Stationary Periodic Solutions Research Articles

Thermocapillary thin films: periodic steady states and film rupture

We study stationary, periodic solutions to the thermocapillary thin-film model 0,\\ x\\in \\mathbb{R},$?> ∂th+∂xh3∂x3h−g∂xh+Mh21+h2∂xh=0,t>0, x∈R, which can be derived from the Bénard–Marangoni problem via a lubrication approximation. When the Marangoni number M increases beyond a critical value M∗ , the constant solution becomes spectrally unstable via a (conserved) long-wave instability and periodic stationary solutions bifurcate. For a fixed period, we find that these solutions lie on a global bifurcation curve of stationary, periodic solutions with a fixed wave number and mass. Furthermore, we show that the stationary periodic solutions on the global bifurcation branch converge to a weak stationary periodic solution which exhibits film rupture. The proofs rely on a Hamiltonian formulation of the stationary problem and the use of analytic global bifurcation theory. Finally, we show the instability of the bifurcating solutions close to the bifurcation point and give a formal derivation of the amplitude equation governing the dynamics close to the onset of instability.

Periodic traveling waves in the ϕ^{4} model: Instability, stability, and localized structures.

We consider the instability and stability of periodic stationary solutions to the classical ϕ^{4} equationnumerically. In the superluminal regime, the model possesses dnoidal and cnoidal waves. The former are modulationally unstable and the spectrum forms a figureeight intersecting at the origin of the spectral plane. The latter can be modulationally stable, and the spectrum near the origin in that case is represented by vertical bands along the purely imaginary axis. The instability of the cnoidal states in that case stems from elliptical bands of complex eigenvalues far from the spectral plane origin. In the subluminal regime, there exist only snoidal waves which are modulationally unstable. Considering the subharmonic perturbations, we show that the snoidal waves in the subluminal regime are spectrally unstable with respect to all subharmonic perturbations, while for the dnoidal and cnoidal waves in the superluminal regime, the transition between the spectrally stable state and the spectrally unstable state occurs through a Hamiltonian Hopf bifurcation. The dynamical evolution of the unstable states is also considered, leading to some interesting localization events on the spatio-temporal backgrounds.

Stationary Distribution and Periodic Solution of a Stochastic n-Species Gilpin–Ayala Competition System with General Saturation Effect and Nonlinear Perturbations

Stationary Distribution and Periodic Solution of a Stochastic n-Species Gilpin–Ayala Competition System with General Saturation Effect and Nonlinear Perturbations

A competition system with nonlinear cross-diffusion: exact periodic patterns

Our concern in this paper is to shed some additional light on the mechanism and the effect caused by the so called cross-diffusion. We consider a two-species reaction–diffusion (RD) system. Both “fluxes” contain the gradients of both unknown solutions. We show that–for some parameter range– there exist two different type of periodic stationary solutions. Using them, we are able to divide into parts the (eight-dimensional) parameter space and indicate the so called Turing domains where our solutions exist. The boundaries of these domains, in analogy with “bifurcation point”, called “bifurcation surfaces”. As it is commonly believed, these solutions are limits as t goes to infinity of the solutions of corresponding evolution system. In a forthcoming paper we shall give a detailed account about our numerical results concerning different kind of stability. Here we also show some numerical calculations making plausible that our solutions are in fact attractors with a large domain of attraction in the space of initial functions.

Ground state solutions and periodic solutions with minimal periods to second-order Hamiltonian systems

In this paper, we study the existence of periodic solutions to second-order Hamiltonian systems. The nonlinear term has a special form, which satisfies a nondecreasing monotone assumption. Making use of a generalized Nehari manifold, we built the homomorphism between the Nehari manifold and the unit sphere of some space. Some sufficient conditions are obtained to guarantee the existence of periodic solutions with the prescribed minimal period to Hamiltonian systems. Our results extend the results in references.

Modulation instability of an electromagnetic wave interacting with a counterpropagating electron beam under condition of cyclotron resonance absorption.

