并数值模拟了相应的实际问题。     

Hopf bifurcation in the Lorenz-type chaotic system

In this paper, we analyze dynamical behaviors of the Lorenz-type system via the complementary-cluster energy-barrier criterion. Moreover the Hopf bifurcation of this system is also investigated by means of the first Lyapunov coefficient. As a consequence, it is proved that this system has three Hopf bifurcation points, at which these Hopf bifurcations are nondegenerate and supercritical.     

Asymptotic solutions of Lagrangian systems with gyroscopic forces

We consider Lagrangian systems in the presence of nondegenerate gyroscopic forces. The problem of stability of a degenerate equilibrium pointO and the existence of asymptotic solutions is studied. In particular we show that nondegenerate gyroscopic forces in general have, at least formally, a stabilizing effect whenO is a strict maximum point of the potential energy. It turns out that when we switch on arbitrary small nondegenerate gyroscopic forces, a bifurcation phenomenon arises: the instability properties ofO are transferred to a compact invariant set which collapses atO when the gyroscopic forces are switched off.Work supported by Russian Fund of Basic Research, the Italian Research Council (CNR) and the Italian Ministery of University (MURST)     

数值离散化对环面分支结构的影响

Parameterized dynamical systems with a simple zero eigenvalue and a couple of purely imaginary eigenvalues are considered. It is proved that this type of eigen-structure leads to torus bifurcation under certain nondegenerate conditions. We show that the discrete systems, obtained by discretizing the ODEs using symmetric, eigen-structure preserving schemes, inherit the similar torus bifurcation properties. Predholm theory in Banach spaces is applied to obtain the global torus bifurcation. Our results complement those on the study of discretization effects of global bifurcation.     

简单高阶分歧点的正则化

构造了计算简单高阶分歧点的扩充系统．     

Double $S$-Breaking Cubic Turning Points and Their Computation

1.Introducti0nManynaturalphen0menapossessm0reorlessexactsymmetries,whicharelikelytobereflectedinanysensiblemathematicalm0del.Idealizationssuchasperiodicboundaryconditi0nscanproduceadditionaJsymmetries.Phen0menawhosemodelsexhibitbothsymmetryandnonlinearityleadtopr0blemswhicharechallengingandrichinc0mplexity.Problemswithsymmetriescanshowarichbifurcationbehavi0ur.Theoccurrenceofmultiplesteadystatebifurcationismostlyduetounderlyingsymmetries.Thisgivesrisetothedifficultiestonumericalcomputation-H…     

Prime Quotients of Jordan Systems and Lie Algebras

We show that, unlike alternative algebras, prime quotients of a nondegenerate Jordan system or a Lie algebra need not be nondegenerate, even if the original Jordan system is primitive, or the Lie algebra is strongly prime, both with nonzero simple hearts. Nevertheless, for Jordan systems and Lie algebras directly linked to associative systems, we prove that even semiprime quotients are necessarily nondegenerate.     

对称破坏分歧点的分裂迭代方法

构造了求解对称破坏分歧点的扩充系统，采用分裂分块迭代方法逼近对称破坏分歧点，并对2.Box Brusselator反应模型进行了数值模拟．     

Codimension one and two bifurcations of a discrete stage-structured population model with self-limitation

In this paper, the bifurcations of a discrete stage-structured population model with self-limitation between the two subgroups are investigated. We explore all possible codimension-one bifurcations associated with transcritical, flip (period doubling) and Neimark-Sacker bifurcations and discuss the stabilities of the fixed points in these non-hyperbolic cases. Meanwhile, we give the explicit approximate expression of the closed invariant curve which is caused by the Neimark-Sacker bifurcation. After that, through the theory of approximation by a flow, we explore the codimension two bifurcations associated with 1:3 strong resonance. We convert the nondegenerate condition of 1:3 resonance into a parametric polynomial, and determine its sign by the theory of complete discrimination system. We introduce new parameters and utilize some variable substitutions to obtain the bifurcation curves around 1:3 resonance, which are returned to the original variables and parameters to express for easy verification. By using a series of complicated approximate identity transformations and polar coordinate transformation, we explore 1:6 weak resonance. Moreover, we calculate the two boundaries of Arnold tongue which are caused by 1:6 weak resonance and defined as the resonance region. Numerical simulations and numerical bifurcation analyzes are made to demonstrate the effective of the theoretical analyzes and to present the relations between these bifurcations. Furthermore, our theoretical analyzes and numerical simulations are explained from the biological point of view.     

Dynamics of Two Point Vortices in an External Compressible Shear Flow

This paper is concerned with a system of equations that describes the motion of two point vortices in a flow possessing constant uniform vorticity and perturbed by an acoustic wave. The system is shown to have both regular and chaotic regimes of motion. In addition, simple and chaotic attractors are found in the system. Attention is given to bifurcations of fixed points of a Poincaré map which lead to the appearance of these regimes. It is shown that, in the case where the total vortex strength changes, the “reversible pitch-fork” bifurcation is a typical scenario of emergence of asymptotically stable fixed and periodic points. As a result of this bifurcation, a saddle point, a stable and an unstable point of the same period emerge from an elliptic point of some period. By constructing and analyzing charts of dynamical regimes and bifurcation diagrams we show that a cascade of period-doubling bifurcations is a typical scenario of transition to chaos in the system under consideration.     

