Liapunov-Schmidt Method Research Articles

The influence of foundation nonlinearity on the post-buckling behavior of a shearable rod near double eigenvalues

The post-buckling behavior of a compressed, simply supported, shearable elastic rod on a nonlinear elastic foundation is studied. For the critical values of auxiliary parameter the eigenvalues are double so that the Liapunov-Schmidt method leads to two bifurcation equations. These equations describe the post-buckling behavior for the values of auxiliary parameter near the critical ones. The occurrence of secondary bifurcations, depending on the nonlinearity of foundation and shear rigidity, is investigated. It is shown that these effects have a significant influence on the post-buckling behavior of the rod. The type of bifurcations, stability, mode interactions and asymptotic expansions of primary and secondary branches are determined. It is shown that the secondary branches can be stable for the nonlinear hardening elastic foundation. The complex bifurcation analysis reveals that for every critical value of the auxiliary parameter, there are eleven different post-buckling behaviors, depending on the nonlinearity of foundation and shear rigidity.

Qualitative analysis on a diffusive predator-prey model with toxins

This paper is concerned with the diffusive predator-prey model with toxins subject to Dirichlet boundary conditions. The uniform persistence of positive solution is given under certain conditions. In addition, by the Liapunov-Schmidt method, the existence and stability of the bifurcation solution from a double eigenvalues are investigated. Moreover, by the fixed point index theory and perturbation theory of eigenvalues, the uniqueness, stability and multiplicity of coexistence states are analyzed when some key parameter changes. Finally, some numerical simulations are presented to verify the theoretical conclusions and further to reflect the importance of parameters to the number of coexistence states.

Number of synchronized and segregated interior spike solutions for nonlinear coupled elliptic systems with continuous potentials

In this paper, we consider the following nonlinear coupled elliptic systems with continuous potentials: $$\left\{ \begin{gathered} - {\varepsilon ^2}\Delta u + \left( {1 + \delta P\left( x \right)} \right)u = {\mu _1}{u^3} + \beta u{v^2}in\Omega , \hfill \\ - {\varepsilon ^2}\Delta v + \left( {1 + \delta Q\left( x \right)} \right)u = {\mu _2}{u^3} + \beta {u^2}vin\Omega , \hfill \\ u > 0,v > 0in\Omega , \hfill \\ \frac{{\partial u}}{{\partial v}} = \frac{{\partial u}}{{\partial v}} = 0on\partial \Omega , \hfill \\ \end{gathered} \right.$$ where Ω is a smooth bounded domain in R N for N = 2, 3, δ, e, μ1 and μ2 are positive parameters, β ∈ R, P(x) and Q(x) are two smooth potentials defined on Ω, the closure of Ω. Due to Liapunov-Schmidt reduction method, we prove that (Ae) has at least O(1/(e|lne|) N ) synchronized and O(1/(e|lne|)2N) segregated vector solutions for e and δ small enough and some β ∈ R. Moreover, for each m ∈ (0,N) there exist synchronized and segregated vector solutions for (Ae) with energies in the order of eN-m. Our results extend the result of Lin et al. (2007) from the Lin-Ni-Takagi problem to the nonlinear Schr¨odinger elliptic systems with continuous potentials.

Secondary bifurcation of a shearable rod with nonlinear spring supports

Abstract The postbuckling behavior of a compressed elastic rod is studied. The rod is supported by two identical hardening spring supports at clamped ends. The constitutive equations of the rod take into account the effect of shear. For local bifurcation analysis the Liapunov-Schmidt method is used. It is shown that for the critical value of auxiliary parameter the lowest eigenvalue becomes double. This fact and double Z 2 symmetry lead to the occurrence of secondary bifurcations for the values of auxiliary parameter near the critical one. For primary and secondary branches asymptotic expansions are constructed. Depending on the values of parameters of nonlinear spring supports and shear rigidity five different groups of bifurcation diagrams are found.

