Stability Of Stationary Solutions Research Articles

Turing Instability and Dynamic Bifurcation for the One‐Dimensional Gray–Scott Model

ABSTRACTWe study the dynamic bifurcation of the one‐dimensional Gray–Scott model by taking the diffusion coefficient of the reactor as a bifurcation parameter. We define a parameter space of for which the Turing instability may happen. Then, we show that it really occurs below the critical number and obtain rigorous formula for the bifurcated stable patterns. When the critical eigenvalue is simple, the bifurcation leads to a continuous (resp. jump) transition for if is negative (resp. positive). We prove that when lies near the Bogdanov–Takens point . When the critical eigenvalue is double, we have a supercritical bifurcation that produces an ‐attractor . We prove that consists of four asymptotically stable static solutions, four saddle static solutions, and orbits connecting them. We also provide numerical results that illustrate the main theorems.

Single transition layer in mass-conserving reaction-diffusion systems with bistable nonlinearity

Abstract Mass-conserving reaction-diffusion systems with bistable nonlinearity are useful models for studying cell polarity formation, which is a key process in cell division and differentiation. We rigorously show the existence and stability of stationary solutions with a single internal transition layer in such reaction-diffusion systems under general assumptions by the singular perturbation theory. Moreover, we present a meaningful model for understanding the existence of an unstable transition layer solution; our numerical simulations show that the unstable solution is a separatrix of the dynamics of the model.

Exact periods and exact critical values for Hopf bifurcations from multi-peak solutions of the shadow Gierer-Meinhardt model in one spatial dimension

We consider the shadow Gierer-Meinhardt model{∂tU=ε2Uxx−U+Upξqfor0<x<1,0<t<T,τ∂tξ=∫01(−ξ+Urξs)dxfor0<t<T,Ux(0,t)=Ux(1,t)=0, where ε>0, τ>0, p>1, q>0, r>0, s≥0 and qr/(p−1)(s+1)>1. Wei and Winter showed that if (p,r)=(3,2) or (2,2), then as τ increases, a stable stationary monotone solution is destabilized by Hopf bifurcation, and hence periodic solutions appear. In this paper we consider two cases (p,r)=(3,1) and (3,3). We show that a Hopf bifurcation occurs for n-mode stationary solutions, n≥1, in a rigorous way, studying eigenvalues in detail. Exact periods and exact critical values of τ can be written by using complete elliptic integrals. A relationship between a period and a shape of a stationary solution is also studied. In particular, a maximum point of the period is studied.

Existence and Stability of a Stationary Solution in a Two-Dimensional Reaction-Diffusion System with Slow and Fast Components

In the paper, the existence of a stable stationary solution in a reaction-diffusion system with slow and fast components in a two-dimensional spatial variable case is investigated. The theorem of the existence of a stationary solution with boundary layers in the case of Dirichlet boundary conditions is proven, its asymptotic approximation is constructed, and conditions for Lyapunov asymptotic stability of this solution are obtained. The research is based on the asymptotic method of differential inequalities, applied to a new class of problems. This result is practically important both for various applications described by similar systems and for the application of numerical stationing methods when solving elliptical boundary value problems.

Resonant chains in triple-planet systems

The mean motion resonance is the most important mechanism that may dominate the dynamics of a planetary system. In a multi-planetary system consisting of $N planets, the planets may form a resonant chain when the ratios of orbital periods of planets can be expressed as the ratios of small integers $T_1: T_2: :T_N=k_1: k_2: k_N$. Due to the high degree of freedom, the motion in such systems could be complex and difficult to depict. In this paper, we investigate the dynamics and possible formation of the resonant chain in a triple-planet system. We defined the appropriate Hamiltonian for a three-planet resonant chain and numerically averaged it over the synodic period. The stable stationary solutions -- apsidal corotational resonances (ACRs) -- of this averaged system, corresponding to the local extrema of the Hamiltonian function, can be searched out numerically. The topology of the Hamiltonian around these ACRs reveals their stabilities. We further constructed the dynamical maps on different representative planes to study the dynamics around the stable ACRs, and we calculated the deviation ($ of the resonant angle in the evolution from the uniformly distributed values, by which we distinguished the behaviour of critical angles. Finally, the formation of the resonant chain via convergent planetary migration was simulated and the stable configurations associated with ACRs were verified. We find that the stable ACR families arising from circular orbits always exist for any resonant chain, and they may extend to a high eccentricity region. Around these ACR solutions, regular motion can be found, typically in two types of resonant configurations. One is characterised by libration of both the two-body resonant angles and the three-body Laplace resonant angle, and the other by libration of only two-body resonant angles. The three-body Laplace resonance does not seem to contribute to the stability of the resonant chain much. The resonant chain can be formed via convergent migration, and the resonant configuration evolves along the ACR families to eccentric orbits once the planets are captured into the chain. Ideally, our methods introduced in this paper can be applied to any resonant chain of any number of planets at any eccentricity.

