Multiple Stationary Solutions Research Articles

Relative Morse index and multiple solutions for a non-periodic Dirac equation with external fields

This paper is concerned with the stationary solutions of the Dirac equation −i∑k=13αk∂ku+aβu+ωu+V(x)u=Gu(x,u),where G is asymptotically quadratic and is not assumed to be C2. We present a new approach to construct an index theory for the associated linear Dirac equation and define the relative Morse index to measure the difference between the nonlinearity at the origin and at infinity. By building upon the idea of combining the index theory and a generalized linking theorem, we obtain existence and multiplicity of stationary solutions.

Event-triggered impulse control on reaction–diffusion Gilpin–Ayala competition model with multiple stationary solutions

In previous literature, the parameter restrictions for the reaction–diffusion Gilpin–Ayala competition ecosystem were limited to the control of the first eigenvalue of the Laplace operator. This paper considers extending the parameter to the control of the second eigenvalue. However, exceeding the first eigenvalue results in a loss of compactness in the energy functional of the determined ecosystem. Consequently, this paper discusses the orthogonal decomposition of a given Sobolev space and variational techniques, obtaining the existence of multiple equilibrium points. The presence of multiple equilibrium points is a common phenomenon in real-world ecosystems and implies that the system cannot be globally stable under normal circumstances. However, this study develops an effective event-triggered impulse mechanism (ETIM) that achieves global stability for the positive equilibrium solution in the ecosystem. Notably, the newly designed ETIM not only eliminates Zeno behaviour but also reduces the cost of impulse control. This is achieved by removing some unnecessary impulses in the fixed impulsive instants mechanism associated with the ecosystem in previous literature. Finally, a numerical simulation validates the effectiveness of the proposed method.

Computing solution landscape of nonlinear space-fractional problems via fast approximation algorithm

The nonlinear space-fractional problems often allow multiple stationary solutions, which can be much more complicated than the corresponding integer-order problems. In this paper, we systematically compute the solution landscapes of nonlinear constant/variable-order space-fractional problems on one- and two-dimensional rectangular domains. A fast approximation algorithm is developed to deal with the variable-order spectral fractional Laplacian by approximating the variable-indexing Fourier modes, and then combined with saddle dynamics to construct the solution landscape of variable-order space-fractional phase field model. Numerical experiments are performed to substantiate the accuracy and efficiency of fast approximation algorithm and elucidate essential features of the stationary solutions of space-fractional phase field model. Furthermore, we demonstrate that the solution landscapes of spectral fractional Laplacian problems can be reconfigured by varying the diffusion coefficients in the corresponding integer-order problems.

Global Stability of Delayed Ecosystem via Impulsive Differential Inequality and Minimax Principle

This paper reports applying Minimax principle and impulsive differential inequality to derive the existence of multiple stationary solutions and the global stability of a positive stationary solution for a delayed feedback Gilpin–Ayala competition model with impulsive disturbance. The conclusion obtained in this paper reduces the conservatism of the algorithm compared with the known literature, for the impulsive disturbance is not limited to impulsive control.

Impulsive stabilization and stability analysis for Gilpin–Ayala competition model involved in harmful species via LMI approach and variational methods

Firstly, a dynamic analysis for reaction–diffusion Gilpin–Ayala competition model involved in harmful species is considered under Dirichlet boundary value condition. Existence of multiple stationary solutions is verified by way of Mountain Pass lemma, and the local stability result of the null solution is obtained by employing linear approximation principle. Secondly, the authors utilize variational methods and linear matrix inequality (LMI) technique to deduce the LMI-based global exponential stability criterion on the null solution which becomes the unique stationary solution of a Markovian jumping ecosystem with delayed feedback under a reasonable boundedness assumption on population densities. Particularly, LMI criterion is involved in free weight coefficient matrix, which reduces the conservatism of the algorithm. In addition, a new impulse control stabilization criterion is also derived, in which no differentiable assumptions on time-delayed functions are proposed. Finally, three numerical examples show the effectiveness of the proposed methods. It is worth mentioning that the obtained stability criteria of null solution presented some useful hints on how to eliminate pests and bacteria.

