Global Bifurcation Analysis Research Articles

GLOBAL AND CHAOTIC DYNAMICS FOR A PARAMETRICALLY EXCITED THIN PLATE

The global bifurcations and chaotic dynamics of a parametrically excited, simply supported rectangular thin plate are analyzed. The formulas of the thin plate are derived by von Karman-type equation and Galerkin's approach. The method of multiple scales is used to obtain the averaged equations. Based on the averaged equations, theory of normal form is used to give the explicit expressions of normal form associated with a double zero and a pair of pure imaginary eigenvalues by Maple program. On the basis of the normal form, global bifurcation analysis of the parametrically excited rectangular thin plate is given by a global perturbation method developed by Kovacic and Wiggins. The chaotic motion of thin plate is found by numerical simulation.

Analysis of global dynamics in a parametrically excited thin plate

The global bifurcations and chaos of a simply supported rectangular thin plate with parametric excitation are analyzed. The formulas of the thin plate are derived by von Karman type equation and Galerkin's approach. The method of multiple scales is used to obtain the averaged equations. Based on the averaged equations, the theory of the normal form is used to give the explicit expressions of the normal form associated with a double zero and a pair of pure imaginary eigenvalues by Maple program. On the basis of the normal form, a global bifurcation analysis of the parametrically excited rectangular thin plate is given by the global perturbation method developed by Kovacic and Wiggins. The chaotic motion of thin plate is also found by numerical simulation.

Global Dynamics of a Parametrically and Externally Excited Thin Plate

Both the local and global bifurcations of a parametrically andexternally excited simply supported rectangular thin plate are analyzed.The formulas of the thin plate are derived from the vonKarman equation and Galerkin's method. The method ofmultiple scales is used to find the averaged equations. The numericalsimulation of local bifurcation is given. The theory of normal form,based on the averaged equations, is used to obtain the explicitexpressions of normal form associated with a double zero and a pair ofpurely imaginary eigenvalues from the Maple program. On the basis of thenormal form, global bifurcation analysis of a parametrically andexternally excited rectangular thin plate is given by the globalperturbation method developed by Kovacic and Wiggins. The chaotic motionof the thin plate is found by numerical simulation.

Discounting and Long-Run Behavior: Global Bifurcation Analysis of a Family of Dynamical Systems

This paper is concerned with the relationship between the discount rate and the nature of long-run behavior in dynamic optimization models. The theory is developed under two conditions. The first is history independence, which rules out multiple limit sets. The second is a condition that avoids the reversion to a stable steady state, as the discount factor is lowered, once cycles have emerged. These conditions appear to be the minimal restrictions that would allow analysis by a bifurcation diagram. The results are illustrated by two well-known examples in this literature, due to Sutherland and Weitzman–Samuelson. Journal of Economic Literature Classification Numbers: C61, D90, 041.

Minimizing Sequences Selected via Singular Perturbations, and their Pattern Formation

We study the Cahn-Hilliard energy E ɛ(u) over the unit square under the constraint of a constant mass m with (ɛ > 0) and without ɛ= 0) interfacial energy. Minimizers of E 0(u) have no preferred pattern and we select patterns via sequences of conditionally critical points of E ɛ(u) converging to minimizers as ɛ tends to zero. Those critical points are not minimizers if the singular limit has no minimal interface. We obtain them by a global bifurcation analysis of the Euler-Lagrange equations for E ɛ(u) where the mass m is the bifurcation parameter. We make use of the symmetry of the unit square, and the elliptic maximum principle, in turn, implies that the location of maxima and minima is fixed for all solutions on global branches. This property is used to guarantee the existence of a singular limit and to verify the Weierstrass-Erdmann corner condition which proves its minimizing property.

Analysis of global bifurcations in a market share attraction model

In this paper we demonstrate how the global dynamics of an economic model can be analyzed. In particular, as an application, we consider a market share attraction model widely used in the analysis of interbrand competition in marketing theory. We analyze the local and global dynamic properties of the resulting two-dimensional noninvertible dynamical system in discrete time. The main result of this paper is given by the study of some global bifurcations that change the structure of the attractors and their basins. These bifurcations are investigated by the use of critical curves, a powerful tool for the analysis of the global properties of noninvertible two-dimensional maps.