Stability of solutions of systems of Volterra integral equations

In this paper, we propose new sufficient conditions for stability of solutions of systems of Volterra linear integral equations and systems of linear integro-differential Volterra equations. Solution stability conditions for systems of Volterra linear integral equations are studied with perturbed right hand sides of the equations. These new sufficient conditions are expressed in terms of logarithmic norms of matrices consisting of coefficients of equations and derivatives of their kernels. For Volterra integro-differential equations we also provide sufficient conditions for stability of solutions when the initial conditions are perturbed.

Computation and analysis of optimal disturbances of stationary solutions of the hepatitis B dynamics model

Abstract Optimal disturbances of a number of typical stationary solutions of the hepatitis B virus infection dynamics model have been found. Specifically optimal disturbances have been found for stationary solutions corresponding to various forms of the chronic course of the disease, including those corresponding to the regime of low-level virus persistence. The influence of small optimal disturbances of individual groups of variables on the stationary solution is studied. The possibility of transition from stable stationary solutions corresponding to chronic forms of hepatitis B to stable stationary solutions corresponding to the state of functional recovery or a healthy organism using optimal disturbances is studied. Optimal disturbances in this study were constructed on the basis of generalized therapeutic drugs characterized by one-compartment and two-compartment pharmacokinetics.

Time-delay-induced spiral chimeras on a spherical surface of globally coupled oscillators.

We consider globally coupled networks of identical oscillators, located on the surface of a sphere with interaction time delays, and show that the distance-dependent time delays play a key role for the spiral chimeras to occur as a generic state in different systems of coupled oscillators. For the phase oscillator system, we analyze the existence and stability of stationary solutions along the Ott-Antonsen invariant manifold to find the bifurcation structure of the spiral chimera state. We demonstrate via an extensive numerical experiment that the time-delay-induced spiral chimeras are also present for coupled networks of the Stuart-Landau and Van der Pol oscillators in the same parameter regime as that of phase oscillators, with a series of evenly spaced band-type regions. It is found that the spiral chimera state occurs as a consequence of a resonant-type interplay between the intrinsic period of an individual oscillator and the interaction time delay as a topological structure property.

Thin-shell wormholes with electromagnetic effects in f(R,T) gravity

The purpose of this paper is to study the feasibility and the appearance of charged thin-shell wormholes using generalized Chaplygin gas under the influence of minimally coupled [Formula: see text] gravitational theory. Here, [Formula: see text] is a generic function of the scalar curvature [Formula: see text] and the trace of stress-energy tensor [Formula: see text]. We explore different components of Lanczos equations in the context of a specific [Formula: see text] functional form, i.e. [Formula: see text], by applying junction conditions on field equations. We study static stable solutions using radial perturbation and the generalized Chaplygin gas equation of state (EoS) with the isotropic environment. We investigate the stable (unstable) static wormhole solutions for various physical parametric values and illustrate them graphically.

