Asymptotic Bifurcation Research Articles

Asymptotic Rules of Equilibrium Desingularization

A local bifurcation analysis of a high-dimensional dynamical system dxdt=f(x) is performed using a good deformation of the polynomial mapping P:Cn→Cn. This theory is used to construct geometric aspects of the resolution of multiple zeros of the polynomial vector field P(x). Asymptotic bifurcation rules are derived from Grothendieck’s theory of residuals. Following the Coxeter–Dynkin classification, the singularity graph is constructed. A detailed study of three types of multidimensional mappings with a large symmetry group has been carried out, namely: 1. A linear singularity (behaves similarly to a one-dimensional complex analysis theory); 2. The lattice singularity (generalized the linear and resembling regular crystal growth models); 3. The fan-shaped singularity (can be split radially like nuclear fission and fusion models).

A paradigmatic system for non-classic interactive buckling

A non-classic double bifurcation of discrete static systems, occurring along a non trivial equilibrium path, is studied. It occurs when, by varying a parameter, a pair of eigenvalues of the tangent stiffness matrix first merge and then disappear. A paradigmatic 2 degrees of freedom system with some symmetry, consisting of an inverted extensible pendulum, so far studied in literature in the linear range only, is proposed. Exact nonlinear solutions are derived, showing the mechanism which leads two distinct bifurcated paths to locally touch each other at the double bifurcation point, and then to detach from the fundamental path. The detrimental effects of the interaction on stability are discussed. An asymptotic bifurcation analysis is also carried out, able to explain how a unique buckling mode existing at the double bifurcation is able to generate, in the perturbation process, two distinct deflections on close bifurcated paths.

Structure and Stability of Steady State Bifurcation in a Cannibalism Model with Cross-Diffusion

This paper deals with spatial patterns of a predator-prey crossdiffusion model with cannibalism. By applying the asymptotic analysis and Rabinowitz bifurcation theorem, we consider the local structure of steady state to the model and determine an explicit formula of the nonconstant steady state. Furthermore, the criteria of the stability/instability for the steady state with small amplitude are established.

Asymptotic bifurcation results for coupled nonlinear wave equations with variable coefficients

This paper is concerned with the bifurcation phenomenon of large amplitude periodic solutions of coupled nonlinear wave equations with either constant or variable coefficients. Such model arises when two waves propagate simultaneously in nonisotropic media. The change of degree argument is used to obtain necessary and sufficient conditions for the existence of asymptotic bifurcation points with respect to the matrix spectrum. The results we get are also valid for uncoupled systems. By changing one of our assumptions on the nonlinear term, we also obtain the necessary and sufficient conditions for the existence of the standard bifurcation points.

Hill-top branching: Its asymptotically expanded and visually solved bifurcation equations

Hill-top branching (HTB) is an exceptional stability problem in which the bifurcation point (BP) coincides exactly with a limit point (LP). In such cases, snap-through and branching instability occur simultaneously. Then, HTB is often obscured in snap-through behavior and is often overlooked in stability design. The compound instability is in fact not well realized in the community of computational mechanics, when the rank deficiency and number of negative eigenvalues of the stiffness matrix are carelessly examined. No report of the relevant literature has established diagnosis of path branching or proposed a branch-switching procedure. Furthermore, few HTB models are available in the literature.This report proposes a set of asymptotic bifurcation equations to address this academically important and exciting issue. There might be a single or two critical eigenvectors of the singular stiffness matrix. Higher-derivatives of the stiffness matrix with respect to nodal displacements can be computed exactly using hyper-dual numbers (HDNs). The resulting simultaneous equations can be solved visually using graphical tools available in popular mathematical software packages such as MATLAB®. From the locations of existing real roots displayed on a graphic monitor, the crossover situation of path branching is visually clear above and below the hill-top. For the present study, three HTB examples are computed to verify the proposed algorithm strategy using HDNs and graphical tools. The obtained numerical results show excellent agreement with analytical and finite element predictions. The proposed asymptotically expanded and visually solved bifurcation equations reliably diagnose asymmetric and unstable symmetric HTB in structural stability problems and function well in computational engineering problems.