Traveling Waves Connecting Equilibrium and Periodic Orbit for a Delayed Population Model on a Two-Dimensional Spatial Lattice

This paper is concerned with the existence of fast traveling waves connecting an equilibrium and a periodic orbit in a delayed population model with stage structure on a two-dimensional spatial lattice, under the assumption that the corresponding ODEs have heteroclinic orbits connecting an equilibrium point and a periodic solution. In this work, we rewrite the mixed functional differential equation as an integral equation in a Banach space and analyze the corresponding linear operator. Our approach eventually reduces a singular perturbation problem to a regular perturbation problem. The existence of traveling wave solution therefore is obtained by using the Liapunov–Schmidt method and implicit function theorem.

A multiscale three-zone reactive mixing model for engineering a scale separation in enzymatic hydrolysis of cellulose

This multiscale three-zone reactive mixing model provides a theoretical framework for engineering a scale separation in batch enzymatic hydrolysis of cellulose to strategize significant leaps in glucose yields. Formulated using the Liapunov–Schmidt method of the classical bifurcation theory, our model explores the multiscale spatiotemporal dynamics between the fundamental processes of macromixing (convection) and micromixing (diffusion) of the enzymes (Endoglucanase, Exoglucanase, β-glucasidase) and reducing sugars, adsorption and desorption of enzymes on the solid cellulosic substrates, and the product-inhibited liquid and solid phase enzymatic reactions that depolymerize microcrystalline cellulose (Avicel). The model is validated for a range of substrate loadings (2–5%) using our experimental results for the two asymptotic cases of no mixing and continuous mixing, as well as for the macro/micro scale-separated optimal mixing strategy that increases the glucose yield by up to 26% by macromixing completely for an initial period followed by micromixing for the remaining duration of the hydrolysis.

Multiscale analysis of simultaneous uptake of two reactive gases in the human lungs and its application to methemoglobin anemia

We present a novel multiscale modeling and simulation methodology for quantifying the simultaneous uptake of two reactive gases in the human lungs, and apply it to predict pulmonary hypoxemia in patients suffering from methemoglobin anemia (resulting from excess inhaled nitric oxide (NO)). We start with the convection–diffusion–reaction equations at each scale of the lung and apply a spatial averaging technique (based on Liapunov–Schmidt method of the classical bifurcation theory) to obtain low-dimensional multiscale models. Our simulations for methemoglobin anemia show that while breathing in room air, the O2 saturation in the patient's hemoglobin falls to below 94% at 50ppm NO, and above 203ppm, NO causes severe hypoxemia by reducing the O2 saturation to below its critical value of 88%. We predict that patients respond to O2 therapy up to inhaled NO levels of 271ppm, above which they are candidates for methylene blue therapy.

Traveling waves connecting equilibrium and periodic orbit for reaction–diffusion equations with time delay and nonlocal response

A class of reaction–diffusion equations with time delay and nonlocal response is considered. Assuming that the corresponding reaction equations have heteroclinic orbits connecting an equilibrium point and a periodic solution, we show the existence of traveling wave solutions of large wave speed joining an equilibrium point and a periodic solution for reaction–diffusion equations. Our approach is based on a transformation of the differential equations to integral equations in a Banach space and the rigorous analysis of the property for a corresponding linear operator. Our approach eventually reduces a singular perturbation problem to a regular perturbation problem. The existence of traveling wave solution therefore is obtained by the application of Liapunov–Schmidt method and the Implicit Function Theorem.

Periodic Traveling Waves for Diffusion Equations with Time Delayed and Non-local Responding Reaction

We develop a singular perturbation technique to study the existence of periodic traveling wave solutions with large wave speed for a class of reaction-diffusion equations with time delay and non-local response. Unlike the classical singular perturbation method, our approach is based on a transformation of the differential equations to integral equations in a Banach space that reduces the singular perturbation problem to a regular perturbation problem. The periodic traveling wave solutions then are obtained by the use of Liapunov-Schmidt method and a generalized implicit function theorem. The general result obtained has been applied to a non-local reaction-diffusion equation derived from an age-structured population model with a logistic type of birth function.