Cyclic Dynamics in Romantic Relationships

Minimal models composed of two ordinary differential equations are considered in this paper to mimic the dynamics of the feelings between two persons. In accordance with attachment theory, individuals are divided into secure and non-secure individuals, and synergic and non-synergic individuals, for a total of four different classes. Then, it is shown that couples composed of secure individuals, as well as those composed of non-synergic individuals can only have stationary modes of behavior. By contrast, couples composed of a secure and synergic individual and a non-secure and non-synergic individual can experience cyclic dynamics. In other words, the coexistence of insecurity and synergism in the couple is the minimum ingredient for cyclic love dynamics. The result is obtained through a detailed local and global bifurcation analysis of the model. Supercritical Hopf, fold and homoclinic bifurcation curves are numerically detected around a Bogdanov–Takens codimension-2 bifurcation point. The existence of a codimension-2 homoclinic bifurcation is also ascertained. The bifurcation structure allows one to identify the role played by individual synergism and reactiveness to partners love and appeal. It also explains why ageing has a stabilizing effect on the dynamics of the feelings. All results are in agreement with common wisdom on the argument. Possible extensions are briefly discussed at the end of the paper.

Pattern formation of the stationary Cahn-Hilliard model

We investigate critical points of the free energy Eε(u) of the Cahn–Hilliard model over the unit square under the constraint of a mean value ü. We show that for any fixed value ü in the so-called spinodal region and to any mode of an infinite class, there are critical points of Eε(u) having the characteristic symmetries of that mode provided ε > 0 is small enough. As ε tends to zero, these critical points have singular limits forming characteristic patterns for each mode. Furthermore, any singular limit is a stable critical point of E0(u)). Our method consists of a global bifurcation analysis of critical points of the energy Eε(u) where the bifurcation parameter is the mean value ü.

Classical economic growth theory: a global bifurcation analysis

This paper presents a global birfurcation analysis for the classical demoeconomic growth model, in which household income must exceed a threshold before children can survive to adulthood and/or until adults are willing to substitute childrearing for consumption. Convergence to a stationary or steady growth state occurs genetically, but persistent fluctuations are also generic (chaotic or cyclic) around a stationary state, or with expanding amplitude around a growth trend, when technology is advancing. Population can also overshoot its domain of viability, experience a drastic contraction, and then die out. These varied possibilities illustrate global dynamic analysis and may help explain some of the vicissitudes of human societies described in the archaeological and historical records.

A group theoretic approach to the global bifurcation analysis of an axially compressed cylindrical shell

The accurate prediction of the buckling load of thin shell structures is an important yet elusive goal. It is particularly important in the aerospace industry where thin shell members are commonly used as structural elements. Due to a lack of adequate analytical results, current practices in industry put heavy reliance on experimental testing and empirical data to supplement theoretical analysis (see D. Bushnell, Computerized Buckling Analysis of Shells, Martinus Nijhoff, Dordrecht, 1985). This paper focuses on recent results of the group theoretic approach to a numerical, global postbuckling analysis of a perfect, axially compressed cylindrical shell with “built-in” end conditions. The “built-in” end conditions obviate the existence of a “trivial” membrane solution branch. The example of an axially compressed cylindrical shell was chosen because it is well known that for thin shells, the primary axisymmetric solution branch is riddled with closely spaced, symmetry-breaking bifurcation points. In a numerical arc-length continuation scheme, the close proximity of the bifurcation points on the primary path manifests itself in severe ill-conditioning of the tangent stiffness matrix. Group theory helps one systematically find an “optimal” set of basis vectors, or symmetry modes, which reflect the symmetry of a given solution path. The immediate payoff in using these symmetry modes as basis vectors is that the tangent stiffness matrix block-diagonalizes and the numerical ill-conditioning is avoided. Thus, an efficient and accurate technique for computing solution branches of a specific type and a subsequent diagnosis for symmetry-breaking bifurcations is made relatively simple. Understanding the global behavior of the perfect structure is crucial in identifying critical imperfections and will ultimately diminish the heavy reliance on expensive experimental verification.

