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Abstract: In a recent paper of Llibre and Rodríguez (J. Differential Equations 198 (2004) 374–380) it is proved that every configuration of cycles in the plane is realizable (up to homeomorphism) by a polynomial vector field of degree at most , where is the number of cycles and the number of primary cycles (a cycle is primary if there are no other cycles contained in the bounded region limited by ). In this letter we prove the same theorem by using an easier construction but with a greater polynomial bound (the vector field we construct has degree at most ). By using the same technique we also construct polynomial vector fields realizing (up to homeomorphism) any configuration of limit cycles which can be linked and knotted in . This answers a question of R. Sverdlove. [Copyright &y& Elsevier]

Abstract: We consider a 2mth order elliptic operator of divergence form in a domain Ω of , whose leading coefficients are uniformly continuous. In the paper [Y. Miyazaki, The theory of divergence form elliptic operators under the Dirichlet condition, J. Differential Equations 215 (2005) 320–356], we developed the theory including the construction of resolvents, assuming that the boundary of Ω is of class . The purpose of this paper is to show that the theory also holds when Ω is a domain, applying the inequalities of Hardy type for the Sobolev spaces. [Copyright &y& Elsevier]

Abstract: The paper deals with generic perturbations from a Hamiltonian planar vector field and more precisely with the number and bifurcation pattern of the limit cycles. In this paper we show that near a 2-saddle cycle, the number of limit cycles produced in unfoldings with one unbroken connection, can exceed the number of zeros of the related Abelian integral, even if the latter represents a stable elementary catastrophe. We however also show that in general, finite codimension of the Abelian integral leads to a finite upper bound on the local cyclicity. In the treatment, we introduce the notion of simple asymptotic scale deformation. [Copyright &y& Elsevier]

In this paper, we give necessary conditions for Lie-stability of the equilibrium solutions of autonomous and periodic Hamiltonian systems with n degrees of freedom which possess multiple resonances of arbitrary order. Necessary conditions for instability in the sense of Lyapunov in some cases of multiple resonances are provided. These conditions depend on the order of the resonance and the coefficients of the Hamiltonian function. [ABSTRACT FROM AUTHOR]

Abstract: The current paper is devoted to the study of spatial spreading dynamics of monostable equations with nonlocal dispersal in spatially periodic habitats. In particular, the existence and characterization of spreading speeds is considered. First, a principal eigenvalue theory for nonlocal dispersal operators with space periodic dependence is developed, which plays an important role in the study of spreading speeds of nonlocal periodic monostable equations and is also of independent interest. In terms of the principal eigenvalue theory it is then shown that the monostable equation with nonlocal dispersal has a spreading speed in every direction in the following cases: the nonlocal dispersal is nearly local; the periodic habitat is nearly globally homogeneous or it is nearly homogeneous in a region where it is most conducive to population growth in the zero-limit population. Moreover, a variational principle for the spreading speeds is established. [Copyright &y& Elsevier]

Abstract: We consider a mechanical system which is controlled by means of moving constraints. Namely, we assume that some of the coordinates can be directly assigned as functions of time by implementing frictionless constraints. This leads to a system of ODE''s whose right hand side depends quadratically on the time derivative of the control. In this paper we introduce a simplified dynamics, described by a differential inclusion. We prove that every trajectory of the differential inclusion can be uniformly approximated by a trajectory of the original system, on a sufficiently large time interval, starting at rest. Under a somewhat stronger assumption, we show this second trajectory reaches exactly the same terminal point. [Copyright &y& Elsevier]

Abstract: This paper deals with existence and exponential decay of homoclinic orbits in the first-order Hamiltonian system where the Hamiltonian function is nonperiodic in and superquadratic in . With certain mild conditions we obtain the solutions via variational methods for strongly indefinite problems. [Copyright &y& Elsevier]

Abstract: For a germ of analytic vector fields, the existence of first integrals, resonance and the convergence of normalization transforming the vector field to a normal form are closely related. In this paper we first provide a link between the number of first integrals and the resonant relations for a quasi-periodic vector field, which generalizes one of the Poincaré''s classical results [H. Poincaré, Sur l''intégration des équations différentielles du premier order et du premier degré I and II, Rend. Circ. Mat. Palermo 5 (1891) 161–191; 11 (1897) 193–239] on autonomous systems and Theorem 5 of [Weigu Li, J. Llibre, Xiang Zhang, Local first integrals of differential systems and diffeomorphism, Z. Angew. Math. Phys. 54 (2003) 235–255] on periodic systems. Then in the space of analytic autonomous systems in with exactly n resonances and n functionally independent first integrals, our results are related to the convergence and generic divergence of the normalizations. Lastly for a planar Hamiltonian system it is well known that the system has an isochronous center if and only if it can be linearizable in a neighborhood of the center. Using the Euler–Lagrange equation we provide a new approach to its proof. [Copyright &y& Elsevier]

