三维Lotka-Volterra 系统正拟周期解的存在性和稳定性

This work focuses on the existence and stability of positive quasi-periodic solutions for the 3-dimensional Lotka-Volterra system. Using KAM （Kolmogorov-Arnold-Moser） theory and Newton iteration, it is shown that there exists a positive quasi-periodic solution in a Cantor family for the 3-dimensional Lotka-Volterra system. On the above basis, we can show the stability of the solution with the help of Lyapunov function.     

Solutions of the generalized Lennard-Jones system

In this paper, we study solution structures of the following generalized Lennard-Jones system in R~n,x=(-α/|x|~(α+2)+β/|x|~(β+2))x,with 0 α β. Considering periodic solutions with zero angular momentum, we prove that the corresponding problem degenerates to 1-dimensional and possesses infinitely many periodic solutions which must be oscillating line solutions or constant solutions. Considering solutions with non-zero angular momentum, we compute Morse indices of the circular solutions first, and then apply the mountain pass theorem to show the existence of non-circular solutions with non-zero topological degrees. We further prove that besides circular solutions the system possesses in fact countably many periodic solutions with arbitrarily large topological degree, infinitely many quasi-periodic solutions, and infinitely many asymptotic solutions.     

KAM for quasi-linear KdV

We prove the existence and stability of Cantor families of quasi-periodic, small-amplitude solutions of quasi-linear autonomous Hamiltonian perturbations of KdV.     

A KAM Result on Compact Lie Groups

Acta Applicandae Mathematicae - We describe some recent results on existence of quasi-periodic solutions of Hamiltonian PDEs on compact manifolds. We prove a linear stability result for the...     

On small perturbation of two-dimensional quasi-periodic systems with hyperbolic-type degenerate equilibrium point

In this paper we consider small quasi-periodic perturbation of two-dimensional nonlinear quasi-periodic system with hyperbolic-type degenerate equilibrium point. By KAM method we prove that it can be reduced to a suitable normal form with zero as equilibrium point by a quasi-periodic transformation. Hence, the perturbed system has a quasi-periodic solution near the equilibrium point.     

Case of a generating family of quasi-periodic solutions in the theory of small parameter : PMM vol. 37, n6, 1973, pp. 990–998

We consider the problem of the existence and the stability in-the-small of periodic solutions of systems of ordinary differential equations with a small parameter μ, which in the generating approximation (μ = 0) admit of a family of quasi-periodic solutions (we are concerned only with the solutions belonging to the indicated family when μ = 0). The case to be investigated is in a specific sense a more general case of the unisolated generating solution in the small parameter theory and, therefore, includes everything previously treated by Malkin [1], Blekhman [2], and others. The main difficulty in the investigation is the presence of a multiple zero root in the characteristic determinant of the problem's generating system, to which both simple as well as quadratic elementary divisors [3] correspond. This fact predestines the presence of three groups of stability criteria for the solution being examined. The method for constructing these criteria, proposed here, assumes, in contrast to a previous one [1], the preliminary determination of not only the generating approximation but also the first one to the desired periodic solution. Particular aspects of the general “mixed” problem treated here were studied earlier in [4, 5].     

Analysis of a Lotka–Volterra food chain chemostat with converting time delays

A model of the food chain chemostat involving predator, prey and growth-limiting nutrients is considered. The model incorporates two discrete time delays in order to describe the time involved in converting processes. The Lotka–Volterra type increasing functions are used to describe the species uptakes. In addition to showing that solutions with positive initial conditions are positive and bounded, we establish sufficient conditions for the (i) local stability and instability of the positive equilibrium and (ii) global stability of the non-negative equilibria. Numerical simulation suggests that the delays have both destabilizing and stabilizing effects, and the system can produce stable periodic solutions, quasi-periodic solutions and strange attractors.     

Construction of quasi-periodic solutions of delay differential equations via KAM techniques

This work focuses on the existence of quasi-periodic solutions for linear autonomous delay differential equation under quasi-periodic time-dependent perturbation near an elliptic-hyperbolic equilibrium point. Using the time-1 map of the solution operator, Newton iteration scheme, space splitting and KAM techniques, it is shown that under appropriate hypothesis, there exist quasi-periodic solutions with the same frequencies as the perturbation for most parameters. We show that if the delay differential equation is analytic, we obtain analytic parameterizations of the solutions.     

Quasi-periodic solutions for perturbed autonomous delay differential equations

This work discusses the persistence of quasi-periodic solutions for delay differential equations. We prove that the perturbed system possesses a quasi-periodic solution under appropriate hypotheses if an unperturbed linear system has quasi-periodic solutions. We extend some well-known results on ordinary differential equations to delay differential equations.     

Quasiperiodic solutions for nonlinear differential equations of second order with symmetry

In this paper, we study the existence of quasi-periodic solutions and the boundedness of solutions for a wide class nonlinear differential equations of second order. Using the KAM theorem of reversible systems and the theory of transformations we obtain the existence of quasi-periodic solutions and the boundedness of solutions under some reasonable conditions.     

