具有四个时滞的平面系统的周期解的存在性

The sufficient condition for the existence of non-constant periodic solutions of the following planar system with four delays are obtained:     

Synchronization of coupled systems to periodic diagonal solutions with synchronized asymptotic phases

Using the change of coordinates, parameterization and characteristic multipliers, we prove the synchronization of a class of coupled nonlinear systems with nontrivial periodic solution. The periodic diagonal solution of the coupled system is asymptotically orbitally stable with asymptotic phase.Examples are given to illustrate the theorem.     

Periodic solutions of perturbed isochronous hamiltonian systems at resonance

We look for periodic solutions of planar systems obtained by adding an asymptotically positively homogeneous nonlinear term to an isochronous hamiltonian system. Precise computations of the topological degree are obtained by elementary phase-plane analysis.     

Abel-like differential equations with no periodic solutions

We present various criteria for the non-existence of positive periodic solutions of generalized Abel differential equations with periodic coefficients that can change sign. As an application, we obtain some families of planar vector fields without limit cycles.     

Existence and uniqueness of periodic solutions for forced Rayleigh-type equations

In this paper, we use the coincidence degree theory to establish new results on the existence and uniqueness of T-periodic solutions for a kind of forced Rayleigh equation of the form

x^″+f(x^′(t))+g(t,x(t))=e(t).     

Periodic perturbations with rotational symmetry of planar systems driven by a central force

We consider periodic perturbations of a central force field having a rotational symmetry, and prove the existence of nearly circular periodic orbits. We thus generalize, in the planar case, some previous bifurcation results obtained by Ambrosetti and Coti Zelati in [1]. Our results apply, in particular, to the classical Kepler problem.     

Positive periodic solutions for a system of anisotropic parabolic equations

We consider a system of two degenerate parabolic equations with nonlocal terms and Dirichlet boundary conditions. More precisely, the degeneracy in each equation of the system is of the type r(x)-Laplacian where r(x) is a function depending on x∈Ω, where Ω is a bounded smooth domain of Rn. The system models the diffusion and the interaction between two different biological species sharing the same territory Ω. The paper provides conditions on the parameters of the problem that guarantee the coexistence of a T-periodic non-negative solution (u,v) with both non-trivial u,v.     

Unbounded solutions of a class of planar systems

In this paper, the unbounded solutions for the following nonlinear planar system:

x′=a+y+−a−y−+f(t),y′=−b+x++b−x−+g(t),     

Computation of periodic solutions of Hamiltonian systems

This paper attempts to give a practical method to compute global periodic solutions of autonomous Hamiltonian systems of arbitrary finite order. The proposed numerical method is based on continuation of solutions branching from equlibrium points and requires no iterations. Moreover, during computation of one-parameter families of periodic orbits, their possible bifurcations are determined as well.     

A continuation principle for a class of periodically perturbed autonomoussystems

The paper deals with a T ‐periodically perturbed autonomous system in ℝⁿ of the form ((PS)) with ε > 0 small. The main goal of the paper is to provide conditions ensuring the existence of T ‐periodic solutions to (PS) belonging to a given open set W ⊂ C ([0, T ],ℝⁿ ). This problem is considered in the case when the boundary ∂W of W contains at most a finite number of nondegenerate T ‐periodic solutions of the autonomous system = ϕ (x). The starting point of our approach is the following property due to Malkin: if for any T ‐periodic limit cycle x 0 of = ϕ (x) belonging to ∂W the so‐called bifurcation function f (θ), θ ∈ [0, T ], associated to x₀, see (1.11), satisfies the condition f(0) ≠ 0 then the integral operator does not have fixed points on ∂W for all ε > 0 sufficiently small. By means of the Malkin's bifurcation function we then establish a formula to evaluate the Leray–Schauder topological degree of I – Q_ε on W. This formula permits to state existence results that generalize or improve several results of the existing literature. In particular, we extend a continuation principle due to Capietto, Mawhin and Zanolin where it is assumed that ∂W does not contain any T ‐periodic solutions of the unperturbed system. Moreover, we obtain generalizations or improvements of some existence results due to Malkin and Loud. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Asymptotic stability of continuum sets of periodic solutions to systems with hysteresis

We consider hysteresis perturbations of a system of ODEs which possesses an asymptotically stable periodic solution z_∗. Where the oscillation of an appropriate projection of this periodic solution is smaller than some threshold number defined by the hysteresis nonlinearity, it is shown that the perturbed system has a continuum of periodic solutions with a rather simple structure in a vicinity of z_∗. The main result is a theorem on the stability of this continuum.     

Existence of two periodic solutions for a non-autonomous SIR epidemic model

A non-autonomous SIR model with periodic transmission rate and a constant removal rate is formulated. By using the continuation theorem of coincidence degree theory, sufficient conditions for the existence of at least two positive periodic solutions are obtained. The stability of the periodic solution for small seasonality is investigated. Numerical simulations are done to show our theoretical results.     

Unboundedness of solutions of a class of planar Hamiltonian systems

In this paper, the unboundedness of solutions for the following planar Hamilton system Ju ′ = ?H (u) + h (t) is discussed, where the function H (u) ∈ C²(R², R) is positive for u ≠ 0 and is positively (q, p)‐quasihomogeneous of quasi‐degree pq, where p > 1 and + = 1, h: S¹ → R² with h ∈ L^∞(0, 2π) is 2π ‐periodic and J is the standard symplectic matrix. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)