Schmidt Reduction Research Articles

Hidden Symmetry in a Kuramoto–Sivashinsky Initial-Boundary Value Problem

We investigate the bifurcation structure of the Kuramoto–Sivashinsky equation with homogeneous Dirichlet boundary conditions. Using hidden symmetry principles, based on an extended problem with periodic boundary conditions and [Formula: see text] symmetry, we show that the zero solution exhibits two kinds of pitchfork bifurcations: one that breaks the reflection symmetry of the system with Dirichlet boundary conditions and one that breaks a shift-reflect symmetry of the extended system. Using Lyapunov–Schmidt reduction, we show both to be supercritical. We extend the primary branches by means of numerical continuation, and show that they lose stability in pitchfork, transcritical or Hopf bifurcations. Tracking the corresponding secondary branches reveals an interval of the viscosity parameter in which up to four stable equilibria and time-periodic solutions coexist. Since the study of the extended problem is indispensable for the explanation of the bifurcation structure, the Kuramoto–Sivashinsky problem with Dirichlet boundary conditions provides an elegant manifestation of hidden symmetry.

Gelfand Type Elliptic Problem Involving Advection

We consider the following Gelfand type elliptic problem involving advection -\Delta u+a(x) \cdot \nabla u=e^{u}\ \ \text{in}\ \mathbb R^{N}, where a(x) is a smooth vector field. According to energy estimates, we obtain the nonexistence results of stable solution for this equation under some restrict conditions about a(x) for N\leq 9 .On the other hand, combining Liapunov–Schmidt reduction method, we prove that it possesses a solution for N\geq 4 . Besides, if a is divergence free and satisfies a smallness condition, then the equation above admits a stable solution for N\geq11 .

Inertial Manifold and Large Deviations Approach to Reduced PDE Dynamics

In this paper a certain type of reaction–diffusion equation—similar to the Allen-Cahn equation—is the starting point for setting up a genuine thermodynamic reduction i.e. involving a finite number of parameters or collective variables of the initial system. We firstly operate a finite Lyapunov–Schmidt reduction of the cited reaction–diffusion equation when reformulated as a variational problem. In this way we gain a finite-dimensional ODE description of the initial system which preserves the gradient structure of the original one and that is exact for the static case and only approximate for the dynamic case. Our main concern is how to deal with this approximate reduced description of the initial PDE. To start with, we note that our approximate reduced ODE is similar to the approximate inertial manifold introduced by Temam and coworkers for Navier–Stokes equations. As a second approach, we take into account the uncertainty (loss of information) introduced with the above mentioned approximate reduction by considering the stochastic version of the ODE. We study this reduced stochastic system using classical tools from large deviations, viscosity solutions and weak KAM Hamilton–Jacobi theory. In the last part we suggest a possible use of a result of our approach in the comprehensive treatment non equilibrium thermodynamics given by Macroscopic Fluctuation Theory.

Bifurcation Phenomena in a Lotka–Volterra Model with Cross-Diffusion and Delay Effect

This paper focuses on a Lotka–Volterra model with delay and cross-diffusion. By using Lyapunov–Schmidt reduction, we investigate the existence, multiplicity, stability and Hopf bifurcation of spatially nonhomogeneous steady-state solutions. Furthermore, we obtain some criteria to determine the bifurcation direction and stability of Hopf bifurcating periodic orbits by using Lyapunov–Schmidt reduction.

Bifurcation of Reaction Cross-Diffusion Systems

This paper is devoted to a reaction cross-diffusion system under Neumann boundary conditions. Firstly, the existence and multiplicity of spatially nonhomogeneous/homogeneous steady-state solutions are investigated by means of Lyapunov–Schmidt reduction. Next, the linear stability and Hopf bifurcations of homogeneous steady-state solutions are described in detail. In particular, the Hopf bifurcation direction and the stability of bifurcating time-periodic solutions are determined by using center manifold reduction and normal form theory. Finally, some of the main results are illustrated by an application to a predator–prey model with Allee effect and one-dimensional spatial domain [Formula: see text].

Clifford Tori and the singularly perturbed Cahn–Hilliard equation

In this paper we construct entire solutions uε to the Cahn–Hilliard equation −ε2Δ(−ε2Δu+W′(u))+W″(u)(−ε2Δu+W′(u))=ε4λε(1−uε), under the volume constraint ∫R3(1−uε)2dx=82π2cε, with cε→1 as ε→0, whose nodal set approaches the Clifford Torus, that is the Torus with radii of ratio 1/2 embedded in R3, as ε→0. It is crucial that the Clifford Torus is a Willmore hypersurface and it is non-degenerate, up to conformal transformations. The proof is based on the Lyapunov–Schmidt reduction and on careful geometric expansions of the Laplacian.

