Stability Of Time-periodic Solutions Research Articles

Rich dynamics of a reaction–diffusion Filippov Leslie–Gower predator–prey model with time delay and discontinuous harvesting

To reflect the harvesting effect, a nonsmooth Filippov Leslie–Gower predator–prey model is proposed. Unlike traditional Filippov models, the time delay and reaction–diffusion under the condition of homogeneous Neumann boundary are considered in our system. The stability of equilibrium and the existence of the spatial Hopf bifurcation of the subsystems at the positive equilibrium are investigated. Furthermore, a comprehensive analysis is conducted on the sliding mode dynamics as well as the regular, virtual, and pseudoequilibria. The findings reveal that our Filippov system exhibits either a globally asymptotically stable regular equilibrium, a globally asymptotically stable time periodic solution, or a globally asymptotically stable pseudoequilibrium, contingent upon the specific values of the time delay and threshold level. A boundary point bifurcation, which transform a stable equilibrium point or periodic solution into a stable pseudoequilibrium, is demonstrated to emphasize the impact of time delay on our Filippov system and the significance of threshold control. Meanwhile, two kinds of global sliding bifurcations are exhibited, which sequentially transform a stable periodic solutions below the threshold into a grazing, sliding switching, and crossing bifurcations, depending on changes in the time delay or threshold level. Our results indicate that bucking bifurcation and crossing bifurcation pose significant challenges to the control of our Filippov system.

Periodic solutions in next generation neural field models

We consider a next generation neural field model which describes the dynamics of a network of theta neurons on a ring. For some parameters the network supports stable time-periodic solutions. Using the fact that the dynamics at each spatial location are described by a complex-valued Riccati equation we derive a self-consistency equation that such periodic solutions must satisfy. We determine the stability of these solutions, and present numerical results to illustrate the usefulness of this technique. The generality of this approach is demonstrated through its application to several other systems involving delays, two-population architecture and networks of Winfree oscillators.

Transitions of zonal flows in a two-layer quasi-geostrophic ocean model

We consider a 2-layer quasi-geostrophic ocean model where the upper layer is forced by a steady Kolmogorov wind stress in a periodic channel domain, which allows to mathematically study the nonlinear development of the resulting flow. The model supports a steady parallel shear flow as a response to the wind stress. As the maximal velocity of the shear flow (equivalently the maximal amplitude of the wind forcing) exceeds a critical threshold, the zonal jet destabilizes due to baroclinic instability and we numerically demonstrate that a first transition occurs. We obtain reduced equations of the system using the formalism of dynamic transition theory and establish two scenarios which completely describe this first transition. The generic scenario is that a conjugate pair of modes loses stability and a Hopf bifurcation occurs as a result. Under an appropriate set of parameters describing related midlatitude oceanic flows, we show that this first transition is continuous: a supercritical Hopf bifurcation occurs and a stable time periodic solution bifurcates. We also investigate the case of double Hopf bifurcations which occur when four modes of the linear stability problem simultaneously destabilize the zonal jet. In this case, we prove that, in the relevant parameter regime, the flow exhibits a continuous transition accompanied by a bifurcated attractor homeomorphic to \(S^3\). The topological structure of this attractor is analyzed in detail and is shown to depend on the system parameters. In particular, this attractor contains (stable or unstable) time-periodic solutions and a quasi-periodic solution.

Spatio-temporal coexistence in the cross-diffusion competition system

<p style='text-indent:20px;'>We study a two component cross-diffusion competition system which describes the population dynamics between two biological species. Since the cross-diffusion competition system possesses the so-called population pressure effects, a variety of solution behaviors can be exhibited compared with the classical diffusion competition system. In particular, we discuss on the existence of spatially non-constant time periodic solutions. Applying the center manifold theory and the standard normal form theory, the cross-diffusion competition system is reduced to a two dimensional dynamical system around a doubly degenerate point. As a result, we show the existence of stable time periodic solutions in the system. This means spatio-temporal coexistence between two biological species.