Computational Use of Group Theory in Bifurcation Analysis of Symmetric Structures

An efficient computational procedure is proposed for identifying singular points in global bifurcation analysis of the static behavior of symmetric discrete structures such as symmetric truss domes. Assuming group equivariance of the system of equations describing the steady state, and making use of group representation theory, the proposed method decomposes the Jacobian matrix (or tangent stiffness matrix) into block-diagonal form, with possibly repeated occurrences of identical blocks, by means of a suitable “local” coordinate transformation. The “local” transformation is computationally favorable in that it requires a small amount of computation and preserves the sparsity of the original Jacobian matrix fairly well. A concrete procedure is described for symmetric truss structures which are equivariant to dihedral groups; an explicit formula is given to the number of diagonal blocks into which the Jacobian matrix splits, and an estimate of the required number of computations shows the efficiency of the proposed method.

Local and global bifurcations in parametrically excited vibrations of nearly square plates

Non-linear flexural vibrations of nearly square plates subject to parametric in-plane excitations are studied. The theoretical results are based on the analysis of a fourth order system of non-linear ordinary differential equations in normal form derived from the dynamic analog of von Karman equations. The equations represent a system with 1:1 resonance and Z 2B + Z 2 or broken D 4 symmetry. Local bifurcation analysis of these equations shows that the system is capable of extremely complex standing as well as travelling waves. A global bifurcation analysis shows the existence of heteroclinic loops which when they break lead to Smale horseshoes and chaotic behavior on an extremely long time scale.

Global bifurcation analysis of an adaptive control system

Global bifurcation analysis of an adaptive control system

Symmetry and nodal properties in the global bifurcation analysis of quasi-linear elliptic equations

Symmetry and nodal properties in the global bifurcation analysis of quasi-linear elliptic equations

Singular global bifurcation problems for the buckling of anisotropic plates

This paper treats a variety of unexpected pathologies that arise in the global bifurcation analysis of axisymmetrie buckled states of anisotropic plates. The geometrically exact plate theory used accounts for flexure, extension and shear. The nonlinear constitutive functions have very general form. As a consequence of the anisotropy the trivial solution may depend discontinuously on the load parameter. Accordingly, the equations for the bifurcation problem have the same character, so that bifurcating branches of solutions become disconnected as the load parameter crosses values at which discontinuities occur. The anisotropy furthermore implies that the governing equations have a singular behaviour much worse than that for isotropic plates. Consequently, a variety of novel constructions are required to demonstrate the validity of the essential results upon which global bifurcation theory stands. (These results include the compactness of certain operators and the uniqueness of solutions of initial value problems for singular ordinary differential equations.) It is shown that in regions of solution-parameter space in which the equations depend continuously on the load parameter there exist connected global branches of solution pairs that have detailed nodal properties inherited from eigenfunctions of the linearized problem. Moreover, these nodal properties are preserved across gaps occurring where discontinuities occur. The methodology used to show this result actually supports constructive methods for finding disconnected branches.

Global bifurcation analysis and uniqueness for a semilinear problem

SynopsisWe study the bifurcation diagram and uniqueness of solutions ofBy using a rescaling technique and the Implicit Function Theorem, we establish the global bifurcation diagram. Uniqueness is proved by a separation argument to complete the bifurcation picture of the problem. Our study suggests that the bifurcation diagrams have different behaviour at λ = 0, depending on whether g(∞) > 0 or g(∞) < 0 in L∞ norm, but quite similar behaviour in Lp or W2,1.

Nonexistence of Global Solutions and Bifurcation Analysis for a Boundary-Value Problem of Parabolic Type

The aim of this paper is to present a bifurcation analysis on the existence of the nonexistence of a global solution for a semilinear parabolic equation and to characterize the local stability and the instability of the corresponding steady-state solutions. The bifurcation result can be described either by a parameter $\lambda$ for a fixed spatial domain $\Omega$ or by varying $\Omega$ for a fixed $\lambda$. The stability analysis gives a result which can be used to determine the stability or instability problem when the system possesses nonintersecting multiple steady-state solutions.

Nonexistence of global solutions and bifurcation analysis for a boundary-value problem of parabolic type

The aim of this paper is to present a bifurcation analysis on the existence of the nonexistence of a global solution for a semilinear parabolic equation and to characterize the local stability and the instability of the corresponding steady-state solutions. The bifurcation result can be described either by a parameter λ \lambda for a fixed spatial domain Ω \Omega or by varying Ω \Omega for a fixed λ \lambda . The stability analysis gives a result which can be used to determine the stability or instability problem when the system possesses nonintersecting multiple steady-state solutions.