Bifurcations in parametric nonlinear programming

Bifurcation and continuation techniques are introduced as a class of methods for investigating the parametric nonlinear programming problem. Motivated by the Fritz John first-order necessary conditions, the parametric programming problem is first reformulated as a closed system of nonlinear equations which contains all Karush-Kuhn-Tucker and Fritz John points, both feasible and infeasible solutions, and relative minima, maxima, and saddle points. Since changes in the structure of the solution set and critical point type can occur only at singularities, necessary and sufficient conditions for the existence of a singularity are developed in terms of the loss of a complementarity condition, the linear dependence constraint qualification, and the singularity of the Hessian of the Lagrangian on a tangent space. After a brief introduction to elementary bifurcation theory, some simple singularities in this parametric problem are analyzed for both branching and persistence of local minima. Finally, a brief introduction to numerical continuation and bifurcation procedures is given to indicate how these facts can be used in a numerical investigation of the problem.This research was supported by the Air force Office of Scientific Research through grant number AFOSR-88-0059.     

Bifurcation problems in nonlinear parametric programming

The nonlinear parametric programming problem is reformulated as a closed system of nonlinear equations so that numerical continuation and bifurcation techniques can be used to investigate the dependence of the optimal solution on the system parameters. This system, which is motivated by the Fritz John first-order necessary conditions, contains all Fritz John and all Karush-Kuhn-Tucker points as well as local minima and maxima, saddle points, feasible and nonfeasible critical points. Necessary and sufficient conditions for a singularity to occur in this system are characterized in terms of the loss of a complementarity condition, the linear dependence of the gradients of the active constraints, and the singularity of the Hessian of the Lagrangian on a tangent space. Any singularity can be placed in one of seven distinct classes depending upon which subset of these three conditions hold true at a solution. For problems with one parameter, we analyze simple and multiple bifurcation of critical points from a singularity arising from the loss of the complementarity condition, and then develop a set of conditions which guarantees the unique persistence of a minimum through this singularity. The research of this author was supported by National Science Foundation through NSF Grant DMS-85-10201 and by the Air Force Office of Scientific Research through instrument number AFOSR-ISSA-85-00079.     

Bifurcation of a three-unit neural network

Considered is a system of delay differential equations modeling a time-delayed connecting network of three neurons without self-feedback. Discussing the change of the number of eigenvalues with zero real part, we locate the boundary of the stability region and finally determine the largest stability region of trivial solution. We investigate the existence of bifurcation phenomena of codimension one/two of the trivial equilibrium by considering the intersections of some parameter curves, which, in the aτ-half parameter plane, correspond to zero root or pure imaginary roots. In particular, the equivariant bifurcation is studied because of the equivariance of the system. We also present numerical simulations to demonstrate the rich dynamical behavior near the equivariant Pitchfork-Hopf bifurcation points, Hopf-Hopf bifurcation points, and some higher codimension bifurcation points.     

Bifurcations and competing coherent structures in the cubic-quintic Ginzburg–Landau equation I: Plane wave (CW) solutions

Singularity theory is used to comprehensively investigate the bifurcations of the steady-states of the traveling wave ODEs of the cubic-quintic Ginzburg–Landau equation (CGLE). These correspond to plane waves of the PDE. In addition to the most general situation, we also derive the degeneracy conditions on the eight coefficients of the CGLE under which the equation for the steady states assumes each of the possible quartic (the quartic fold and an unnamed form), cubic (the pitchfork and the winged cusp), and quadratic (four possible cases) normal forms for singularities of codimension up to three. Since the actual governing equations are employed, all results are globally valid, and not just of local applicability. In each case, the recognition problem for the unfolded singularity is treated. The transition varieties, i.e. the hysteresis, isola, and double limit curves are presented for each normal form. For both the most general case, as well as for various combinations of coefficients relevant to the particular cases, the bifurcation curves are mapped out in the various regions of parameter space delimited by these varieties. The multiplicities and interactions of the plane wave solutions are then comprehensively deduced from the bifurcation plots in each regime, and include features such as regimes of hysteresis among co-existing states, domains featuring more than one interval of hysteresis, and isola behavior featuring dynamics unrelated to the primary solution branch in limited ranges of parameter space.     