Hopf bifurcation in a reaction-diffusive two-species model with nonlocal delay effect and general functional response

A nonlocal delayed reaction-diffusive two-species model with Dirichlet boundary condition and general functional response is investigated in this paper. Based on the Lyapunov–Schmidt reduction, the existence, bifurcation direction and stability of Hopf bifurcating periodic orbits near the positive spatially nonhomogeneous steady-state solution are obtained, where the time delay is taken as the bifurcation parameter. Moreover, the general results are applied to a diffusive Lotka–Volterra type food-limited population model with nonlocal delay effect, and it is found that diffusion and nonlocal delay can also affect the other dynamic behavior of the system by numerical experiments.

Bifurcation of Bounded Solutions of Impulsive Differential Equations

In this article, we examine nonautonomous bifurcation patterns in nonlinear systems of impulsive differential equations. The approach is based on Lyapunov–Schmidt reduction applied to the linearization of a particular nonlinear integral operator whose zeroes coincide with bounded solutions of the impulsive differential equation in question. This leads to sufficient conditions for the presence of fold, transcritical and pitchfork bifurcations. Additionally, we provide a computable necessary condition for bifurcation in nonlinear scalar impulsive differential equations. Several examples are provided illustrating the results.

Existence and stability of coexistence states in a competition unstirred chemostat

In this paper, we consider the two similar competing species in a competition unstirred chemostat model with diffusion. The two competing species are assumed to be identical except for their maximal growth rates. In particular, we study the existence and stability of the coexistence states, and the semi-trivial equilibria or the unique coexistence state is the global attractor can be established under some suitable conditions. Our mathematical approach is based on Lyapunov–Schmidt reduction, the implicit function theory and spectral theory.

Stability analysis of a stage structure model with spatiotemporal delay effect

This paper is concerned with a stage structure model with spatiotemporal delay and homogeneous Dirichlet boundary condition. The existence of steady state solution bifurcating from the trivial equilibrium is obtained by using Lyapunov–Schmidt reduction. The stability analysis of the positive spatially nonhomogeneous steady state solution is investigated by a detailed analysis of the characteristic equation. Using the properties of the omega limit set, we obtain the global convergence of the solution with finite delay.

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

In this paper, we consider the following nonlinear coupled elliptic systems(Aε){−ε2Δu+u=μ1u3+βuv2in Ω,−ε2Δv+v=μ2v3+βu2vin Ω,u>0,v>0in Ω,∂u∂ν=∂v∂ν=0on ∂Ω, where ε>0, μ1>0, μ2>0, β∈R, and Ω is a bounded domain with smooth boundary in R3. Due to Lyapunov–Schmidt reduction method, we proved that (Aε) has at least O(1ε3|ln⁡ε|) synchronized and segregated vector solutions for ε small enough and some β∈R. Moreover, for each m∈(0,3) there exist synchronized and segregated vector solutions for (Aε) with energies in the order of ε3−m. Our result extends the result of Lin, Ni and Wei [20], from the Lin–Ni–Takagi problem to the nonlinear elliptic systems.

Standing Waves in Near-Parallel Vortex Filaments

A model derived in (Klein et al., J Fluid Mech 288:201–248, 1995) for n near-parallel vortex filaments in a three dimensional fluid region takes into consideration the pairwise interaction between the filaments along with an approximation for motion by self-induction. The same system of equations appears in descriptions of the fine structure of vortex filaments in the Gross–Pitaevski model of Bose–Einstein condensates. In this paper we construct families of standing waves for this model, in the form of n co-rotating near-parallel vortex filaments that are situated in a central configuration. This result applies to any pair of vortex filaments with the same circulation, corresponding to the case n = 2. The model equations can be formulated as a system of Hamiltonian PDEs, and the construction of standing waves is a small divisor problem. The methods are a combination of the analysis of infinite dimensional Hamiltonian dynamical systems and linear theory related to Anderson localization. The main technique of the construction is the Nash–Moser method applied to a Lyapunov–Schmidt reduction, giving rise to a bifurcation equation over a Cantor set of parameters.

Small amplitude periodic solutions of Klein–Gordon equations

We consider a class of nonlinear Klein–Gordon equations utt=uxx−u+f(u) and obtain a family of small amplitude periodic solutions, where the temporal and spatial period have different scales. The proof is based on a combination of Lyapunov–Schmidt reduction, averaging and Nash–Moser iteration.

