Types Of Critical Points Research Articles

An MP-Newton method for computing nonlinear eigenpairs and its application for solving a semilinear Schrödinger equation

ln Yao and Zhou (2008), a minimax method for computing nonlinear eigenpairs by calculating critical points of the Lagrange multiplier function is presented. But, the method is slow and can find limited amount of eigenpairs. In this paper, a new general characterization, orthogonal-max characterization, for critical points of the Lagrange multiplier function is suggested. An MP-Newton method for finding orthogonal-max type critical points is designed through analyzing how the minimax method works. The new method becomes fast and able to calculate more nonlinear eigenpairs. Numerical experiment confirms these two progresses. Also, the MP-Newton method inherits the advantages of the minimax method. A convergence result for the method is established. Finally, an application for solving a semilinear Schrödinger equation is discussed.

An MP-Newton method for finding orthogonal-max type critical points and its application for calculating multiple solutions of semilinear elliptic equations: part I—method

An MP-Newton method for finding orthogonal-max type critical points and its application for calculating multiple solutions of semilinear elliptic equations: part I—method

Crossing Limit Cycles in Planar Piecewise Linear Systems Separated by a Nonregular Line with Node–Node Type Critical Points

In this paper, we investigate the existence and number of crossing limit cycles in a class of planar piecewise linear systems with node–node type critical points defined in two zones separated by a nonregular line formed by two rays emanated from the origin [Formula: see text], which are the positive [Formula: see text]- and [Formula: see text]-axes. We focus our attention on the existence of two-point crossing limit cycles, which intersect the switching line at two points. We obtain sufficient conditions under which the system has two two-point crossing limit cycles which intersect only one of the two rays. Moreover, we construct examples to show that this class of systems can have two, three or four two-point crossing limit cycles.

On the Number of Limit Cycles of Planar Piecewise Smooth Quadratic Systems with Focus-Parabolic Type Critical Point

On the Number of Limit Cycles of Planar Piecewise Smooth Quadratic Systems with Focus-Parabolic Type Critical Point

Separable solutions to nonlinear anisotropic diffusion equation in elliptic coordinates.

Two- and three-dimensional exact solutions of the nonlinear diffusion equation are proved to exist in elliptic coordinates subject to an arbitrary piecewise constant azimuthal anisotropy. Degrees of freedom traditionally used to satisfy boundary conditions are instead employed to ensure continuity and conservation of mass across contiguity surfaces between subdomains of distinct diffusivities. Not all degrees of freedom are exhausted thereby, and conditions are given for the inclusion of higher harmonics. Degrees of freedom associated with one isotropic subdomain are always available to satisfy boundary conditions. The second harmonic is pivotal in the solution construction as well as the identification of partial symmetries in the domain partition. The anisotropy gives rise to an unconventional mixed type critical point that combines saddle and node-like characteristics. This article is part of the theme issue 'New trends in pattern formation and nonlinear dynamics of extended systems'.

Attractors of a Weakly Dissipative System Allowing Transition to the Stochastic Web in the Conservative Limit

This article deals with the dynamics of a pulse-driven self-oscillating system — the Van der Pol oscillator — with the pulse amplitude depending on the oscillator coordinate. In the conservative limit the “stochastic web” can be obtained in the phase space when the function defining this dependence is a harmonic one. The paper focuses on the case where the frequency of external pulses is four times greater than the frequency of the autonomous system. The results of a numerical study of the structure of both parameter and phase planes are presented for systems with different forms of external pulses: the harmonic amplitude function and its power series expansions. Complication of the pulse amplitude function results in the complication of the parameter plane structure, while typical scenarios of transition to chaos visible in the parameter plane remain the same in different cases. In all cases the structure of bifurcation lines near the border of chaos is typical of the existence of the Hamiltonian type critical point. Changes in the number and the relative position of coexisting attractors are investigated while the system approaches the conservative limit. A typical scenario of destruction of attractors with a decrease in nonlinear dissipation is revealed, and it is shown to be in good agreement with the theory of 1:4 resonance. The number of attractors of period 4 seems to grow infinitely with the decrease of dissipation when the pulse amplitude function is harmonic, while in other cases all attractors undergo destruction at certain values of dissipation parameters after the birth of high-period periodic attractors.