Exponential Dichotomies, the Fredholm Alternative, and Transverse Homoclinic Orbits in Partial Functional Differential Equations

In this paper we investigate the transversality of homoclinic orbits in partial functional differential equations. We first discuss the exponential dichotomies for linear operator equations. Then we show that the Fredholm Alternative holds if the homogeneous equation has exponential dichotomies on R. Transversality of homoclinic orbits for periodically perturbed partial functional differential equations is studied using the Liapunov-Schmidt method and the Melnikov integral.

Multi-mode low-dimensional models for non-isothermal homogeneous and catalytic reactors

A systematic procedure based on the Liapunov–Schmidt method of bifurcation theory is used to derive low-dimensional models for different types of non-isothermal homogeneous, catalytic and coupled homogeneous–heterogeneous reactors. These low-dimensional models are described by multiple concentration and temperature modes (variables), each of which is representative of a physical scale of the system. These “multi-mode models” capture mass and thermal micromixing as exchange of material and energy, respectively, between the modes (scales). The multi-mode models retain all the parameters and most of the qualitative features of the full convection–diffusion–reaction equations. While in the limit of vanishingly small local heat and mass diffusion times, they reduce to the classical ideal pseudo-homogeneous reactor models, they are also capable of capturing the mixing or mass (and/or heat) transfer-limited asymptotes for the case of fast reactions. We illustrate the usefulness of the multi-mode models in predicting mixing and selectivity effects on reactor performance and the influence of local transport effects on reactor runaway and bifurcation behavior for the case of non-isothermal homogeneous and catalytic reactors.

Hopf Bifurcation for a Class of Partial Differential Equation with Delay

We study a kind of delayed reaction-diffusion equation with Dirichlet boundary condition. We show the existence of a sequence of values {τkn}n=0,1,2,... of the parameter τ such that a Hopf bifurcation occurs when the delay passes through each value {τkn}. The main techniques used here are some results on nonlinear eigenvalue problems, the analysis of the characteristic equation of the linearized problem, the Liapunov-Schmidt method and the implicit function theorem.

Bifurcation property and persistence of configurations for parallel mechanisms

The configuration of parallel mechanisms at the singularity position is uncertain. How to control the mechanism through the singularity position with a given configuration is one of the key problems of the robot controlling. In this paper the bifurcation property and persistence of configurations at the singularity position is investigated for 3-DOF parallel mechanisms. The dimension of the bifurcation equations is reduced by Liapunov-Schmidt reduction method. According to the strong equivalence condition, the normal form which is consistent with the bifurcation condition of the original equation is selected. Through universal unfolding of the bifurcation equation, the influences of the disturbance factors, such as the influence of length of the input component on the configuration persistence at the bifurcation position, are analyzed. Using this method we can obtain the bifurcation curve in which the configuration will be held when the mechanism passes through the singularity position. Therefore, the configuration is under control in this way.

ON THE NORMAL FORMS OF CERTAIN PARAMETRICALLY EXCITED SYSTEMS

In this paper, a modified normal form approach for obtaining normal forms of parametrically excited systems is presented. This approach provides a number of significant advantages over the existing normal form approaches, and improves the associated calculations. The approach lends itself more readily to symbolic calculations, like MAPLE, and the calculations of normal forms, together with the associated coefficients, are carried out much more conveniently. Four examples are presented to illustrate the approach. All examples include a comparison of the results obtained by the methods of normal forms and averaging. Example 4 contains a comparison of the results obtained by the normal form approach and Liapunov–Schmidt method as well.