Computing Solution Landscape of Nonlinear Space-Fractional Problems Via Fast Approximation Algorithm

The nonlinear space-fractional problems often allow multiple stationary solutions, which can be much more complicated than the corresponding integer-order problems. In this paper, we systematically compute the solution landscapes of nonlinear constant/variable-order space-fractional problems. A fast approximation algorithm is developed to deal with the variable-order spectral fractional Laplacian by approximating the variable-indexing Fourier modes, and then combined with saddle dynamics to construct the solution landscape of variable-order space-fractional phase field model. Numerical experiments are performed to substantiate the accuracy and efficiency of fast approximation algorithm and elucidate essential features of the stationary solutions of space-fractional phase field model. Furthermore, we demonstrate that the solution landscapes of spectral fractional Laplacian problems can be reconfigured by varying the diffusion coefficients in the corresponding integer-order problems.

Multiple stationary solutions of parabolic problem with discontinuous nonlinearities and their stability

In this paper, we discuss several results regarding existence, multiplicity and stability of solutions of differential equation with discontinuous nonlinearities relevant in several applications. Our problem can be formulated as a free boundary problem with unknown boundaries to be characterized. We first show that the bifurcation diagram is a generalized ‘S-shaped’ curve. Then we study the stability of different parts of the bifurcation curve. Our approach is based on the comparison principle and some monotonicity arguments.

Construction of a Pathway Map on a Complicated Energy Landscape

How do we search for the entire family tree of possible intermediate states, without unwanted random guesses, starting from a stationary state on the energy landscape all the way down to energy minima? Here we introduce a general numerical method that constructs the pathway map, which guides our understanding of how a physical system moves on the energy landscape. The method identifies the transition state between energy minima and the energy barrier associated with such a state. As an example, we solve the Landau-de Gennes energy incorporating the Dirichlet boundary conditions to model a liquid crystal confined in a square box; we illustrate the basic concepts by examining the multiple stationary solutions and the connected pathway maps of the model.

Human Capital, Polarisation and Pareto-improving Activating Welfare

Human capital not only earns income but also is a source of direct felicity. The interaction between these two motives for effort can generate multiple stationary solutions for individual household optimisation. This multiplicity freezes any sufficiently unequal initial skill distribution and persistently segregates households into two separate groups: one rich and educated, one poor and uneducated. If public goods are to be financed, polarisation is typically Pareto-inefficient. While unconditional transfers only reduce the incentive to accumulate skills, activating tax-transfer systems that Pareto-dominate any non-redistributing system exist. Transfers are transitory and a negative marginal income tax is imposed on poor households’ income.

Multiplicity results by shooting reproducing kernel Hilbert space method for the catalytic reaction in a flat particle

In this paper, a model of simultaneous mass and heat transfer within a porous catalyst in a flat particle is considered. A new modification of the shooting reproducing kernel Hilbert space (SRKHS) method is proposed, which is also capable of handling the system of nonlinear boundary value problems by employing Newtons method. The proposed method is a well-performance technique in both predicting and calculating multiple solutions of the nonlinear boundary value problems. Applying the SRKHS method shows that the mentioned model might admit multiple stationary solutions (unique, dual or triple solutions) depending on the values of the parameters of the model. Furthermore, the convergence of the method is proved and some numerical tests reveal the high efficiency of this new version of SRKHS method.

Multiple critical points of saddle geometry functionals

We study the multiplicity of critical points for continuously differentiable functionals on real Banach spaces. We prove that a functional which satisfies the assumptions of the Saddle Point Theorem and moreover is bounded from below has at least three critical points. Apparently, there is a global minimizer and a saddle point and we show the existence of a third critical point. The idea of the proof is based on the minus-gradient flow. This result is closely related to the three critical points theorem of H. Brezis and L. Nirenberg which assumes a local linking. Finally, we apply the result on the Dirichlet problem for semilinear stationary PDEs. The analysis includes, for example, the existence of multiple stationary solutions of bistable (or Allen–Cahn) equation and semipositone problems.