In this paper, transmission of a monochromatic wave through a counterpropagating electron beam under the condition of cyclotron resonance absorption is studied by theoretical analysis and numerical simulation. Conditions of the modulation instability (MI) are analyzed. The MI strongly affects the regimes of transmission. We also derive explicit periodic stationary solutions expressed in terms of elliptic Jacobi functions, as well as bright- and dark-soliton solutions. Analysis of these solutions allows obtaining threshold values of the driving power and frequency for the different regimes of transmission, such as cyclotron absorption, multifrequency self-modulation oscillations, and stationary single-frequency propagation. The theoretical predictions are verified by numerical simulation. In this way, we obtain the conditions at which a continuous-wave driving signal disintegrates into a close-to-periodic train of microwave soliton pulses.

FINITE ELEMENT METHOD FOR NONLINEAR EDDY CURRENT PROBLEMS IN POWER TRANSFORMERS

An efficient finite element method to take account of the nonlinearity of the magnetic materials when analyzing three dimensional eddy current problems is presented in this paper. The problem is formulated in terms of vector and scalar potentials approximated by edge and node based finite element basis functions. The application of Galerkin techniques leads to a large, nonlinear system of ordinary differential equations in the time domain. The excitations are assumed to be time-periodic and the steady state periodic solution is of interest only. This is represented in the frequency domain as a Fourier series for each finite element degree of freedom and a finite number of harmonics is to be determined, i.e. a harmonic balance method is applied. Due to the nonlinearity, all harmonics are coupled to each other, so the size of the equation system is the number of harmonics times the number of degrees of freedom. The harmonics would be decoupled if the problem were linear, therefore, a special nonlinear iteration technique, the fixed-point method is used to linearize the equations by selecting a timeindependent permeability distribution, the so called fixed-point permeability in each nonlinear iteration step. This leads to uncoupled harmonics within these steps resulting in two advantages. One is that each harmonic is obtained by solving a system of algebraic equations with only as many unknowns as there are finite element degrees of freedom. A second benefit is that these systems are independent of each other and can be solved in parallel. The appropriate selection of the fixed point permeability accelerates the convergence of the nonlinear iteration. The method is applied to the analysis of a large power transformer. The solution of the electromagnetic field allows the computation of various losses like eddy current losses in the massive conducting parts (tank, clamping plates, tie bars, etc.) as well as the specific losses in the laminated parts (core, tank shielding, etc.). The effect of the presence of higher harmonics on these losses is investigated.

Stationary distribution and periodic solution of a stochastic Nicholson's blowflies model with distributed delay

Due to the complex set of abiotic and biotic interaction variables in the ecosystems, the dynamics of animal populations are generally considered to be driven in astate‐dependent manner. This paper presents an analysis of a stochastic Nicholson's blowflies model with distributed delay, which consists of the existence and uniqueness of globally positive solution, extinction, existence of positive T‐periodic solution and positive recurrence. Our results suggest that a smaller white noise is necessary to cause the existence of positive T‐periodic solution and positive recurrence of the stochastic model, while a larger white noise will accelerate the extinction of the population. In addition, numerical simulation is used to support our main results.

A sign-function receiving scheme for sine signals enhanced by stochastic resonance* *Project supported by the Natural Science Foundation of Hebei Province of China (Grant Nos. F2019506031, F2019506037, and F2020506036), the Frontier Innovation Program of Army Engineering University (Grant No. KYSZJQZL2005), the Basic Frontier Science and Technology Innovation Program of Army Engineering University (Grant No. KYSZJQZL2020).