Low-dimensional models for describing mixing effects in laminar flow tubular reactors

The Liapunov–Schmidt (LS) technique of bifurcation theory is used to average the convective-diffusion equation in the transverse direction and obtain low-dimensional two-mode models that describe mixing effects in laminar flow tubular reactors. For the isothermal case, these models are described by a pair of equations involving two modes, namely, the spatially averaged (〈 C 〉) and the mixing-cup ( C m) concentration vectors. The first equation traces the evolution of C m with residence time, while the second is a local balance equation that describes local mixing as an exchange between the reaction scale (represented by 〈 C 〉) and the convection scale (represented by C m ) in terms of the local mixing time. The LS method also shows that such low-dimensional description is possible only if the local Damköhler number (ratio of local mixing time to reaction time) satisfies the convergence criteria of being less than 0.858. It is shown that the two-mode models have the same accuracy as the infinite (radial) mode convection model, within the range of validity of the latter. The two-mode models for homogeneous reactors have many similarities with the classical two-phase models for heterogeneous catalytic reactors, with the transfer coefficient concept (between surface and mixing cup concentrations, C S and C m , respectively) being replaced by that of an exchange coefficient (between 〈 C 〉 and C m ). Examples are presented to illustrate the usefulness of the two-mode models in predicting the effects of non-identical local mixing times, non-uniform reactant feeding and non-linear kinetics on conversion and yields of products for single and multiple reactions in laminar flow tubular reactors.

Transversal homoclinic orbits for higher dimensional difference equations

In this paper, we use the functional analytic method (theory of exponential dichotomies and Liapunov-Schmidt method) to study the homoclinic bifurcations of higher dimensional difference equations in a degenerate case. We obtain a Melnikov vector mapping for difference equations with the help of which the existence of transversal homoclinic orbits can be detected.

Uniqueness of the Bistable Traveling Wave for Mutualist Species

For a system of reaction–diffusion equations that models the interaction of n mutualist species, the existence of the bistable traveling wave solution has been proved where the nonlinear reaction terms possess a certain type of monotonicity. However the problem of whether there can be two distinct traveling waves remains open. In this paper we use a homotopy approach incorporated with the Liapunov–Schmidt method to show that the bistable traveling wave solution is unique. Our method developed in this paper can also be applied to study the existence and uniqueness of traveling wave solutions for some competition models.

Multiple boundary peak solutions for some singularly perturbed Neumann problems

We consider the problem {ɛ2Δu-u+f(u)=0inΩ,u>0inΩ,∂u/∂v=0on∂Ω,where Ω is a bounded smooth domain in RN, ɛ > 0 is a small parameter and f is a superlinear, subcritical nonlinearity. It is known that this equation possesses boundary spike solutions such that the spike concentrates, as ε approaches zero, at a critical point of the mean curvature function H(P),P ∈ ∂ Ω. It is also known that this equation has multiple boundary spike solutions at multiple nondegenerate critical points of H(P) or multiple local maximum points of H(P).In this paper, we prove that for any fixed positive integer K there exist boundary K-peak solutions at a local minimum point of H(P). This implies that for any smooth and bounded domain there always exist boundary K-peak solutions.We first use the Liapunov–Schmidt method to reduce the problem to finite dimensions. Then we use a maximizing procedure to obtain multiple boundary spikes.

MELNIKOV VECTOR AND HOMOCLINIC BIFURCATIONS IN A DEGENERATE CASE

The main purpose of this paper is to study homoclinic bifurcation in a degenerate case of differential equations of form x˙=f(x⋅ɛ) and x˙=g(x)+h(t,x⋅ɛ). By using theory of exponential dichotomies and Liapunov-Schmidt method, the authors obtain the Melnikov-like vectors by which the existence of homoclinic orbits can be detected.

Buckling of a beam subjected to electromagnetic force and its stabilization by controlling the perturbation of the bifurcation

For a beam subjected to electromagnetic force, magnetoelastic buckling due to the increase of such force is theoretically investigated by taking account of the nonlinearity of the electromagnetic force and the elastic force of the beam. Using Liapunov-Schmidt method and center manifold theory, the equilibrium space, the bifurcation set and the bifurcation diagram are theoretically derived. Also, the effect of the higher modes other than the buckling mode on the mode shape of the postbuckling state is discussed. Furthermore, a control method to stabilize the magnetoelastic buckling is proposed, and the unstable equilibrium state of the beam in the postbuckling state, i.e., the straight position of the beam, is stabilized by controlling the perturbation of the bifurcation.