Numerical continuation of solution at a singular point of high codimension for systems of nonlinear algebraic or transcendental equations

Numerical continuation of solution through certain singular points of the curve of the set of solutions to a system of nonlinear algebraic or transcendental equations with a parameter is considered. Bifurcation points of codimension two and three are investigated. Algorithms and computer programs are developed that implement the procedure of discrete parametric continuation of the solution and find all branches at simple bifurcation points of codimension two and three. Corresponding theorems are proved, and each algorithm is rigorously justified. A novel algorithm for the estimation of errors of tangential vectors at simple bifurcation points of a finite codimension m is proposed. The operation of the computer programs is demonstrated by test examples, which allows one to estimate their efficiency and confirm the theoretical results.     

分支问题中转接解支的开折方法

分支问题中转接解支的开折方法朱正佑，丛玉豪（兰州大学数学系）ＴＨＥＵＮＦＯＬＤＩＮＧＴＥＣＨＮＩＱＵＥＦＯＲＳＷＩＴＣＨＩＮＧＳＯＬＵＴＩＯＮＢＲＡＮＣＨＥＳＩＮＢＩＦＵＲＣＡＴＩＯＮＰＲＯＢＬＥＭＳ￥ＺｈｕＺｈｅｎｇ－ｙｏｕ；ＣｏｎｇＹｕ－ｈａｏ（...     

Bifurcations of a micro-electromechanical nonlinear coupling system

The dynamical behavior of a micro-electromechanical nonlinear coupling system – deformable micromirror device, is investigated in this paper. In the literature some nonlinear phenomena have been explored by using the numerical method, and saddle-node bifurcation and periodic motions were discovered numerically. Overcoming the obstacle of the unsolvable of the equilibrium points, we analytically obtain the number and stability of the equilibrium points of the system discussed. The saddle-node bifurcation is obtained through the analytic method. Further, both codimension two bifurcations are revealed by the rigorous analysis. Finally, numerical simulations are in good agreement with the theoretical analysis.     

Bifurcation analysis on a generalized recurrent neural network with two interconnected three-neuron components

A class of recurrent neural networks is constructed by generalizing a specific class of n-neuron networks. It is shown that the newly constructed network experiences generic pitchfork and Hopf codimension one bifurcations. It is also proved that the emergence of generic Bogdanov–Takens, pitchfork–Hopf and Hopf–Hopf codimension two, and the degenerate Bogdanov–Takens bifurcation points in the parameter space is possible due to the intersections of codimension one bifurcation curves. The occurrence of bifurcations of higher codimensions significantly increases the capability of the newly constructed recurrent neural network to learn broader families of periodic signals.     

Bifurcation analysis of a Leslie-type predator–prey system with simplified Holling type IV functional response and strong Allee effect on prey

In this paper, a Leslie-type predator–prey system with simplified Holling type IV functional response and strong Allee effect on prey is proposed. The dissipativity of the system and the existence of all possible equilibria are investigated. The investigation emphasizes the exploring of bifurcation. It is shown that the system exists several non-hyperbolic positive equilibria, such as a weak focus of multiplicities one and two, (degenerate) saddle–nodes and Bogdanov–Takens singularities (cusp case) of codimensions 2 and 3. At these equilibria, it is proved that the system undergoes various kinds of bifurcations, such as saddle–node bifurcation, Hopf bifurcation, degenerate Hopf bifurcation and Bogdanov–Takens bifurcation of codimensions 2 and 3. With the parameters selected properly, there exhibits a limit cycle, a homoclinic loop, two limit cycles, a semistable limit cycle, or the simultaneous occurrence of a homoclinic loop and a limit cycle in the system. Moreover, it is also proved that the system has a cusp of codimension at least 4. Hence, there may exist three limit cycles generated from Hopf bifurcation of codimension 3. Numerical simulations are done to support the theoretical results.     