Abstract: This paper investigates the number and distribution of the limit cycles bifurcated from several graphics and ensembles through a saddle-node and two hyperbolic saddles and for the non-generic cases of , and , , where and are the hyperbolicity ratio of the saddles and , respectively. For the case of , , we suppose that the connection from to and the connection from to keep unbroken. We prove that these graphics and ensembles are of finite cyclicity respectively. Moreover, the cyclicity is linearly dependent on the order of the neutral saddle if is contractive and . We also show that the nearer is close to 1, the more the limit cycles are bifurcated. For the case of , , we obtain that these graphics and ensembles are of finite cyclicity respectively if is of finite order and the hp-connection from to keeps unbroken. [Copyright &y& Elsevier]

Abstract: The purpose of this paper is to investigate the asymptotic behavior of positive solutions of nonautonomous and random competitive Kolmogorov systems via the skew-product flows approach. It is shown that there exists an unordered carrying simplex which attracts all nontrivial positive orbits of the skew-product flow associated with a nonautonomous (random) competitive Kolmogorov system. [Copyright &y& Elsevier]

Abstract: In this paper we prove the existence and multiplicity of homoclinic orbits for first order Hamiltonian systems of the form where is asymptotically linear at ∞ and is not assumed to be periodic. [Copyright &y& Elsevier]

Abstract: In this paper, one-dimensional (1D) nonlinear Schrödinger equation with the periodic boundary condition is considered. It is proved that for each given constant potential m and each prescribed integer , the equation admits a Whitney smooth family of small amplitude, time quasi-periodic solutions with N Diophantine frequencies. The proof is based on a partial Birkhoff normal form reduction and an improved KAM method. [Copyright &y& Elsevier]

Abstract: In this paper we deal with discontinuous vector fields on and we prove that the analysis of their local behavior around a typical singularity can be treated via singular perturbation. The regularization process developed by Sotomayor and Teixeira is crucial for the development of this work. [Copyright &y& Elsevier]

Abstract: This paper is concerned with Hamiltonian systems of linear differential equations with periodic coefficients under a small perturbation. It is well known that Krein''s formula determines the behavior of definite multipliers on the unit circle and is quite useful in studying the (strong) stability of Hamiltonian system. Our aim is to give a simple formula that determines the behavior of indefinite multipliers with two multiplicity, which is generic case. The result does not require analyticity and is proved directly. Applying this formula, we obtain instability criteria for solutions with periodic structure in nonlinear dissipative systems such as the Swift–Hohenberg equation and reaction–diffusion systems of activator–inhibitor type. [Copyright &y& Elsevier]

Abstract: This paper is devoted to the study of differential inclusions, particularly discontinuous perturbed sweeping processes in the infinite-dimensional setting. On the one hand, the sets involved are assumed to be prox-regular and to have a variation given by a function which is of bounded variation and right continuous. On the other hand, the perturbation satisfies a linear growth condition with respect to a fixed compact subset. Finally, the case where the sets move in an absolutely continuous way is recovered as a consequence. [Copyright &y& Elsevier]

Abstract: In this paper, we study the asymptotic behavior of solutions for the plate equation with a localized damping and a critical exponent. We prove the existence, regularity and finite dimensionality of a global attractor in . [Copyright &y& Elsevier]

Abstract: In this paper we consider the differential inclusion problemwhere is radially symmetric, and stands for the generalized gradient of a locally Lipschitz function . Under suitable oscillatory assumptions on the potential at zero or at infinity, we show the existence of infinitely many, radially symmetric solutions of (DI). No symmetry requirement on is needed. Our approach is based on a non-smooth Ricceri-type variational principle, developed by Marano and Motreanu (J. Differential Equations 182 (2002) 108–120). [Copyright &y& Elsevier]

The purpose of this paper is to investigate uniform persistence for nonautonomous and random parabolic Kolmogorov systems via the skew-product semiflows approach. It is first shown that the uniform persistence of the skew-product semiflow associated with a nonautonomous (random) parabolic Kolmogorov system implies that of the system. Various sufficient conditions in terms of the so-called unsaturatedness and/or Lyapunov exponents for uniform persistence of the skew-product semiflows are then provided. Among others, it is shown that if the associated skew-product semiflow has a global attractor and its restriction to the boundary of the state space has a Morse decomposition which is unsaturated or whose external Lyapunov exponents are positive, then it is uniformly persistent. More specific conditions are discussed for uniform persistence in n-species, particularly 3-species, random competitive systems. [Copyright &y& Elsevier]

In this paper, we present a new technique of proving the existence of an infinite number of symmetric periodic solutions. This technique combines the covering relations method introduced by Zgliczyn´ski (J. Differential Equations 171 (2001) 173; Methods in Nonlinear Anal. 8 (1996) 169; Nonlinearity 10 (1997) 243) with fixed set iteration method described in Lamb (Reversing symmetries in dynamical systems, Ph.D. Thesis, Universiteit van Amsterdam, 1994). As an example we present a computer-assisted proof of symbolic dynamics in the Kuramoto–Sivashinsky equations. Moreover, we prove the existence of an infinite number of symmetric and an infinite number of nonsymmetric periodic solutions. [Copyright &y& Elsevier]