An Algorithm for Construction of the Asymptotic Approximation of a Stable Stationary Solution to a Diffusion Equation System with a Discontinuous Source Function

An algorithm is presented for the construction of an asymptotic approximation of a stable stationary solution to a diffusion equation system in a two-dimensional domain with a smooth boundary and a source function that is discontinuous along some smooth curve lying entirely inside the domain. Each of the equations contains a small parameter as a factor in front of the Laplace operator, and as a result, the system is singularly perturbed. In the vicinity of the curve, the solution of the system has a large gradient. Such a problem statement is used in the model of urban development in metropolitan areas. The discontinuity curves in this model are the boundaries of urban biocenoses or large water pools, which prevent the spread of urban development. The small parameter is the ratio of the city’s outskirts linear size to the whole metropolis linear size. The algorithm includes the construction of an asymptotic approximation to a solution with a large gradient at the media interface as well as the steps for obtaining the existence conditions. To prove the existence and stability theorems, we use the upper and lower solutions, which are constructed as modifications of the asymptotic approximation to the solution. The latter is constructed using the Vasil’yeva algorithm as an expansion of a small parameter exponent.

Stable transition layer for the Allen–Cahn equation when the spatial inhomogeneity vanishes on a nonsmooth hypersurface in ℝn

In this paper, we consider an inhomogeneous Allen–Cahn problem where ( ) is a bounded set with smooth boundary and is a non‐negative Lipschitz‐continuous function in . Let be an ‐dimensional hypersurface that divides into two disjoints components ( ) such that on and in . Using the variational concept of ‐convergence, we prove the existence of stable stationary solutions developing a transition layer on as .

Global stability of stationary solutions for a class of semilinear stochastic functional differential equations with additive white noise

This paper gives a criterion for the existence of a stationary solution for a class of semilinear stochastic functional differential equations with additive white noise and its global stability. Under the condition that the global Lipschitz constant of nonlinear term f is less than the absolute value of the top Lyapunov exponent for the linear flow Φ with f being monotone or anti-monotone, and the time delay is not very big, we show that the infinite-dimensional stochastic flow possesses a unique globally attracting random equilibrium in the state space of continuous functions, which produces the globally stable stationary solution. Compared to the result of Jiang and Lv (2016) [24], we remove the assumption of boundedness for f.

Stable discontinuous stationary solutions to reaction-diffusion-ODE systems

A general system of n ordinary differential equations coupled with one reaction-diffusion equation, considered in a bounded N-dimensional domain, with no-flux boundary condition is studied in a context of pattern formation. Such initial boundary value problems may have different types of stationary solutions. In our parallel work [Instability of all regular stationary solutions to reaction-diffusion-ODE systems (2021)], regular (i.e. sufficiently smooth) stationary solutions are shown to exist, however, all of them are unstable. The goal of this work is to construct discontinuous stationary solutions to general reaction-diffusion-ODE systems and to find sufficient conditions for their stability.

Stabilization of the Moving Front Solution of the Reaction-Diffusion-Advection Problem

We consider the initial-boundary value problem of reaction-diffusion-advection that has a solution of a front form. The statement comes from the theory of wave physics. We study the question of the solution stabilizing to the stationary one. Proof of the stabilization theorem is based on the concepts of upper and lower solutions and corollaries from comparison theorems. The upper and lower solutions with large gradients are constructed as modifications of the formal moving front asymptotic approximation in a small parameter. The main idea of the proof is to show that the upper and lower solutions of the initial-boundary value problem get into the attraction domain of the asymptotically stable stationary solution on a sufficiently large time interval. The study conducted in this work gives an answer about the non-local attraction domain of the stationary solution and can give some stationing criteria. The results are illustrated by computational examples.

Stability of a nonideally excited Duffing oscillator

This paper investigates the dynamics of a Duffing oscillator excited by an unbalanced motor. The interaction between motor and vibrating system is considered as nonideal, which means that the excitation provided by the motor can be influenced by the vibrating response, as is the case in general for real systems. This constitutes an important difference with respect to the classical (ideally excited) Duffing oscillator, where the amplitude and frequency of the external forcing are assumed to be known a priori. Starting from pre-resonant initial conditions, we investigate the phenomena of passage through resonance (the system evolves towards a post-resonant state after some transient near-resonant oscillations) and resonant capture (the system gets locked into a near-resonant stationary oscillation). The stability of stationary solutions is analytically studied in detail through averaging procedures, and the obtained results are confirmed by numerical simulations.