Hopf Bifurcation and Hopf-Pitchfork Bifurcation in an Integro-Differential Reaction-Diffusion System

We study the bifurcations of small amplitude time-periodic solutions and chaotic solutions of a two-component integro-differential reaction-diffusion system in one spatial dimension. The system has doubly degenerate points and triply degenerate points. The following results are obtained. (I) Around the doubly degenerate points, a reduced two-dimensional dynamical system on the center manifold is obtained. We find that the small amplitude stable time-periodic solutions can bifurcate from the non-uniform stationary solutions through the Hopf bifurcations for all $n$. (II) Around the triply degenerate point, a three-dimensional dynamical system on the center manifold is obtained. The reduced system can be transformed into normal form for the Hopf-Pitchfork bifurcation. The truncated normal form can possess the invariant tori and the heteroclinic loop. Furthermore, the system under the non $S^{1}$-symmetric perturbation may possess the Shil'nikov type homoclinic orbit. Numerical results for the integro-differential reaction-diffusion system are presented and found to be convincing.

Time periodic solutions to the beam equation with weak damping

In this paper, we study the beam equation with weak damping in n-dimensional space. In the case of time-dependent periodic external force imposed, we prove the existence and uniqueness of time periodic solutions that have the same period as the time-dependent periodic external force in some suitable function space for all space dimensions n ≥ 1. The proof is based on the spectral analysis for the solution operators and the contraction mapping theorem. Moreover, we show the time asymptotic stability of time periodic solutions by continuous argument.

Existence and asymptotic stability of time periodic solutions to the generalized double dispersion equation with periodic external force

In this paper, we investigate the generalized double dispersion equation with periodic external force in Rn. We prove the existence and uniqueness of time periodic solutions that have the same period as the external force in some suitable function space for the space dimension n≥3. The proof is based on the spectral analysis for the solution operator and the contraction mapping theorem. In addition, we also discuss the time asymptotic stability of the time periodic solutions by continuous argument.

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.

Maximum Entropy Closure of Balance Equations for Miniband Semiconductor Superlattices

Charge transport in nanosized electronic systems is described by semiclassical or quantum kinetic equations that are often costly to solve numerically and difficult to reduce systematically to macroscopic balance equations for densities, currents, temperatures and other moments of macroscopic variables. The maximum entropy principle can be used to close the system of equations for the moments but its accuracy or range of validity are not always clear. In this paper, we compare numerical solutions of balance equations for nonlinear electron transport in semiconductor superlattices. The equations have been obtained from Boltzmann–Poisson kinetic equations very far from equilibrium for strong fields, either by the maximum entropy principle or by a systematic Chapman–Enskog perturbation procedure. Both approaches produce the same current-voltage characteristic curve for uniform fields. When the superlattices are DC voltage biased in a region where there are stable time periodic solutions corresponding to recycling and motion of electric field pulses, the differences between the numerical solutions produced by numerically solving both types of balance equations are smaller than the expansion parameter used in the perturbation procedure. These results and possible new research venues are discussed.

Stability of time periodic solution of the Navier–Stokes equation on the half-space under oscillatory moving boundary condition

Navier–Stokes system on the half space with periodically oscillating boundary has a time periodic solution which depends on time variable and vertical variable only. It is proved that the time periodic solution is asymptotically stable when the Reynolds number is sufficiently small; and the decay estimates of the perturbations are established in the frameworks of both strong and weak solutions.

A Newton--Galerkin Method for Fluid Flow Exhibiting Uncertain Periodic Dynamics

The determination of limit-cycles plays an important role in characterizing complex dynamical systems, such as unsteady fluid flows. In practice, dynamical systems are described by models equations involving parameters which are seldom exactly known, leading to parametric uncertainties. These parameters can be suitably modeled as random variables, so if the system possesses almost surely a stable time periodic solution, limit-cycles become stochastic, too. This paper introduces a novel numerical method for the computation of stable stochastic limit-cycles based on the spectral stochastic finite element method with polynomial chaos (PC) expansions. The method is designed to overcome the limitation of PC expansions related to convergence breakdown for long term integration. First, a stochastic time scaling of the model equations is determined to control the phase-drift of the stochastic trajectories and allowing for accurate low order PC expansions. Second, using the rescaled governing equations, we aim at ...

On the Existence and Stability of Time Periodic Solution to the Compressible Navier–Stokes Equation on the Whole Space

The existence of a time periodic solution of the compressible Navier–Stokes equation on the whole space is proved for a sufficiently small time periodic external force when the space dimension is greater than or equal to 3. The proof is based on the spectral properties of the time-T-map associated with the linearized problem around the motionless state with constant density in some weighted L∞ and Sobolev spaces. The time periodic solution is shown to be asymptotically stable under sufficiently small initial perturbations and the L∞ norm of the perturbation decays as time goes to infinity.

