Positive Time and Almost Time Periodic Solutions for the Quasigeostrophic Motions

Existence and stability of time periodic traveling waves for a periodic bistable Lotka–Volterra competition system

Time periodic strong solutions to the Keller-Segel system coupled to Navier-Stokes equation

Turing instabilities and patterns near a Hopf bifurcation

In suitable bounded regions immersed in vacuum, time periodic wave solutions solving a full set of electrodynamics equations can be explicitly computed. Analytical expressions are available in special cases, whereas numerical simulations are necessary in more complex situations. The attention here is given to selected three-dimensional geometries, which are topologically equivalent to a toroid, where the behavior of the waves is similar to that of fluid-dynamics vortex rings. The results show that the shape of the sections of these rings depends on the behavior of the eigenvalues of a certain elliptic differential operator. Time-periodic solutions are obtained when at least two of such eigenvalues attain the same value. The solutions obtained are discussed in view of possible applications in electromagnetic whispering galleries or plasma physics.

The current paper is devoted to the study of the stability of space–time periodic traveling wave solutions and positive space–time periodic entire solutions of nonlocal dispersal cooperative systems in space–time periodic habitats. We first show the existence, uniqueness and stability of positive space–time periodic entire solution $${\mathbf {u}}^{*}(t,x)$$ for such nonlocal dispersal cooperative system. The existence of space–time periodic traveling wave solution connecting $${\mathbf {0}}$$ and positive space–time periodic entire solution $${\mathbf {u}}^{*}(t,x)$$ has been established by Bao, Shen and Shen (Commun. Pure Appl. Anal. 18: 361–396, 2019). In this paper, by using comparison principle and a weight function, we further show that the space–time periodic traveling wave solution for nonlocal dispersal cooperative system is asymptotically stable, as long as the initial value is uniformly bounded in a weighted space.

In a laser with an injected signal (LIS), the phase of the laser field can be constant (phase locking) or grow unbounded (phase drift) or vary periodically (phase entrainment). We analyze these three responses by studying the bifurcation diagram of the time-periodic solutions in the limit of small values of the population relaxation rate (small \ensuremath{\gamma}), small values of the detuning parameters (small \ensuremath{\Delta} and \ensuremath{\Theta}), and small values of the injection field (small y). Our bifurcation analysis has led to the following results: (1) We determine analytically the conditions for a Hopf bifurcation to stable time-periodic solutions. This bifurcation appears at y=${\mathit{y}}_{\mathit{H}}$ and is possible only if the detuning parameters are sufficiently large compared to \ensuremath{\gamma} [specifically, \ensuremath{\Delta} and \ensuremath{\Theta} must be O(${\ensuremath{\gamma}}^{1/2}$) quantities]. (2) We construct the periodic solutions in the vicinity of y=${\mathit{y}}_{\mathit{H}}$. We show that the phase of the laser field varies periodically. We then follow numerically this branch of periodic solutions from y=${\mathit{y}}_{\mathit{H}}$ to y=0. Near y=0, the phase becomes unbounded. (3) A transition between bounded and unbounded phase time-periodic solutions appears at y=${\mathit{y}}_{\mathit{c}}$ (0${\mathit{y}}_{\mathit{c}}$${\mathit{y}}_{\mathit{H}}$). This transition does not appear at a bifurcation point. We characterize this transition by analyzing the behavior of the phase as y\ensuremath{\rightarrow}${\mathit{y}}_{\mathit{c}}^{+}$ and as y\ensuremath{\rightarrow}${\mathit{y}}_{\mathit{c}}^{\mathrm{\ensuremath{-}}}$. (4) We find conditions for a secondary bifurcation from the periodic solutions to quasiperiodic solutions. The secondary branch of solutions is then investigated numerically and is shown to terminate at the limit point of the steady states. (5) We develop a singular perturbation analysis of the LIS equations valid as \ensuremath{\gamma}\ensuremath{\rightarrow}0. This analysis allows the determination of a complete branch of bounded time-periodic solutions. We show that the amplitude of these time-periodic solutions becomes unbounded as y\ensuremath{\rightarrow}0 provided that \ensuremath{\Delta} and \ensuremath{\Theta} are sufficiently small compared to \ensuremath{\gamma} [specifically, \ensuremath{\Delta} and \ensuremath{\Theta} are zero or O(\ensuremath{\gamma}) quantities].