Codimension two bifurcations of fixed points in a class of vibratory systems with symmetrical rigid stops

A vibratory system having symmetrically placed rigid stops and subjected to periodic excitation is considered. Local codimension two bifurcations of the vibratory system with symmetrical rigid stops, associated with double Hopf bifurcation and interaction of Hopf and pitchfork bifurcation, are analyzed by using the center manifold theorem technique and normal form method of maps. Dynamic behavior of the system, near the points of codimension two bifurcations, is investigated by using qualitative analysis and numerical simulation. Hopf-flip bifurcation of fixed points in the vibratory system with a single stop are briefly analyzed by comparison with unfoldings analyses of Hopf-pitchfork bifurcation of the vibratory system with symmetrical rigid stops. Near the value of double Hopf bifurcation there exist period-one double-impact symmetrical motion and quasi-periodic impact motions. The quasi-periodic impact motions are represented by the closed circle and “tire-like” attractor in projected Poincaré sections. With change of system parameters, the quasi-periodic impact motions usually lead to chaos via “tire-like” torus doubling.     

Stable cycles in a Cournot duopoly model of Kopel

We consider a discrete map proposed by M. Kopel that models a nonlinear Cournot duopoly consisting of a market structure between the two opposite cases of monopoly and competition. The stability of the fixed points of the discrete dynamical system is analyzed. Synchronization of two dynamics parameters of the Cournot duopoly is considered in the computation of stability boundaries formed by parts of codim-1 bifurcation curves. We discover more on the dynamics of the map by computing numerically the critical normal form coefficients of all codim-1 and codim-2 bifurcation points and computing the associated two-parameter codim-1 curves rooted in some codim-2 points. It enables us to compute the stability domains of the low-order iterates of the map. We concentrate in particular on the second, third and fourth iterates and their relation to the period doubling, 1:3 and 1:4 resonant Neimark–Sacker points.     

周期激励Stuart-Landau方程的某些动力学行为

研究了周期激励Stuart-Landau方程的锁频周期解．利用奇异性理论分别研究了这些解关于外部激励振幅和频率的分岔行为．结果表明:关于外部激励振幅的普适开折具有余维3,在某些条件下,得到了转迁集及分岔图．另外还证明:关于频率的分岔问题具有无穷余维,因此该情形下的动力学分岔行为非常复杂．发现了一些新的动力学现象,它们是孙亮等所获结果的补充．     

Singularities of Families of Chords

A chord is a straight line joining two points of a pair of hypersurfaces in an affine space such that the tangent hyperplanes at these points are parallel. We classify the singularities of envelopes of the families of chords determined by generic pairs of plane curves and surfaces in three-space. The list contains all bifurcation diagrams of simple boundary singularities (of the corresponding multiplicity).     

An observation concerning the vanishing topology of certain isolated determinantal singularities

We extend the results about the vanishing topology of isolated Cohen–Macaulay codimension 2 singularities from a previous article by Frühbis-Krüger and Zach (On the vanishing topology of isolated Cohen–Macaulay codimension 2 singularities. arXiv:1501.01915, 2015). Due to the Hilbert–Burch theorem, these singularities have a canonical determinantal structure and a well behaved deformation theory, which, in particular, yields a unique Milnor fiber. Studying the case of possibly non-isolated singularities in the Tjurina transform, as introduced in Frühbis-Krüger and Zach (2015), we reveal that in dimension 3 and 2 there is always exactly one special vanishing cycle in degree 2 closely related to the determinantal structure of the singularity.