On Some Classes of Time-Periodic Solutions for the Navier--Stokes Equations in the Whole Space

For the three-dimensional Navier--Stokes problem on the whole space, we study the existence, regularity, and stability of time-periodic solutions in Lebesgue, Lorentz, or Sobolev spaces, when the periodic forcing belongs to critical classes of forces.

Time periodic solutions of non-isentropic compressible magnetohydrodynamic system

In this paper, we study the non-isentropic compressible magnetohydrodynamic system with a time periodic external force in ℝn. Under the condition that the optimal time decay rates are obtained by spectral analysis, we show that the existence, uniqueness and time-asymptotic stability of time periodic solutions when the space dimension n ≥ 5. Our proof is based on a combination of the energy method and the contraction mapping theorem.

Time periodic solutions of the non-isentropic compressible fluid models of Korteweg type

In this paper, the non-isentropic compressible Navier-Stokes-Korteweg system with a time periodic external force is considered in $\mathbb{R}^n$. The optimal time decay rates are obtained by spectral analysis. Using the optimal decay estimates, we show that the existence, uniqueness and time-asymptotic stability of time periodic solutions when the space dimension $n\geq 5$. Our proof is based on a combination of the energy method and the contraction mapping theorem.

Existence and stability of time periodic solution to the compressible Navier–Stokes equation for time periodic external force with symmetry

Time periodic problem for the compressible Navier–Stokes equation for barotropic flow on the whole space is studied. The existence of a time periodic solution is proved for sufficiently small time periodic external force with some symmetry when the space dimension is greater than or equal to 3. The proof is based on the spectral properties of the time-T-map associated with the linearized problem around the motionless state with constant density in some weighted Sobolev space. The stability of the time periodic solution is also proved and the decay estimate of the perturbation is established.

Periodic Solutions with a Boundary Layer of Reaction–Diffusion Equations with Singularly Perturbed Neumann Boundary Conditions

We consider singularly perturbed reaction–diffusion equations with singularly perturbed Neumann boundary conditions. We establish the existence of a time-periodic solution u(x, t, ε) with boundary layers and derive conditions for their asymptotic stability. The boundary layer part of u(x, t, ε) is of order one, which distinguishes our case from the case of regularly perturbed Neumann boundary conditions, where the boundary layer is of order ε. Another peculiarity of our problem is that — in contrast to the case of Dirichlet boundary conditions — it may have several asymptotically stable time-periodic solutions, where these solutions differ only in the description of the boundary layers. Our approach is based on the construction of sufficiently precise lower and upper solutions.

Collective phase description of oscillatory convection

We formulate a theory for the collective phase description of oscillatory convection in Hele-Shaw cells. It enables us to describe the dynamics of the oscillatory convection by a single degree of freedom which we call the collective phase. The theory can be considered as a phase reduction method for limit-cycle solutions in infinite-dimensional dynamical systems, namely, stable time-periodic solutions to partial differential equations, representing the oscillatory convection. We derive the phase sensitivity function, which quantifies the phase response of the oscillatory convection to weak perturbations applied at each spatial point, and analyze the phase synchronization between two weakly coupled Hele-Shaw cells exhibiting oscillatory convection on the basis of the derived phase equations.

Time periodic solutions of the compressible magnetohydrodynamic equations

The compressible magnetohydrodynamic equations with a time periodic external force in Rn are studied in this paper. We establish the existence, uniqueness and time-asymptotic stability of time periodic solutions when n≥5. Our proof is essentially based on a combination of the energy method and the time decay estimates of solutions to the linearized system.

Dynamics of solitary waves generated by subcritical instability under the action of delayed feedback control

We consider the influence of a global delayed feedback control which acts on a system governed by a subcritical Ginzburg–Landau equation. The method based on a variational principle is applied for the derivation of a low-dimensional evolution model. In the framework of this model a one-pulse solution is found, and its linear and nonlinear stability analysis is carried out. The existence region for a stable time-periodic pulse solution is found between the boundaries in the parameter space corresponding to a Hopf bifurcation and a saddle-node bifurcation. The obtained results are compared with the results of an analytical linear theory and direct numerical simulations of the original problem.