To address the problem that it is difficult to detect an intermediate frequency (IF) signal at the receiving end of a communication system under extremely low signal-to-noise ratio (SNR) conditions, we propose a stochastic resonance (SR)-enhanced sine-signal detection method based on the sign function. By analyzing the SR mechanism of the sine signal and combining it with the characteristics of a dual-sequence frequency-hopping (DSFH) receiver, a periodic stationary solution of the Fokker–Planck equation (FPE) with a time parameter is obtained. The extreme point of the sine signal is selected as the decision time, and the force law of the electromagnetic particles is analyzed. A receiving structure based on the sign function is proposed to maximize the output difference of the system, and the value condition of the sign function is determined. In order to further improve the detection performance, in combination with the central-limit theorem, the sampling points are averaged N times, and the signal-detection problem is transformed into a hypothesis-testing problem under a Gaussian distribution. The theoretical analysis and simulation experiment results confirm that when N is 100 and the SNR is greater than 20 dB, the bit-error ratio (BER) is less than 1.5 × 10−2 under conditions in which the signal conforms to the optimal SR parameters.

Thermocapillary instability and wave formation on a viscous film flowing down an inclined plane with linear temperature variation: Effect of odd viscosity

Thermocapillary instability and formation of waves for a thin viscous falling film with broken time-reversal-symmetry have been discussed in this present study. The film is flowing over a flat, rigid inclined plate with linear temperature variation. The presence of the odd part of the Cauchy stress tensor with odd viscosity coefficient brings new characteristics in thin film flow. The nonlinear evolution model, which tracks the free surface formation for this problem, has been developed using classical long-wave expansion method. Due to the presence of odd viscosity in the liquid, the evolution equation modified significantly. Both spatial and temporal analysis for linear stability has been performed along with the investigation of weakly nonlinear waves, leading the free-surface equation into the famous Kuramoto–Sivashinsky equation. Periodic stationary wave solutions of the full evolution equation confirm the existence of two significant wave families, γ1 and γ2, which are discussed in detail. Formation of the dynamical system and study of different bifurcation analysis and phases diagrams are also studied. The spatiotemporal evolution of the model has been analyzed numerically for different values of the odd viscosity parameter and Marangoni number. The odd viscosity gives stabilizing effect while the increase in Marangoni number increases the instability.

Существование и устойчивость периодических решений уравнения нейронного поля

We study the existence and linear stability of stationary periodic solutions to a neural field model, an intergo-differential equation of the Hammerstein type. Under the assumption that the activation function is a discontinuous step function and the kernel is decaying sufficiently fast, we formulate necessary and sufficient conditions for the existence of a special class of solutions that we call 1-bump periodic solutions. We then analyze the stability of these solutions by studying the spectrum of the Frechet derivative of the corresponding Hammerstein operator. We prove that the spectrum of this operator agrees up to zero with the spectrum of a block Laurent operator. We show that the non-zero spectrum consists of only eigenvalues and obtain an analytical expression for the eigenvalues and the eigenfunctions. The results are illustrated by multiple examples.

Periodic stationary solutions of the Nagumo lattice differential equation: existence regions and their number

The Nagumo lattice differential equation admits stationary solutions with arbitrary spatial period for sufficiently small diffusion rate. The continuation from the stationary solutions of the decoupled system (a system of isolated nodes) is used to determine their types; the solutions are labelled by words from a three-letter alphabet. Each stationary solution type can be assigned a parameter region in which the solution can be uniquely identified. Numerous symmetries present in the equation cause some of the regions to have identical or similar shape. With the help of combinatorial enumeration, we derive formulas determining the number of qualitatively different existence regions. We also discuss possible extensions to other systems with more general nonlinear terms and/or spatial structure.

Global potential, topology, and pattern selection in a noisy stabilized Kuramoto–Sivashinsky equation

We formulate a general method to extend the decomposition of stochastic dynamics developed by Ao et al. [J. Phys. Math. Gen. 37, L25-L30 (2004)] to nonlinear partial differential equations which are nonvariational in nature and construct the global potential or Lyapunov functional for a noisy stabilized Kuramoto-Sivashinsky equation. For values of the control parameter where singly periodic stationary solutions exist, we find a topological network of a web of saddle points of stationary states interconnected by unstable eigenmodes flowing between them. With this topology, a global landscape of the steady states is found. We show how to predict the noise-selected pattern which agrees with those from stochastic simulations. Our formalism and the topology might offer an approach to explore similar systems, such as the Navier Stokes equation.