Infinitely many solutions for a class of superlinear Dirac–Poisson system

This paper is concerned with the nonlinear Dirac–Poisson system −i∑k=13αk∂ku+(V(x)+a)βu+ωu−ϕu=Fu(x,u),−Δϕ=4π|u|2,inR3where V is an external potential and F is a superlinear nonlinearity modeling various types of interactions. Existence and multiplicity of stationary solutions are obtained for the system with periodicity condition via variational methods.

Poisson–Nernst–Planck equations with steric effects — non-convexity and multiple stationary solutions

We study the existence and stability of stationary solutions of Poisson–Nernst–Planck equations with steric effects (PNP-steric equations) with two counter-charged species. We show that within a range of parameters, steric effects give rise to multiple solutions of the corresponding stationary equation that are smooth. The PNP-steric equation, however, is found to be ill-posed at the parameter regime where multiple solutions arise. Following these findings, we introduce a novel PNP–Cahn–Hilliard model, show that it is well-posed and that it admits multiple stationary solutions that are smooth and stable. The various branches of stationary solutions and their stability are mapped utilizing bifurcation analysis and numerical continuation methods.

The existence of multiple equilibrium points in global attractors for some symmetric dynamical systems II

This paper is a continuation of Zhong et al. (2014), we go on discussing some properties of the global attractor for some symmetric dynamical system with a Lyapunov function F in a Banach space. The difference between this paper and Zhong et al. (2014) is that the origin is not a local minimum point but a saddle point of F. Under some suitable assumptions, we establish an abstract result about the existence of the multiple equilibrium points in the global attractor by estimating the lower bound of Z2-index of the global attractor. As an application of this abstract result, we consider the existence of multiple stationary solutions for some semilinear reaction–diffusion equation when the origin is an unstable equilibrium point.

Phase field simulations of plastic strain-induced phase transformations under high pressure and large shear

Pressure and shear strain-induced phase transformations (PTs) in a nanograined bicrystal at the evolving dislocations pile-up have been studied utilizing a phase field approach (PFA). The complete system of PFA equations for coupled martensitic PT, dislocation evolution, and mechanics at large strains is presented and solved using the finite element method (FEM). The nucleation pressure for the high-pressure phase (HPP) under hydrostatic conditions near a single dislocation was determined to be 15.9 GPa. Under shear, a dislocation pile-up that appears in the left grain creates strong stress concentration near its tip and significantly increases the local thermodynamic driving force for PT, which causes nucleation of HPP even at zero pressure. At pressures of 1.59 and 5 GPa and shear, a major part of a grain transforms to HPP. When dislocations are considered in the transforming grain as well, they relax stresses and lead to a slightly smaller stationary HPP region than without dislocations. However, they strongly suppress nucleation of HPP and require larger shear. Unexpectedly, the stationary HPP morphology is governed by the simplest thermodynamic equilibrium conditions, which do not contain contributions from plasticity and surface energy. These equilibrium conditions are fulfilled either for the majority of points of phase interfaces or (approximately) in terms of stresses averaged over the HPP region or for the entire grain, despite the strong heterogeneity of stress fields. The major part of the driving force for PT in the stationary state is due to deviatoric stresses rather than pressure. While the least number of dislocations in a pile-up to nucleate HPP linearly decreases with increasing applied pressure, the least corresponding shear strain depends on pressure nonmonotonously. Surprisingly, the ratio of kinetic coefficients for PT and dislocations affect the stationary solution and the nanostructure. Consequently, there are multiple stationary solutions under the same applied load and PT, and deformation processes are path dependent. With an increase in the size of the sample by a factor of two, no effect was found on the average pressure and shear stress and HPP nanostructure, despite the different number of dislocations in a pile-up. The obtained results represent a nanoscale basis for understanding and description of PTs under compression and shear in a rotational diamond anvil cell and high-pressure torsion.