Bifurcation of limit cycles from a parabolic–parabolic type critical point in a class of planar piecewise smooth quadratic systems

In this paper we study small amplitude limit cycles bifurcated from a parabolic–parabolic (PP) type critical point of planar piecewise smooth quadratic systems having exactly one switching line given by the x-axis. Besides the non-smoothness, more difficulties arise when the critical point has a parabolic contact. For such kind of critical point, to make sure that the return map is analytic, one has to use the generalized polar coordinates instead of the classic polar coordinates to compute the corresponding Lyapunov constants. Consequently, much more complicated computations are involved. By the recent results of Novaes and Silva, the index of the first nonzero Lyapunov constant for a PP type critical point is always an even number 2ℓ+2 with ℓ≥1. We call the corresponding weak focus (0,0) is of order ℓ. We obtain eight center conditions and six conditions under which (0,0) is a weak focus of order 6. Furthermore, we prove that at least seven limit cycles can bifurcate from (0,0).

Limit Cycles of Planar Piecewise Smooth Quadratic Systems with Focus-Parabolic Type Critical Points

It is very important to determine the maximum number of limit cycles of planar piecewise smooth quadratic systems and it has become a focal subject in recent years. Almost all of the previous studies on this problem focused on systems with focus–focus type critical points. In this paper, we consider planar piecewise smooth quadratic systems with focus-parabolic type critical points. By using the generalized polar coordinates to compute the corresponding Lyapunov constants, we construct a class of planar piecewise smooth quadratic systems with focus-parabolic type critical points having six limit cycles. Our results improve the results obtained by Coll, Gasull and Prohens in 2001, who constructed a class of such systems with four limit cycles.

Ab initio compressibility of metastable low albite: revealing a lambda-type singularity at pressures of the Earth’s upper mantle

The elastic behavior of low albite is investigated ab initio under hydrostatic pressure up to 16 GPa. Our calculations complement and extend previous studies confirming a highly anisotropic character of the feldspar cell compression and, more importantly, revealing a clear change of all structure deformation trends around 8–9 GPa pressure. We correlate this change to the trend of the bulk modulus of low albite as a function of pressure, which we compute in different and independent ways using (1) the Birch–Murnaghan equation of state, (2) the analytical Voigt–Reuss–Hill averaging scheme of calculated elastic constants, and (3) a pressure–volume numerical differentiation procedure. The latter, in particular, uncovers a singularity in the bulk modulus between 8 and 9 GPa pressure which is evocative of a $$\lambda$$ -type critical point. We find that the same behavior emerges when comparing with pressure–volume datasets from the experimental literature, where it has been so far overlooked due to the misleading use of a fourth-order Birch–Murnaghan equation of state. Indeed, we show that the equation of state must be extended up to at least the sixth power of the Eulerian strain to approximate the complex elastic behavior of feldspars. The low albite structure softens under increasing pressure between 5 and 8 GPa, as a result of the initiation of auxiliary compression mechanisms—notably, the squeezing of the crankshaft chains along b—and then abruptly resumes a stiffening trend in association with a displacive transformation of the O–O pair interactions. Whether this is an isosymmetric phase transition or a supercritical crossover, it suggests a compatibility with seismological profiles indicating a low wave-velocity anomaly in correspondence of the upper portion of the subducting Pacific plate and the disappearance of such anomaly at greater depths, assuming the alkali feldspar survives as a metastable phase. The data and methodology described here can enable the exploration of important, potentially overlooked features in other minerals, and inspire future high-pressure research in mineral physics.