     

Nowhere minimal CR submanifolds and Levi-flat hypersurfaces

A local uniqueness property of holomorphic functions on real-analytic nowhere minimal CR submanifolds of higher codimension is investigated. A sufficient condition called almost minimality is given and studied. A weaker necessary condition, being contained a possibly singular real-analytic Levi-flat hypersurface is studied and characterized. This question is completely resolved for algebraic submanifolds of codimension 2 and a sufficient condition for noncontainment is given for non algebraic submanifolds. As a consequence, an example of a submanifold of codimension 2, not biholomorphically equivalent to an algebraic one, is given. We also investigate the structure of singularities of Levi-flat hypersurfaces.     

Bifurcation of degenerate homoclinics

We analyze the continuation and bifurcation of homoclinic orbits near a given degenerate homoclinic orbit. We show that the existence of such degenerate homoclinic orbit is a codimension three phenomenon, and that generically the set of parametervalues at which a nearby homoclinic exists forms a codimension one surface which shows a singularity of Whitney umbrella type at the critical parametervalue. The line of self-intersecting points of such surface corresponds to systems which have two nearby homoclinics.     

A Genealogy of Convex Solids Via Local and Global Bifurcations of Gradient Vector Fields

Three-dimensional convex bodies can be classified in terms of the number and stability types of critical points on which they can balance at rest on a horizontal plane. For typical bodies, these are non-degenerate maxima, minima, and saddle points, the numbers of which provide a primary classification. Secondary and tertiary classifications use graphs to describe orbits connecting these critical points in the gradient vector field associated with each body. In previous work, it was shown that these classifications are complete in that no class is empty. Here, we construct 1- and 2-parameter families of convex bodies connecting members of adjacent primary and secondary classes and show that transitions between them can be realized by codimension 1 saddle-node and saddle–saddle (heteroclinic) bifurcations in the gradient vector fields. Our results indicate that all combinatorially possible transitions can be realized in physical shape evolution processes, e.g., by abrasion of sedimentary particles.     

Local analysis near a folded saddle-node singularity

Folded saddle-nodes occur generically in one parameter families of singularly perturbed systems with two slow variables. We show that these folded singularities are the organizing centers for two main delay phenomena in singular perturbation problems: canards and delayed Hopf bifurcations. We combine techniques from geometric singular perturbation theory—the blow-up technique—and from delayed Hopf bifurcation theory—complex time path analysis—to analyze the flow near such folded saddle-nodes. In particular, we show the existence of canards as intersections of stable and unstable slow manifolds. To derive these canard results, we extend the singularly perturbed vector field into the complex domain and study it along elliptic paths. This enables us to extend the invariant slow manifolds beyond points where normal hyperbolicity is lost. Furthermore, we define a way-in/way-out function describing the maximal delay expected for generic solutions passing through a folded saddle-node singularity. Branch points associated with the change from a complex to a real eigenvalue structure in the variational equation along the critical (slow) manifold make our analysis significantly different from the classical delayed Hopf bifurcation analysis where these eigenvalues are complex only.     

Coupling leads to chaos

The main goal of this paper is to prove analytically the existence of strange attractors in a family of vector fields consisting of two Brusselators linearly coupled by diffusion. We will show that such a family contains a generic unfolding of a 4-dimensional nilpotent singularity of codimension 4. On the other hand, we will prove that in any generic unfolding Xμ of an n-dimensional nilpotent singularity of codimension n there are bifurcation curves of (n−1)-dimensional nilpotent singularities of codimension n−1 which are in turn generically unfolded by Xμ. Arguments conclude recalling that any generic unfolding of the 3-dimensional nilpotent singularity of codimension 3 exhibits strange attractors.     

零点不可导时锥映射的歧点与正特征元的全局结构

研究了半序Banach空间中一类非线性锥映射歧点的存在性与正特征元的全局结构.与已知文献不同的是,不要求算子在零点沿着锥Frechet可微. 作为应用,讨论了一类椭圆型偏微分方程边值问题正解的歧点与全局结构.