Boundary-induced pattern formation from temporal oscillation: Spatial map analysis

Boundary-induced pattern formation is investigated using spatially one-dimensional, two-component reaction-diffusion equations. Temporal oscillation is successively transformed into a spatial pattern, triggered by diffusion from the fixed boundary. We introduced a spatial map, whose temporal sequence, under selection criteria from multiple stationary solutions, can completely reproduce the emergent pattern, by replacing the time with space. The relationship of the pattern wavelength with the period of oscillation is also obtained The pattern wavelength is proportional to the period of the oscillation when the emergent pattern is periodic. The generality of the pattern selection process and algorithm is discussed with possible relevance to biological morphogenesis.

On the exact results of a non-linear model arising in catalytic reaction in a flat particle with power kinetics reaction rate

The latest study of a model of heat and mass transfer for a catalytic reaction within a porous catalyst flat particle (Journal of the Taiwan Institute of Chemical Engineers 48 (2015) 49–55 [7]) shows that this model can be exactly solved when the reaction rate is taken linear term as f(y)=y, where y is heat within porous catalyst particle. In this paper, the mentioned model is revisited by considering the power kinetics nth order reaction rate instead of a linear reaction term. It is shown that the problem is still exactly tractable. Furthermore, it is revealed that previous results can be recovered from this exactly solvable generalization case. It is also proved that the problem might have multiple stationary solutions (unique, dual and triple solutions) depending on the values of the parameters of the model.

Multiple solutions and numerical analysis to the dynamic and stationary models coupling a delayed energy balance model involving latent heat and discontinuous albedo with a deep ocean.

We study a climatologically important interaction of two of the main components of the geophysical system by adding an energy balance model for the averaged atmospheric temperature as dynamic boundary condition to a diagnostic ocean model having an additional spatial dimension. In this work, we give deeper insight than previous papers in the literature, mainly with respect to the 1990 pioneering model by Watts and Morantine. We are taking into consideration the latent heat for the two phase ocean as well as a possible delayed term. Non-uniqueness for the initial boundary value problem, uniqueness under a non-degeneracy condition and the existence of multiple stationary solutions are proved here. These multiplicity results suggest that an S-shaped bifurcation diagram should be expected to occur in this class of models generalizing previous energy balance models. The numerical method applied to the model is based on a finite volume scheme with nonlinear weighted essentially non-oscillatory reconstruction and Runge-Kutta total variation diminishing for time integration.

The existence of multiple equilibrium points in a global attractor for some symmetric dynamical systems

In this paper, we are mainly concerned with some properties of the global attractor for some symmetric dynamical systems with a Lyapunov function in a Banach space. Under some suitable assumptions, we give an estimate of lower bound of Z2 index of the global attractor and get the existence of the multiple equilibrium points in the global attractor for the symmetric dynamical systems. As an application of these results, we consider the existence of multiple stationary solutions for some semilinear reaction–diffusion equations.

Extremum seeking control of the CANON process—Existence of multiple stationary solutions

The paper considers the use of extremum seeking control (ESC) for real-time optimization of the completely autotrophic nitrogen removal over nitrite (CANON) process. In particular, the feasibility of employing relatively fast estimation and control, compared to the open-loop dynamics, is considered. From simulations it is found that the ESC can lock onto stationary solutions far removed from the optimum and that, for some control parameters, multiple stationary solutions may exist. To verify the simulation results we analyze a general dynamic model for ESC and show that all stationary solutions can be characterized by simple conditions on the gain and phase lag of the frequency response of the locally linearized system. In order to relate the derived conditions to conditions for optimality, we consider characteristic dynamic properties of systems displaying extremum points and show that the phase lag condition implies that a stationary solution in the vicinity of the optimum is likely to exist even with relatively fast estimation and control. However, the phase lag condition can also be fulfilled at operating points with no relation to the optimum whatsoever. Thus, the ESC can yield multiple stationary solutions and for some initial conditions lock onto solutions far removed from the optimum. A simple stability analysis is presented to verify that multiple solutions can be locally asymptotically stable. A simple bioreactor model is considered to illustrate the results and we show that the ESC provides three stationary solutions, of which two are stable and one unstable, even for relatively slow estimation schemes. We also confirm that the stationary solutions of the CANON process, found through simulations, fulfill both the phase lag condition and the stability criterion derived in the paper.