Localized nodal solutions of higher topological type for nonlinear Schrödinger–Poisson system

In this paper, we focus our attention on following Schrödinger–Poisson system: −ε2Δu+V(x)u+λψu=|u|p−2u,x∈R3,−ε2Δψ=u2,u∈H1(R3).where 4<p<6 and ε,λ>0 are small positive parameters. Under a local condition imposed on the potential V, we study above system and obtain an infinite sequence of localized sign-changing solutions by applying the symmetric mountain pass theorem. Precisely, these solutions are constructed as higher topological type critical points of the energy functional and concentrated at a local minimum set of the potential V. Our method follows the same spirit of Chen and Wang’s method (Chen and Wang, 2017) which does not need any non-degeneracy condition of the limiting equations, but we cannot use it directly due to the presence of nonlocal term ψ(x)=14π∫R3u2(y)|x−y|dy. We employ some new analytical skills to overcome the obstacles caused by the nonlocal term, our results improve and extend some related ones in the literature.

On the Stability of the Nonlinear Center under Quasi-periodic Perturbations

The problem of stability of the zero solution of a system with a “center”-type critical point at the origin of coordinates is considered. For the first time, such a problem for autonomous systems was investigated by A.M. Lyapunov. We continued Lyapunov’s investigations for systems with a periodic dependence on time. In this paper, systems with a quasi-periodic time dependence are considered. It is assumed that the basic frequencies of quasi-periodic functions satisfy the standard Diophantine-type condition. The problem under consideration can be interpreted as the problem of stability of the state of equilibrium of the oscillator $$\ddot {x} + {{x}^{{2n - 1}}}$$ = 0, n is an integer number, n ≥ 2, under “small” quasi-periodic perturbations.

Electroreflectance study on optical anisotropy in β-Ga2O3

Electroreflectance (ER) spectra of β-Ga2O3 were measured using Schottky barrier diodes. The ER spectra were well fitted to a third derivative line shape function by assuming three excitonic transitions. The transition energies exhibited moderate blueshifts with an external reverse bias by reflecting the contribution of transitions at M0 type critical points. Optical anisotropy was selectively observed in the polarized ER spectra. The fittings for the third derivative line shape function gave relatively large broadening parameters (0.11–0.89 eV) due mainly to the large exciton-longitudinal-optical-phonon interaction in β-Ga2O3. The ER spectra exhibited a third derivative line shape with a Franz-Keldysh oscillation superimposed on it. The results demonstrate that ER measurements are useful for investigating optical anisotropy in β-Ga2O3 and related alloys.

Functionality of Disorder in Muscle Mechanics.

A salient feature of skeletal muscles is their ability to take up an applied slack in a microsecond timescale. Behind this fast adaptation is a collective folding in a bundle of elastically interacting bistable elements. Since this interaction has a long-range character, the behavior of the system in force and length controlled ensembles is different; in particular, it can have two distinct order-disorder-type critical points. We show that the account of the disregistry between myosin and actin filaments places the elementary force-producing units of skeletal muscles close to both such critical points. The ensuing "double criticality" contributes to the system's ability to perform robustly and suggests that the disregistry is functional.

Existence and uniqueness of normalized solutions for the Kirchhoff equation

For a class of Kirchhoff functional, we first give a complete classification with respect to the exponent p for its L 2 -normalized critical points, and show that the minimizer of the functional, if exists, is unique up to translations. Secondly, we search for the mountain pass type critical points for the functional on the L 2 -normalized manifold, and also prove that this type critical point is unique up to translations. Our proof relies only on some simple energy estimates and avoids using the concentration-compactness principles. These conclusions extend some known results in previous papers.