Bifurcation from infinity for elliptic problems on {mathbb {R}}^N

In the paper the asymptotic bifurcation of solutions to a parameterized stationary semilinear Schrödinger equation involving a potential of the Kato-Rellich type is studied. It is shown that the bifurcation from infinity occurs if the parameter is an eigenvalue of the hamiltonian lying below the asymptotic bottom of the bounded part of the potential. Thus the bifurcating solution are related to bound states of the corresponding Schrödinger equation. The argument relies on the use of the (generalized) Conley index due to Rybakowski and resonance assumptions of the Landesman-Lazer or sign-condition type.

Formulation of a computational asymptotic bifurcation theory applicable to hill-top branching and multiple bifurcation analyses

Formulation of a computational asymptotic bifurcation theory applicable to hill-top branching and multiple bifurcation analyses

Computational two-mode asymptotic bifurcation theory combined with hyper dual numbers and applied to plate/shell buckling

For error-free computation of higher-order derivatives of a complex mathematical expression composed of elementary functions, hyper-dual numbers (HDNs) are receiving close attention in computational engineering and sciences. Differently from conventional finite differences, HDNs numerically evaluate, for instance, the exact first-order and second-order derivatives of the stiffness matrix with respect to nodal degrees-of-freedom (dof) in nonlinear finite element (FE) analysis.One of the most promising applications of HDNs is the asymptotic structural stability theory, which usually requires, in conventional series expansions, an unrealistically large number of second-order derivative of the stiffness matrix to predict the instability behavior of large-scale FE models. This computational bottleneck may be avoided along with so-called classical Lyapunov–Schmidt–Koiter condensation by a novel idea proposed in the present study. The proposed two-mode computational asymptotic theory is implementable in engineering practice to algebraically identify snap-through and path-branching in stability problems. One more associated innovation is a graphical solution of two simultaneous equations resulting from the asymptotic expansions. This sophisticated solution idea utilizing a popular mathematical tool (MATLAB) is widely applicable to quickly visualize the location and number of existing solutions.More specifically, the present paper formulates a two-mode asymptotic bifurcation theory in combination with HDNs and presents an examination of the proposed computational theory on stability problems of plates and shells. Near a precisely computed singular point on the equilibrium path, the incremental displacement field is represented in two modes: θ and δ. One mode (θ) is the critical eigenvector corresponding to the zero eigenvalue of the singular tangent stiffness matrix. The other one (δ) is a linear combination of all other non-critical eigenvectors. θ and δ respectively constitute the homogeneous and particular solutions of the linearized equilibrium equations established in the singular point. Two perturbation parameters, σ and τ, work as weights of θ and δ, respectively, to find possible equilibria around the singular point. The nonlinear equilibrium equations are expanded into asymptotic series of σ and τ to use HDNs to compute 20 coefficients including first-order and second-order derivatives of the stiffness matrix in polynomials. The simultaneous polynomial equations of σ and τ are solved by using the MATLAB graphic operations.The obtained results in numerical examples well predict the stability behavior in excellent agreement with solutions available in conventional FE stability analysis. The two-mode asymptotic bifurcation theory combined with HDNs is sufficient to diagnose snap-through, asymmetric branching, unstable and stable symmetric bifurcation in structural stability problems.