Dynamics in coupled oscillators with recurrent/random forcing, a PDE approach

The current paper is devoted to the study of coupled oscillators with recurrent/random forcing. Special attention is given to the solutions having the same recurrence/randomness as that of the forcing (recurrent/random solutions for short). By embedding coupled oscillators into coupled parabolic equations, it establishes a general theorem on the existence of recurrent/random solutions. It also finds conditions under which such solutions are unique. When the recurrent forcing is actually quasi-periodic or almost periodic, recurrent solutions are refereed to as quasi-periodic or almost periodic solutions in a weak sense and they are quasi-periodic or almost periodic in the classical sense under the uniqueness conditions. In addition, applications of the general theory to coupled Duffing type oscillators and Josephson junctions are considered and the results obtained extend several existing ones for quasi-periodic Duffing oscillators.     

On Quasi-Periodic Solutions of Differential Equations with Piecewise Constant Argument

We study the existence of quasi-periodic solutions to differential equations with piecewise constant argument (EPCA, for short). It is shown that EPCA with periodic perturbations possess a quasi-periodic solution and no periodic solution. The appearance of quasi-periodic rather than periodic solutions is due to the piecewise constant argument. This new phenomenon illustrates a crucial difference between ODE and EPCA. The results are extended to nonlinear equations.     

Quasi-periodic solutions of nonlinear Schrödinger equations of higher dimension

It is shown that there are plenty of quasi-periodic solutions of nonlinear Schrödinger equations of higher spatial dimension, where the dimension of the frequency vectors of the quasi-periodic solutions are equal to that of the space.     

乘法双准周期解析函数的一些引理*

本文证明乘法双准周期解析函数即使在区域边界上实部等于零,本身仍可能不恒为零,且指出了出现这种情况的条件,并用实例说明确实存在这种情况.最后并讨论了乘数不事先指定时间题的一般解.     

The coexistence of quasi-periodic and blow-up solutions in a class of Hamiltonian systems

The coexistence of quasi-periodic solutions and blow-up phenomena in a class of higher dimensional Duffing-type equations is proved in this paper. Moreover, we show that the initial point sets for both kinds of solutions are of infinite Lebesgue measure in the phase space. For the part of quasi-periodic solutions, the tool we used is the small twist theorem for higher dimensional cases.     

Construction of quasi-periodic solutions of guaranteed accuracy by the small parameter method

Theorems on the localization of exact solutions are proved for a quasilinear mathematical model describing quasi-periodic processes. Based on these theorems, constructive algorithms are proposed for calculating quasi-periodic solutions with guaranteed accuracy. Quasi-periodic motions play an important role in engineering and physics, where they often represent determining states. Quasi-periodic motions can be found in many ecological, biological, and economic processes.     

Traveling waves in the complex Ginzburg-Landau equation

Summary In this paper we consider a modulation (or amplitude) equation that appears in the nonlinear stability analysis of reversible or nearly reversible systems. This equation is the complex Ginzburg-Landau equation with coefficients with small imaginary parts. We regard this equation as a perturbation of the real Ginzburg-Landau equation and study the persistence of the properties of the stationary solutions of the real equation under this perturbation. First we show that it is necessary to consider a two-parameter family of traveling solutions with wave speedυ and (temporal) frequencyθ; these solutions are the natural continuations of the stationary solutions of the real equation. We show that there exists a two-parameter family of traveling quasiperiodic solutions that can be regarded as a direct continuation of the two-parameter family of spatially quasi-periodic solutions of the integrable stationary real Ginzburg-Landau equation. We explicitly determine a region in the (wave speedυ, frequencyθ)-parameter space in which the weakly complex Ginzburg-Landau equation has traveling quasi-periodic solutions. There are two different one-parameter families of heteroclinic solutions in the weakly complex case. One of them consists of slowly varying plane waves; the other is directly related to the analytical solutions due to Bekki & Nozaki [3]. These solutions correspond to traveling localized structures that connect two different periodic patterns. The connections correspond to a one-parameter family of heteroclinic cycles in an o.d.e. reduction. This family of cycles is obtained by determining the limit behaviour of the traveling quasi-periodic solutions as the period of the amplitude goes to ∞. Therefore, the heteroclinic cycles merge into the stationary homoclinic solution of the real Ginzburg-Landau equation in the limit in which the imaginary terms disappear.     

Quasi-periodic solutions of forced isochronous oscillators at resonance

We deal with the existence of quasi-periodic solutions of forced isochronous oscillators with a repulsive singularity, the nonlinearity is a bounded perturbation. Using a variant of Moser's twist theorem of invariant curves, due to Ortega [R. Ortega, Boundedness in a piecewise linear oscillator and a variant of the small twist theorem, Proc. London Math. Soc. 79 (1999) 381-413], we show that there are many quasi-periodic solutions and the boundedness of all solutions.     

Preservation of stability under discretization of systems of ordinary differential equations

We study the stability preservation problem while passing from ordinary differential to difference equations. Using the method of Lyapunov functions, we determine the conditions under which the asymptotic stability of the zero solutions to systems of differential equations implies that the zero solutions to the corresponding difference systems are asymptotically stable as well. We prove a theorem on the stability of perturbed systems, estimate the duration of transition processes for some class of systems of nonlinear difference equations, and study the conditions of the stability of complex systems in nonlinear approximation.