Instability of all regular stationary solutions to reaction-diffusion-ODE systems

A general system of several ordinary differential equations coupled with a reaction-diffusion equation in a bounded domain with zero-flux boundary condition is studied in the context of pattern formation. These initial-boundary value problems may have regular (i.e. sufficiently smooth) stationary solutions. This class of close-to-equilibrium patterns includes stationary solutions that emerge due to the Turing instability of a spatially constant stationary solution. The main result of this work is instability of all regular patterns. It suggests that stable stationary solutions arising in models with non-diffusive components must be far-from-equilibrium exhibiting singularities. Such discontinuous stationary solutions have been considered in our parallel work (Cygan et al., 2021 [4]).

Delta synchronization of Poincaré chaos in gas discharge-semiconductor systems.

We introduce a new type of chaos synchronization, specifically the delta synchronization of Poincaré chaos. The method is demonstrated for the irregular dynamics in coupled gas discharge-semiconductor systems (GDSSs). It is remarkable that the processes are not generally synchronized. Our approach entirely relies on ingredients of the Poincaré chaos, which in its own turn is a consequence of the unpredictability in Poisson stable motions. The drive and response systems are in the connection, such that the latter is processed through the electric potential of the former. The absence of generalized synchronization between these systems is indicated by utilizing the conservative auxiliary system. However, the existence of common sequences of moments for finite convergence and separation confirms the delta synchronization. This can be useful for complex dynamics generation and control in electromagnetic devices. A bifurcation diagram is constructed to separate stable stationary solutions from non-trivial oscillatory ones. Phase portraits of the drive and response systems for a specific regime are provided. The results of the sequential test application to indicate the unpredictability and the delta synchronization of chaos are demonstrated in tables. The computations of the dynamical characteristics for GDSSs are carried out by using COMSOL Multiphysics version 5.6 and MATLAB version R2021b.

Traveling pulses with oscillatory tails, figure-eight-like stack of isolas, and dynamics in heterogeneous media

The interplay between 1D traveling pulses with oscillatory tails (TPOs) and heterogeneities of the bump type is studied to formulate a generalized three-component FitzHugh–Nagumo system of equations. We present that stationary pulses with oscillatory tails (SPOs) form a “snaky” structure in homogeneous spaces and then TPO branches form a “figure-eight-like stack of isolas” located adjacent to the snaky structure of the SPO. Here, we adopt input resources such as the voltage-difference as a bifurcation parameter. A drift bifurcation from the SPO to TPO is observed by introducing another parameter at which these two solution sheets merge. In contrast to the monotone tail case, in the heterogeneous problem, a nonlocal interaction appears between the TPO and the heterogeneity, creating infinitely many saddle solutions and finitely many stable stationary solutions distributed across the line. The response of the TPO shows various dynamics including pinning and depinning processes along with penetration (PEN) and rebound (REB). The stable/unstable manifolds of these saddles interact with the TPO in a complex manner, creating a subtle dependence on the initial condition, and a difficulty to predict the behavior after collision even in 1D space. For the 1D case, a systematic global exploration of solution branches induced by heterogeneities (heterogeneity-induced-ordered patterns; HIOPs) reveals that HIOPs contain all the asymptotic states after collision to facilitate the prediction of the solution results without solving the PDEs. The reduction method of finite-dimensional system enables the clarification of the detailed mechanism of the transitions from PEN to pinning and pinning to REB based on a dynamical system perspective. Consequently, the basin boundary between two distinct outputs against the heterogeneities yields infinitely many successive reconnections of heteroclinic orbits among those saddles when the strength of heterogeneity increases. This causes the aforementioned subtle dependence of initial condition.

Existence and stability of a stable stationary solution with a boundary layer for a system of reaction–diffusion equations with Neumann boundary conditions

Existence and stability of a stable stationary solution with a boundary layer for a system of reaction–diffusion equations with Neumann boundary conditions

Existence and stability of a stable stationary solution with a boundary layer for a system of reaction-diffusion equations with Neumann boundary conditions

Рассматривается начально-краевая задача для сингулярно возмущенной параболической системы двух уравнений типа реакция-диффузия с условиями Неймана с коэффициентами диффузии различной степени малости без требования квазимонотонности правых частей. Получено асимптотическое приближение стационарного решения с пограничным слоем и доказаны теоремы существования, асимптотической устойчивости по Ляпунову и локальной единственности такого решения. Полученный результат применен к классу задач химической кинетики.