Targeted Energy Transfers for Suppressing Regenerative Machine Tool Vibrations

We study the dynamics of targeted energy transfers in suppressing chatter instability in a single-degree-of-freedom (SDOF) machine tool system. The nonlinear regenerative (time-delayed) cutting force is a main source of machine tool vibrations (chatter). We introduce an ungrounded nonlinear energy sink (NES) coupled to the tool, by which energy transfers from the tool to the NES and efficient dissipation can be realized during chatter. Studying variations of a transition curve with respect to the NES parameters, we analytically show that the location of the Hopf bifurcation point is influenced only by the NES mass and damping coefficient. We demonstrate that application of a well-designed NES renders the subcritical limit cycle oscillations (LCOs) into supercritical ones, followed by Neimark–Sacker and saddle-node bifurcations, which help to increase the stability margin in machining. Numerical and asymptotic bifurcation analyses are performed and three suppression mechanisms are identified. The asymptotic stability analysis is performed to study the domains of attraction for these suppression mechanisms which exhibit good agreement with the bifurcations sets obtained from the numerical continuation methods. The results will help to design nonlinear energy sinks for passive control of regenerative instabilities in machining.

Dynamics of a Wind Turbine with the Working Element Based on an Antiparallel Link Mechanism

A wind turbine with the working element based on an antiparallel link mechanism is proposed. A blade of the turbine is attached to the connecting rod. The flow acting on the blade is supposed to have a constant speed. A quasi-static model of an aerodynamic action is used. A rotor of the electric generator is attached to the axis of the crank. Electromechanical torque is supposed to be a linear function of the angular speed of the rotor of the generator. The mechanical system has one degree of freedom. The angle of rotation of the crank is chosen as a generalized coordinate. A mathematical model of the system was constructed. The attracting 2p- periodic trajectory of the dynamical system corresponds to the working periodic regime of the wind turbine. The Poincare-Pontryagin asymptotic approach is used to obtain the sufficient conditions for the existence and stability of the 2p-periodic trajectories and to estimate the mechanical power in the corresponding modes. Two types of asymptotic bifurcation diagrams of the trapped power were constructed: depending on the average speed of the center of pressure in the working modes, and depending on the coefficient of the external mechanical load. An asymptotic dependence of the average aerodynamic torque on the average speed of the center of pressure was constructed. A laboratory prototype of the turbine was constructed and tested. The prototype proved to be self-starting at the wind speed higher than 4m/s and with a sufficient initial position of a wind-receiving plate.

Localized bulging in an inflated cylindrical tube of arbitrary thickness – the effect of bending stiffness

We study localized bulging of a cylindrical hyperelastic tube of arbitrary thickness when it is subjected to the combined action of inflation and axial extension. It is shown that with the internal pressure P and resultant axial force F viewed as functions of the azimuthal stretch on the inner surface and the axial stretch, the bifurcation condition for the initiation of a localized bulge is that the Jacobian of the vector function (P,F) should vanish. This is established using the dynamical systems theory by first computing the eigenvalues of a certain eigenvalue problem governing incremental deformations, and then deriving the bifurcation condition explicitly. The bifurcation condition is valid for all loading conditions, and in the special case of fixed resultant axial force it gives the expected result that the initiation pressure for localized bulging is precisely the maximum pressure in uniform inflation. It is shown that even if localized bulging cannot take place when the axial force is fixed, it is still possible if the axial stretch is fixed instead. The explicit bifurcation condition also provides a means to quantify precisely the effect of bending stiffness on the initiation pressure. It is shown that the (approximate) membrane theory gives good predictions for the initiation pressure, with a relative error less than 5%, for thickness/radius ratios up to 0.67. A two-term asymptotic bifurcation condition for localized bulging that incorporates the effect of bending stiffness is proposed, and is shown to be capable of giving extremely accurate predictions for the initiation pressure for thickness/radius ratios up to as large as 1.2.

On a criterion for the complete continuity of the Fréchet derivative

In this paper we introduce, by means of the Hausdorff measure of noncompactness χ, two new classes of operators (not necessarily linear): operators locally strongly χ-condensing at a point and operators strongly χ-condensing at infinity (on spherical interlayers). These classes include all completely continuous operators and some noncondensing operators. Necessary and sufficient conditions for the complete continuity of a Frechet derivative at a point and of an asymptotic derivative (if they exist) are proved. M. A. Krasnosel′skii’s theorem on asymptotic bifurcation points for completely continuous vector fields is generalized to the class of vector fields strongly χ-condensing at infinity.