In this paper, we deal with the time periodic problem to coupled chemotaxis-fluid models. We prove the existence of large time periodic strong solutions for the full chemotaxis-Navier–Stokes system in spatial dimension \(N=2\), and the existence of large time periodic strong solutions for the chemotaxis-Stokes system in spatial dimension \(N=3\). On the basis of these, the regularity of the solutions can be further improved. More precisely speaking, if the time periodic source g and the potential force \(\nabla \varphi \) belong to \(C^{\alpha , \frac{\alpha }{2}}(\overline{\Omega }\times \mathbb R)\), the solutions we obtained are also classical solutions.

Time periodic solutions of Hamilton-Jacobi equations with autonomous Hamiltonian on the circle

On a problem of forced nonlinear oscillations. numerical example of bifurcation into an invariant torus

In this article, we investigate the time periodic solutions for two-dimensional Navier-Stokes equations with nontrivial time periodic force terms. Under the time periodic assumption of the force term, the existence of time periodic solutions for two-dimensional Navier-Stokes equations has received extensive attention from many authors. With the smallness assumption of the time periodic force, we show that there exists only one time periodic solution and this time periodic solution is globally asymptotically stable in the $$H^1$$H1 sense. Without smallness assumption of the force term, there is no stability analysis theory addressed. It is expected that when the amplitude of the force term is increasing, the time periodic solution is no longer asymptotically stable. In the last part of the article, we use numerical experiments to study the bifurcation of the time periodic solutions when the amplitude of the force is increasing. Extrapolating to the heating of the earth by the sun, the bifurcation diagram hints that when the earth receives a relatively small amount of solar energy regularly, the time periodic fluid patterns are asymptotically stable; while/when the earth receives too much solar energy even though in a time periodic way, the time periodic pattern of the fluid motions will lose its stability.

We consider the coupled chemotaxis-fluid model for periodic pattern formation on two- and three-dimensional domains with mixed nonhomogeneous boundary value conditions, and prove the existence of nontrivial time periodic solutions. It is worth noticing that this system admits more than one periodic solution. In fact, it is not difficult to verify that (0, c, 0, 0) is a time periodic solution. Our purpose is to obtain a time periodic solution with nonconstant bacterial density.

In this paper, we study the time periodic traveling wave solutions for a Kermack–McKendrick SIR epidemic model with individuals diffusion and environment heterogeneity. In terms of the basic reproduction number $$R_0$$ of the corresponding periodic ordinary differential model and the minimal wave speed $$c^*$$ , we establish the existence of periodic traveling wave solutions by the method of super- and sub-solutions, the fixed-point theorem, as applied to a truncated problem on a large but finite interval, and the limiting arguments. We further obtain the nonexistence of periodic traveling wave solutions for two cases involved with $$R_0$$ and $$c^*$$ .

Dynamics of time-periodic reaction-diffusion equations with compact initial support on [formula omitted

Time periodic traveling wave solutions for periodic advection–reaction–diffusion systems

In this paper we construct time-periodic solutions of the nonlinear Schrodinger equation $$ - i{\partial _t}u = \left( { - {\partial _{xx}}u + v\left( x \right)} \right)u + g\left( {u,\bar u,x} \right),0 < x < \pi ,t \in \mathbb{R}. $$ (1.1) satisfying either periodic or Dirichlet boundary conditions. We assume that the non-linearity g is of the form \( g\left( {u,{\mkern 1mu} \bar u,{\mkern 1mu} x} \right) = {\partial _{\bar u}}\mathcal{G}\left( {u,{\mkern 1mu} \bar u,{\mkern 1mu} x} \right) \), for some real valued function G. This equation can then be viewed as a Hamiltonian system with infinitely many degrees of freedom, and the problem of time periodic solutions is related to perturbation theory in the neighborhood of elliptic stationary points of such systems. In contrast to the case of finite dimensional Hamiltonian systems, even the problem of periodic solutions for (1.1) exhibits the phenomenon of small denominators. Our results imply that the effect of these small denominators may be overcome, resulting in the construction of Cantor-like families of periodic solutions in a neighborhood of the equilibrium u = 0. We will present the details here only for the case of periodic boundary conditions, since this case is technically more difficult, allows for interesting resonances between the linear modes, and because the case of Dirichlet boundary conditions has already been treated by Kuksin (1988, 1993) using KAM methods. We note that in the special case in which g depends on u and ū only through the combination |u|2, one does not encounter small denominators, and one can construct periodic solutions with an ordinary implicit function theorem.