Dynamical system approach to scalar–vector–tensor cosmology

Using scalar-vector-tensor Brans Dicke (VBD) gravity [3] in presence of self interaction BD potential $V(\phi)$ and perfect fluid matter field action we solve corresponding field equations via dynamical system approach for flat Friedmann Robertson Walker metric (FRW). We obtained 3 type critical points for $\Lambda CDM$ vacuum de Sitter era where stability of our solutions are depended to choose particular values of BD parameter $\omega.$ One of these fixed points is supported by a constant potential which is stable for $\omega<0$ and behaves as saddle (quasi stable) for $\omega\geq0.$ Two other ones are supported by a linear potential $V(\phi)\sim\phi$ which one of them is stable for $\omega=0.27647.$ For a fixed value of $\omega$ there is at least 2 out of 3 critical points reaching to a unique critical point. Namely for $\omega=-0.16856(-0.56038)$ the second (third) critical point become unique with the first critical point. In dust and radiation eras we obtained 1 critical point which never become unique fixed point. In the latter case coordinates of fixed points are also depended to $\omega.$ To determine stability of our solutions we calculate eigenvalues of Jacobi matrix of 4D phase space dynamical field equations for de Sitter, dust and radiation eras. We should be point also potentials which support dust and radiation eras must be similar to $V(\phi)\sim\phi^{-\frac{1}{2}}$ and $V(\phi)\sim\phi^{-1}$ respectively. In short our study predicts that radiation and dust eras of our VBD-FRW cosmology transmit to stable de Sitter state via non-constant potential (effective variable cosmological parameter) by choosing $\omega=0.27647$.

Multiple positive standing wave solutions for schrödinger equations with oscillating state-dependent potentials

Motivated by relevant physical applications, we study Schrodinger equations with state-dependent potentials. Existence, localization and multiplicity results are established for positive standing wave solutions in the case of oscillating potentials. To this aim, a localized Pucci-Serrin type critical point theorem is first obtained. Two examples are then given to illustrate the new theory.

Infinitely many periodic solutions for discrete second order Hamiltonian systems with oscillating potential

In this article we obtain two sequence of infinitely many periodic solutions for discrete second order Hamiltonian systems with an oscillating potential. One sequence of solutions are local minimizers of the functional corresponding to the system, the other sequence are minimax type critical points of the functional.

Dielectric and Heat Capacity Study of (Pb(Mg1/3Nb2/3)O3)0.74-(PbTiO3)0.26 Ferroelectric Relaxor Near the Cubic-Tetragonal-Rhombohedral Triple Point

We investigate PMN(1-x)-PTx single crystal with the composition x = 0.26, near the cubic-tetragonal-rhombohedral (C-T-R) triple point. We show the dielectric spectroscopy and heat capacity results obtained in PMN0.74-PT0.26 as a function of the bias electric field for [100] crystal axis. Dielectric and heat capacity data show that the electric field induced monoclinic phase sequence appears between tetragonal and rhombohedral phases. The impact of the bias field on the monoclinic phase sequence is studied as well as the impact on the critical behavior of the piezoelectric coefficient near the liquid-vapor type critical point.

Multiple solutions for an eigenvalue problem involving [formula omitted]-Laplacian type operators

In this paper we establish the existence of two nontrivial weak solutions of some eigenvalue problems, involving a general elliptic operator in divergence form. First we study a one parameter problem under homogeneous Dirichlet boundary conditions in bounded domains, and then a two parameters problem under inhomogeneous nonlinear Robin boundary conditions in regular bounded domains. These results complete in several directions some recent papers, where a Ricceri type critical point theorem was used. The main argument of this paper is based on two recent theorems due to Arcoya and Carmona (2007) in [2] which we use in a slightly modified version, see Section 2.

Electrocaloric effect in relaxor ferroelectrics

A theoretical model for the electrocaloric effect (ECE) in relaxor ferroelectrics is presented. By solving a self-consistent relation for the ECE temperature change ΔT and minimizing numerically the mean field free energy for relaxors, the field and temperature dependence of ΔT is calculated. The corresponding harmonic Landau coefficient a=a(T), which differs from the ferroelectric case by always being positive, is derived from the spherical random bond-random field model, and the fourth-order coefficient b is treated as a phenomenological parameter, which can be either positive or negative. For b&amp;lt;0, a line of field-induced first-order relaxor-to-ferroelectric phase transitions exists in relaxors, which terminates at a liquid-vapor type critical point ECP,TCP. The critical behavior close to ECP,TCP is analyzed. It is shown that near the first-order phase transition a temperature or field interval or gap formally appears, where ΔT cannot be found. However, domain formation in the coexistence range should restore the continuous behavior of the ECE observed in real systems. Finally, it is shown that the ECE responsivity R1=ΔT/E reaches a maximum near the critical point, in agreement with recent experiments.