A Splitting Iteration Method for a Simple Corank-2 Bifurcation Problem

A splitting iteration method is introduced to approximate a simple corank-2 bifurcation point of a nonlinear equation with small extended systems. This iteration method converges linearly with an adjustable speed and needs little extra computational work.     

二重高阶对称破缺分歧点和它们的数值确定

本文考虑带Z_××Z₂对称性的两参数非线性的二重高阶对称破缺分歧点。利用对称性,我们提出了相应的正则扩张系统来确定这类分歧点。同时指出存在两条平方音叉式分歧点的道路通过该点。     

COMPUTATION OF HOPF BRANCHES BIFURCATING FROM A HOPF/PITCHFORK POINT FOR PROBLEMS WITH Z_2-SYMMETRY

1.IntroductionThispaperisdevotedtothecalculatiollofT,ranchesofHopfpointswhichemallatefromacertaillsingularPointofatwopar:lmeterII()nlinearsystemwhereXisarealHillersspac(},Aabifllrcationparallleter,oranadditionalcontrolparameter,alldgisasnlootllmapping.\Nreassllllle(HI)gisA--synlnletric:thereexistsalillearoperators:X-Xsatisfying(I:identicaloperatorillX)Itiswellknottrnthat(1.2)irldllcestilesplittillgl)Th.firstauthorhasbeedsupportedbyar(>searchgralltoftileV'Olkswagen-StiftungWesaythatxiss…     

Center,Centroid, Extended Centroid,and Quotients of Jordan Systems

In this article we prove that the extended centroid of a nondegenerate Jordan system is isomorphic to the centroid (and to the center in the case of Jordan algebras) of its maximal Martindale-like system of quotients with respect to the filter of all essential ideals.     

Stability and Neimark-Sacker bifurcation of a semi-discrete population model

In this paper, a semi-discrete model is derived for a nonlinear simple population model, and its stability and bifurcation are investigated by invoking a key lemma we present. Our results display that a Neimark-Sacker bifurcation occurs in the positive fixed point of this system under certain parametric conditions. By using the Center Manifold Theorem and bifurcation theory, the stability of invariant closed orbits bifurcated is also obtained. The numerical simulation results not only show the correctness of our theoretical analysis, but also exhibit new and interesting dynamics of this system, which do not exist in its corresponding continuous version.     

Multiple duffing problem in a folding structure with hill-top bifurcation

This paper reviews the theoretical basis and its application for a multiple type of Duffing oscillation. This paper uses a suitable theoretical model to examine the structural instability of a folding truss which is limited so that only vertical displacements are possible for each nodal point supported by both sides. The equilibrium path in this ideal model has been found to have a type of “hill-top bifurcation” from the theoretical work of bifurcation analysis. Dynamic analysis allows for geometrical non-linearity based upon static bifurcation theory. We have found that a simple folding structure based on Multi-Folding-Microstructures theory is more interesting when there is a strange trajectory in multiple homo/hetero-clinic orbits than a well-known ordinary homoclinic orbit, as a model of an extended multiple degrees-of-freedom Duffing oscillation. We found that there are both globally and locally dynamic behaviours for a folding multi-layered truss which corresponds to the structure of the multiple homo/hetero-clinic orbits. This means the numerical solution depends on the dynamic behaviour of the system subjected to the forced cyclic loading such as folding or expanding action. The author suggests simplified theoretical models for hill-top bifurcation that help us to understand globally and locally dynamic behaviours, which depends on the static bifurcation problem. Such models are very useful for forecasting simulations of the extended Duffing oscillation model as essential and invariant nonlinear phenomena.     

Period-doubling bifurcation in an extended van der Pol system with bounded random parameter

An extended van der Pol system with bounded random parameter subjected to harmonic excitation is investigated by Chebyshev polynomial approximation. Firstly the stochastic extended van der Pol system is reduced into its equivalent deterministic one, solvable by suitable numerical methods. Then we explored nonlinear dynamical behavior about period-doubling bifurcation in stochastic system. Numerical simulations show that similar to the conventional period-doubling phenomenon in deterministic extended van der Pol system, stochastic period-doubling bifurcation may also occur in the stochastic extended van der Pol system. Besides, different from the deterministic case, in addition to the conventional bifurcation parameters, i.e. the amplitude and frequency of harmonic excitation, in the stochastic case the intensity of random parameter should also be taken as a new bifurcation parameter.     

N阶分歧问题的数值分析

构造了求解一类非退化分歧点及相关参数的扩充系统，给出了拟牛顿迭代法并证明了收敛性．     

Hopf bifurcations in a Ricardo-Malthus model

Brander and Taylor presented a simple and basic framework for discussing the problem on human population and renewable natural resources in the year 1998, and D’Alessandro recently extended this work mainly by introducing a nonlinear term into the model, if seeing from the mathematical point of view. A limit cycle in this new model was reported by the author via numerically simulated drawing. In this paper, we show that this limit cycle actually is a bifurcating limit cycle of a one-parameter Hopf bifurcation.