Control of time-periodic systems via symbolic computation with application to chaos control

A general method for the control of linear time-periodic systems employing symbolic computation of Floquet transition matrix is considered in this work. It is shown that this method is applicable to chaos control. Nonlinear chaotic systems can be driven to a desired periodic motion by designing a combination of a feedforward controller and a feedback controller. The design of the feedback controller is achieved through the symbolic computation of fundamental solution matrix of linear time-periodic systems in terms of unknown control gains. Then, the Floquet transition matrix (state transition matrix evaluated at the end of the principal period) can determine the stability of the system owing to classical techniques such as pole placement, Routh–Hurwitz criteria, etc. Thus it is possible to place the Floquet multipliers in the desired locations to determine the control gains. This method can be applied to systems without small parameters. The Duffing’s oscillator, the Rössler system and the nonautonomous parametrically forced Lorenz equations are chosen as illustrative examples to demonstrate the application.     

Periodic Solutions of a Nonlinear Wave Equation with Nonconstant Coefficients

The existence of time-periodic solutions of a nonlinear equation for forced oscillations of a bounded string is proved when the d'Alembert operator has nonconstant coefficients and the nonlinear term has power-law growth.     

Existence and stability of time-periodic solutions in a model for spherical flames with time-periodic heat losses

The problem under investigation is the study of the effect of periodicity when time-periodic heat losses are added in a model for spherical flames introduced by G. Joulin. In the present context, the flame either quenches in finite or infinite time, or its radius converges to an upper stable time-periodic solution, or to a lower unstable time-periodic solution.     

On the time-periodic problem for the Stokes system in domains with cylindrical outlets to infinity

We study the time-periodic Stokes problem in the domain with cylindrical outlets to infinity in weighted function spaces. We prove that there exists a unique solution with prescribed fluxes over the sections of outlets to infinity and that, in each outlet, this solution tends to the corresponding time-periodic Poiseuille flow. Published in Lietuvos Matematikos Rinkinys, Vol. 47, No. 2, pp. 177–195, April–June, 2007.     

人口问题中一广义Ginzburg-Landau模型方程的时间周期解

该文应用Galerkin方法证明人口问题中一广义 Ginzburg-Landau模型方程的时间周期问题广义时间周期解与古典时间周期解的存在性与唯一性.     

Asymptotics of Small Exterior Navier-Stokes Flows with Non-Decaying Boundary Data

We prove the existence of unique solutions for the 3D incompressible Navier-Stokes equations in an exterior domain with small boundary data which do not necessarily decay in time. As a corollary, the existence of unique small time-periodic solutions is shown. We next show that the spatial asymptotics of the periodic solution is given by the same Landau solution at all times. Lastly we show that if the boundary datum is time-periodic and the initial datum is asymptotically self-similar, then the solution converges to the sum of a time-periodic vector field and a forward self-similar vector field as time goes to infinity.     

On the Cauchy problem for the transport equation with random noise

We establish the existence and uniqueness of a solution to the Cauchy problem for the transport equation with random noise in Rd. When the vector field is time-periodic and the noise is multiplicative and nondegenerate, we show the existence of a time-periodic measure, which includes an invariant measure as a special case.     

Time-periodic Solutions to Systems of Conservation Laws with Ellipticity

We study the time-periodic solutions to systems of conservation laws with ellipticity. It is shown that under time-periodic boundary condition, the system admits at least one global time-periodic solution bounded uniformly with the same period.     

Time-periodic Stokes equations with inhomogeneous Dirichlet boundary conditions in a half-space

The time-periodic Stokes problem in a half-space with fully inhomogeneous right-hand side is investigated. Maximal regularity in a time-periodic L^p setting is established. A method based on Fourier multipliers is employed that leads to a decomposition of the solution into a steady-state and a purely oscillatory part in order to identify the suitable function spaces.     

Time-Periodic Solution to the Compressible Navier–Stokes/Allen–Cahn System

In this paper,we investigate the time-periodic solution to a coupled compressible Navier–Stokes/Allen–Cahn system which describes the motion of a mixture of two viscous compressible fluids with a time periodic external force in a periodic domain in R^N.The existence of the time-periodic solution to the system is established by using an approach of parabolic regularization and combining with the topology degree theory,and then the uniqueness of the period solution is obtained under some smallness and symmetry assumptions on the external force.     