On the existence of positive solutions and solutions with compact support for a spectral nonlinear elliptic problem with strong absorption

We study a semilinear elliptic equation with a strong absorption term given by a non-Lipschitz function. The motivation is related with study of the linear Schrödinger equation with an infinite well potential. We start by proving a general existence result for non-negative solutions. We use also variational methods, more precisely Nehari manifolds, to prove that for any λ>λ1 (the first eigenvalue of the Laplacian operator) there exists (at least) a non-negative solution. These solutions bifurcate from infinity at λ1 and we obtain some interesting additional information. We sketch also an asymptotic bifurcation approach, in particular this shows that there exists an unbounded continuum of non-negative solutions bifurcating from infinity at λ=λ1. We prove that for some neighborhood of (λ1,+∞) the positive solutions are unique. Then Pohozaev’s identity is introduced and we study the existence (or not) of free boundary solutions and compact support solutions. We obtain several properties of the energy functional and associated quantities for the ground states, together with asymptotic estimates in λ, mostly for λ↗λ1. We also consider the existence of solutions with compact support in Ω for λ large enough.

The Applications of Algebraic Methods on Stable Analysis for General Differential Dynamical Systems with Multidelays

The distribution of purely imaginary eigenvalues and stabilities of generally singular or neutral differential dynamical systems with multidelays are discussed. Choosing delays as parameters, firstly with commensurate case, we find new algebraic criteria to determine the distribution of purely imaginary eigenvalues by using matrix pencil, linear operator, matrix polynomial eigenvalues problem, and the Kronecker product. Additionally, we get practical checkable conditions to verdict the asymptotic stability and Hopf bifurcation of differential dynamical systems. At last, with more general case, the incommensurate, we mainly study critical delays when the system appears purely imaginary eigenvalue.

Asymptotic bifurcation and second order elliptic equations on [formula omitted

This paper deals with asymptotic bifurcation, first in the abstract setting of an equation G(u)=λu, where G acts between real Hilbert spaces and λ∈R, and then for square-integrable solutions of a second order non-linear elliptic equation on RN. The novel feature of this work is that G is not required to be asymptotically linear in the usual sense since this condition is not appropriate for the application to the elliptic problem. Instead, G is only required to be Hadamard asymptotically linear and we give conditions ensuring that there is asymptotic bifurcation at eigenvalues of odd multiplicity of the H-asymptotic derivative which are sufficiently far from the essential spectrum. The latter restriction is justified since we also show that for some elliptic equations there is no asymptotic bifurcation at a simple eigenvalue of the H-asymptotic derivative if it is too close to the essential spectrum.

Bifurcation and stability analysis of steady states to a Brusselator model

In this paper we consider the structure of the nonnegative steady state solutions of a Brusselator model. We prove the existence and boundedness of continua of the steady state solutions by applying the theorems of global bifurcation theory. The concentration of one intermediary reactant is treated as a bifurcation parameter. Through the asymptotic bifurcation analysis, we derive an explicit formula for the non-constant steady states (i.e., stationary patterns). Using it as a base, we establish the stability criteria and find a selection mechanism of the wave modes for the stable patterns by computing the leading term of the principal eigenvalue. Our results demonstrate that all of the bifurcations other than the one at the first bifurcation location are unstable, and if the pattern is stable, then its mode must be a positive integer which minimizes the bifurcation parameter. A specific example is presented to illustrate our analytical results.

Dynamics of a discontinuous discrete Beverton–Holt model

Our aim is to investigate the global asymptotic behaviour, oscillation, periodicity and bifurcation in discontinuous Beverton–Holt type difference equationwhere x0>0, functions k and r are discontinuous piecewise constant functionssatisfyingand h is Heaviside function.