On homoclinic orbits to center manifolds of elliptic–hyperbolic equilibria in Hamiltonian systems

We consider a Hamiltonian system which has an elliptic–hyperbolic equilibrium with a homoclinic loop. We identify the set of orbits which are homoclinic to the center manifold of the equilibrium via a Lyapunov–Schmidt reduction procedure. This leads to the study of a singularity which inherits a certain structure from the Hamiltonian nature of the system. Under non-degeneracy assumptions, we classify the possible Morse indices of this singularity, permitting a local description of the set of homoclinic orbits. We also consider the case of time-reversible Hamiltonian systems.

Stability analysis of a nonlinear electromechanical coupling transmission system with time delay feedback

The static bifurcation characteristic and the stability of a nonlinear electromechanical coupling system with time delay are studied in this paper. Considering the energy in air-gap field of AC motor, the dynamic equation of electromechanical coupling transmission system is deduced by using the dissipation Lagrange equation. The equivalent low-dimensional bifurcation equation is obtained by using the method of Lyapunov–Schmidt reduction. And the bifurcation characteristic is analyzed by the method of the singularity theory. In order to control the dynamic behavior of the system, a time delay feedback is introduced and the distribution of the roots is studied. The analytical conditions that determine the feedback gain and time delay are given based on the delay differential equation stability theory. Under certain conditions, it is shown that the zero solution is locally asymptotically stable when the time delay is suitably small, while the change of stability of zero solution will cause a bifurcating periodic solution as the time delay passes through a certain critical value. Numerical simulations are also performed, which confirm the analytical results.

Bifurcation and stability of a Mimura–Tsujikawa model with nonlocal delay effect

In this paper, we investigate a Mimura–Tsujikawa model with nonlocal delay effect under the homogeneous Neumann boundary condition. By using Lyapunov–Schmidt reduction, we investigate the existence, multiplicity, stability, and Hopf bifurcation of nontrivial steady‐state solutions bifurcating from the nonzero steady‐state solution. Moreover, we illustrate our general results by applications to models with a one‐dimensional spatial domain. Copyright © 2016 John Wiley & Sons, Ltd.

Bifurcation results for a fractional elliptic equation with critical exponent in $$\mathbb {R}^n$$ R n

In this paper we study some nonlinear elliptic equations in $${\mathbb R}^n$$ obtained as a perturbation of the problem with the fractional critical Sobolev exponent, that is $$\begin{aligned} (-\Delta )^s u = \varepsilon \,h\,u_+^q + u_+^p { \text{ in } }{\mathbb R}^n, \end{aligned}$$ where $$s\in (0,1)$$ , $$n>4s$$ , $$\varepsilon >0$$ is a small parameter, $$p=\frac{n+2s}{n-2s}$$ , $$0<q<p$$ and h is a continuous and compactly supported function. To construct solutions to this equation, we use the Lyapunov–Schmidt reduction, that takes advantage of the variational structure of the problem. For this, the case $$0<q<1$$ is particularly difficult, due to the lack of regularity of the associated energy functional, and we need to introduce a new functional setting and develop an appropriate fractional elliptic regularity theory.

Lyapunov–Schmidt Reduction for Fractional Differential Systems

Lyapunov–Schmidt reduction is one of the powerful and significant methods to simplify nonlinear ordinary differential systems (ODSs) with standard formulation. In this paper, we extend Lyapunov–Schmidt reduction to fractional ordinary differential systems (FODSs) with Caputo derivatives.

Stability and bifurcation in a diffusive Lotka–Volterra system with delay

In this paper, we investigate the dynamics of a class of diffusive Lotka–Volterra equation with time delay subject to the homogeneous Dirichlet boundary condition in a bounded domain. The existence of spatially nonhomogeneous steady state solution is investigated by applying Lyapunov–Schmidt reduction. The stability and nonexistence of Hopf bifurcation at the spatially nonhomogeneous steady-state solution with the changes of a specific parameter are obtained by analyzing the distribution of the eigenvalues. Moreover, we illustrate our general results by applications to models with a single delay and one-dimensional spatial domain.

Stability and Bifurcation in a Delayed Reaction–Diffusion Equation with Dirichlet Boundary Condition

In this paper, we study the dynamics of a diffusive equation with time delay subject to Dirichlet boundary condition in a bounded domain. The existence of spatially nonhomogeneous steady-state solution is investigated by applying Lyapunov–Schmidt reduction. The existence of Hopf bifurcation at the spatially nonhomogeneous steady-state solution is derived by analyzing the distribution of the eigenvalues. The direction of Hopf bifurcation and stability of the bifurcating periodic solution are also investigated by means of normal form theory and center manifold reduction. Moreover, we illustrate our general results by applications to the Nicholson’s blowflies models with one- dimensional spatial domain.