Trigonometric spline and spectral bounds for the solution of linear time-periodic systems

Linear time-periodic systems arise whenever a nonlinear system is linearized about a periodic trajectory. Examples include anisotropic rotor-bearing systems and parametrically excited systems. The structure of the solution to linear time-periodic systems is known due to Floquet’s Theorem. We use this information to derive a new norm which yields two-sided bounds on the solution and in this norm vibrations of the solution are suppressed. The obtained results are a generalization for linear time-invariant systems. Since Floquet’s Theorem is non-constructive, the applicability of the aforementioned results suffers in general from an unknown Floquet normal form. Hence, we discuss trigonometric splines and spectral methods that are both equipped with rigorous bounds on the solution. The methodology differs systematically for the two methods. While in the first method the solution is approximated by trigonometric splines and the upper bound depends on the approximation quality, in the second method the linear time-periodic system is approximated and its solution is represented as an infinite series. Depending on the smoothness of the time-periodic system, we formulate two upper bounds which incorporate the approximation error of the linear time-periodic system and the truncation error of the series representation. Rigorous bounds on the solution are necessary whenever reliable results are needed, and hence they can support the analysis and, e.g., stability or robustness of the solution may be proven or falsified. The theoretical results are illustrated and compared to trigonometric spline bounds and spectral bounds by means of three examples that include an anisotropic rotor-bearing system and a parametrically excited Cantilever beam.     

The phase velocity of acoustic perturbations near a vibration excitation source

The plane problem of the forced vibrations of an elastic half-space due to the action of a time-periodic force, acting along the tangent to the surface of a medium is considered. It is shown that in the near zone the phase velocity of surface acoustic perturbations varies considerably with distance from the vibration source and considerably exceeds the Rayleigh wave velocity.     

Time-periodic solutions of a nonlinear wave equation

The existence of a time-periodic solution of an n-dimensional nonlinear wave equation is established with n=2 and 3.     

A remark on stable subharmonic solutions of time-periodic reaction-diffusion equations

This paper is concerned with a time-periodic reaction-diffusion equation. It is known that typical trajectories approach periodic solutions with possibly longer period than that of the equation. Such solutions are called subharmonic solutions. In this paper, for any domain Ω, time-period τ>0 and integer n?2, we construct an example of a time-periodic reaction-diffusion equation on Ω with a minimal period τ which possesses a stable solution of minimal period nτ.     

一类广义Swift-Hohenberg模型方程的时间周期问题

本文研究一类广义Swift-Hohenberg模型方程的时间周期问题，证明问题周期广义解和周期古典解的存在性与唯一性．     

TIME-PERIODIC SOLUTIONS OF THE VLASOV-POISSON-FOKKER-PLANCK SYSTEM

In this note it is shown that the Vlasov-Poisson-Fokker-Planck system in the three-dimensional whole space driven by a time-periodic background profile near a positive constant state admits a time-periodic small-amplitude solution with the same period. The proof follows by the Serrin's method on the basis of the exponential time-decay property of the linearized system in the case of the constant background profile.     

Reproductive solution of a second-grade fluid system

We treat the existence and uniqueness of reproductive solution (weak time-periodic solution) of a second-grade fluid system for small enough source terms, by using the Galerkin approximation method and compactness arguments.     

Existence and stability of periodic solutions for parabolic systems with time delays

This paper is concerned with the existence and stability time-periodic solutions for a class of coupled parabolic equations with time delay, and time delays may appear in the nonlinear reaction functions. The existence of time-periodic solutions is for a class of locally Lipschitz continuous reaction functions without any quasimonotone requirement. Our approach to the problem is by the method of upper and lower solution and using Schauder fixed point theorem. Some methods for proving the stability of the periodic solution are also given. The results for the general system can be applied to the standard parabolic equations without time delay and corresponding ordinary differential system. Finally, a model arising from chemistry is used to illustrate the obtained results.     

Absence of positive solutions of a second-order semilinear parabolic equation with time-periodic coefficients

We consider a parabolic equation with time-periodic measurable bounded coefficients and with a nonlinear source. The equation is studied on the time line and in a domain of x that does not contain the closed unit ball. The definition of a weak solution of the equation is given.     

一类高阶非线性波动方程的时间周期问题

本文利用临界点理论证明一类高阶非线性波动方程时间周期弱解的存在性．