Global bifurcation and local multiplicity results for elliptic equations with singular nonlinearity of super exponential growth in $\mathbb{R}^2$

In this paper, we study the solutions to the following singular elliptic problem of exponential type growth posed in a bounded smooth domain $\Omega \subset {{\mathbb R}}^2$: \begin{eqnarray*}( P_\lambda)\qquad \left\{\begin{array} {ll} & - \Delta u = \lambda (u^{-\delta}+h(u)e^{u^\alpha}) \quad\mbox{ in }\,\Omega,\\ & u> 0\quad\mbox{ in }\,\Omega, \;\;u\vert_{\partial\Omega} = 0. \end{array}\right. \end{eqnarray*} Here, $1\leq\alpha\leq 2$, $0<\delta<3$, ${\lambda}\geq 0$ and $h(t)$ is assumed to be a smooth ``perturbation" of $e^{t^{\alpha}}$ as $t \rightarrow \infty$ (see ${\bf (H1)-(H2)}$ below). We show the existence of an unbounded connected branch of solutions to $(P_{\lambda})$ emanating from the trivial solution at ${\lambda}=0$. In the radial case (i.e., when $\Omega =B_1$ and $u$ is radially symmetric) we make a detailed study of the blow-up/convergence of the solution branch as it approaches the asymptotic bifurcation point at infinity. In the critical case $\alpha=2$, we interpret the multiplicity results in terms of the corresponding bifurcation diagrams and the asymptotic profile of large solutions along the branch at infinity.

Global bifurcation for asymptotically linear Schrödinger equations

We prove global asymptotic bifurcation for a very general class of asymptotically linear Schrödinger equations $${\left\{\begin{array}{lll}\Delta u + f(x, u)u = \lambda u \quad {\rm in} \; \mathbb{R}^N,\\ u \in H^1(\mathbb{R}^N) \backslash \{0\}, \quad N \; \geqslant \; 1.\qquad\qquad\qquad(1)\end{array}\right.}$$ The method is topological, based on recent developments of degree theory. We use the inversion $${u\to v:= u/\Vert u\Vert_X^2}$$ in an appropriate Sobolev space $${X=W^{2,p}(\mathbb{R}^{N}),}$$ and we first obtain bifurcation from the line of trivial solutions for an auxiliary problem in the variables $${(\lambda,v) \in {\mathbb R}\times X.}$$ This problem has a lack of compactness and of regularity, requiring a truncation procedure. Going back to the original problem, we obtain global branches of positive/negative solutions ‘bifurcating from infinity’. We believe that, for the values of λ covered by our bifurcation approach, the existence result we obtain for positive solutions of (1) is the most general so far.

Stationary Solutions of a Volume-Filling Chemotaxis Model with Logistic Growth and Their Stability

In this paper, we derive the conditions for the existence of stationary solutions (i.e., nonconstant steady states) of a volume-filling chemotaxis model with logistic growth over a bounded domain subject to homogeneous Neumann boundary conditions. At the same time, we show that the same system without the chemotaxis term does not admit pattern formations. Moreover, based on an explicit formula for the stationary solutions, which is derived by asymptotic bifurcation analysis, we establish the stability criteria and find a selection mechanism of the principal wave modes for the stable stationary solution by estimating the leading term of the principal eigenvalue. We show that all bifurcations except the one at the first location of the bifurcation parameter are unstable, and if the pattern is stable, then its principal wave mode must be a positive integer which minimizes the bifurcation parameter. For a special case where the carrying capacity is one half, we find a necessary and sufficient condition for the stability of pattern solutions. Numerical simulations are presented, on the one hand, to illustrate and fit our analytical results and, on the other hand, to demonstrate a variety of interesting spatio-temporal patterns, such as chaotic dynamics and the merging process, which motivate an interesting direction to pursue in the future.