Instabilities in certain one-dimensional singular interfacial equation

We re-examine a generalized singular equation to discuss the coarsening of growing interfaces, in the presence of Ehrlich–Schwoebel–Villain barrier that induces a pyramidal or mound-type structure without slope selection. Our aim is to obtain analytically results on the coarsening process by inspecting the behavior of branch of the steady state periodic solutions.

Multistability of Almost Periodic Solution for Memristive Cohen-Grossberg Neural Networks With Mixed Delays.

This paper presents the multistability analysis of almost periodic state solutions for memristive Cohen-Grossberg neural networks (MCGNNs) with both distributed delay and discrete delay. The activation function of the considered MCGNNs is generalized to be nonmonotonic and nonpiecewise linear. It is shown that the MCGNNs with n -neuron have (K+1)n locally exponentially stable almost periodic solutions, where nature number K depends on the geometrical structure of the considered activation function. Compared with the previous related works, the number of almost periodic state solutions of the MCGNNs is extensively increased. The obtained conclusions in this paper are also capable of studying the multistability of equilibrium points or periodic solutions of the MCGNNs. Moreover, the enlarged attraction basins of attractors are estimated based on original partition. Some comparisons and convincing numerical examples are provided to substantiate the superiority and efficiency of obtained results.

Stationary distribution and periodic solution of stochastic chemostat models with single-species growth on two nutrients

In this paper, we consider two chemostat models with random perturbation, in which single species depends on two perfectly substitutable resources for growth. For the autonomous system, we first prove that the solution of the system is positive and global. Then we establish sufficient conditions for the existence of an ergodic stationary distribution by constructing appropriate Lyapunov functions. For the non-autonomous system, by using Has’minskii theory on periodic Markov processes, we derive it admits a nontrivial positive periodic solution. Finally, numerical simulations are carried out to illustrate our main results.

Counting and ordering periodic stationary solutions of lattice Nagumo equations

We study the rich structure of periodic stationary solutions of Nagumo reaction diffusion equation on lattices. By exploring the relationship with Nagumo equations on cyclic graphs we are able to divide these periodic solutions into equivalence classes that can be partially ordered and counted. In order to accomplish this, we use combinatorial concepts such as necklaces, bracelets and Lyndon words.

Hopf-Hopf bifurcation and chaotic attractors in a delayed diffusive predator-prey model with fear effect

We investigate a diffusive predator-prey model by incorporating the fear effect into prey population, since the fear of predators could visibly reduce the reproduction of prey. By introducing the mature delay as bifurcation parameter, we find this makes the predator-prey system more complicated and usually induces Hopf and Hopf-Hopf bifurcations. The formulas determining the properties of Hopf and Hopf-Hopf bifurcations by computing the normal form on the center manifold are given. Near the Hopf-Hopf bifurcation point we give the detailed bifurcation set by investigating the universal unfoldings. Moreover, we show the existence of quasi-periodic orbits on three-torus near a Hopf-Hopf bifurcation point, leading to a strange attractor when further varying the parameter. We also find the existence of Bautin bifurcation numerically, then simulate the coexistence of stable constant stationary solution and periodic solution near this Bautin bifurcation point.

Cortical waves and post‐stroke brain stimulation

A neural field model with different activation and inhibition connectivity and response functions is considered. Stability analysis of a homogeneous in space solution determines the conditions of the emergence of stationary periodic solutions and of periodic travelling waves. Various regimes of wave propagation are illustrated in numerical simulations. The influence of external stimulation on the wave properties is investigated.

Stationary Distribution and Periodic Solution for the Stochastic Chemostat with Perturbations on Multiple Parameters

Stationary Distribution and Periodic Solution for the Stochastic Chemostat with Perturbations on Multiple Parameters