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Let X = ( X , B , μ ) be a σ -finite measure space and f : X → X be a measurable transformation such that the composition operator T f : φ ↦ φ ∘ f is a bounded linear operator acting on L p ( X , B , μ ) , 1 ≤ p < ∞ . We provide a necessary and sufficient condition on f for T f to be topologically transitive or topologically mixing. We also characterize the topological dynamics of composition operators induced by weighted shifts, non-singular odometers and inner functions. The results provided in this article hold for composition operators acting on more general Banach spaces of functions. [ABSTRACT FROM AUTHOR]

Abstract: Let be a topological dynamical system. A pair of points is called Banach proximal if for any , the set has Banach density one. We study the structure of the Banach proximal relation. A useful tool is the notion of the support of a topological dynamical system. We show that a dynamical system is strongly proximal if and only if every pair in is Banach proximal. A subset S of X is Banach scrambled if every two distinct points in S form a Banach proximal pair but not asymptotic. We construct a dynamical system with the whole space being a Banach scrambled set. Even though the Banach proximal relation of the full shift is of first category, it has a dense Mycielski invariant Banach scrambled set. We also show that for an interval map it is Li–Yorke chaotic if and only if it has a Cantor Banach scrambled set. [Copyright &y& Elsevier]

Abstract: In recent years many new definitions of shadowing, using a notion of ergodic (or average) pseudo-orbit were introduced. While, under the assumption of chain mixing, average pseudo-orbit usually can be equivalently replaced by pseudo-orbit in these definitions of shadowing, it is not completely clear when these shadowing properties (i.e. approximation of a pseudo-orbit by a real orbit on a sufficiently large set of indices) can or cannot occur. In this paper we analyze necessary and sufficient conditions for shadowing over a set with positive density (or syndetic). While we do not provide full characterization, a few relations to standard notions from topological dynamics are obtained. [Copyright &y& Elsevier]

Abstract: In recent years many new definitions of shadowing, using a notion of ergodic (or average) pseudo-orbit were introduced. While, under the assumption of chain mixing, average pseudo-orbit usually can be equivalently replaced by pseudo-orbit in these definitions of shadowing, it is not completely clear when these shadowing properties (i.e. approximation of a pseudo-orbit by a real orbit on a sufficiently large set of indices) can or cannot occur. In this paper we analyze necessary and sufficient conditions for shadowing over a set with positive density (or syndetic). While we do not provide full characterization, a few relations to standard notions from topological dynamics are obtained. [Copyright &y& Elsevier]

Abstract: We develop further the topological theory of regular variation of [N.H. Bingham, A.J. Ostaszewski, Topological regular variation: I. Slow variation, LSE-CDAM-2008-11]. There we established the uniform convergence theorem (UCT) in the setting of topological dynamics (i.e. with a group T acting on a homogenous space X), thereby unifying and extending the multivariate regular variation literature. Here, working with real-time topological flows on homogeneous spaces, we identify an index of regular variation, which in a normed-vector space context may be specified using the Riesz representation theorem, and in a locally compact group setting may be connected with Haar measure. [Copyright &y& Elsevier]

Abstract: This paper concerns with limit cycles through Hopf and homoclinic bifurcations for near-Hamiltonian systems. By using the coefficients appeared in Melnikov functions at the centers and homoclinic loops, some sufficient conditions are obtained to find limit cycles. [Copyright &y& Elsevier]

Abstract: Among the six classes of Zeeman''s classification for three-dimensional Lotka–Volterra competitive systems with limit cycles, besides the classes 26, 27, 28 and 29, multiple limit cycles are found in classes 30 and 31 by an algorithmic method proposed by Hofbauer and So [J. Hofbauer, J.W. So, Multiple limit cycles for three-dimensional Lotka–Volterra equations, Appl. Math. Lett. 7 (1994) 65–70]. This also gives an answer to a problem proposed in [J. Hofbauer, J.W. So, Multiple limit cycles for three-dimensional Lotka–Volterra equations, Appl. Math. Lett. 7 (1994) 65–70]. [Copyright &y& Elsevier]

Abstract: There exists a set S with three elements such that if a meromorphic function f, having at most finitely many simple poles, shares the set S CM with its derivative , then . [Copyright &y& Elsevier]

Abstract: This paper discusses a class of inhomogeneous nonlinear Schrödinger equation where , satisfies some assumptions. By a constrained variational problem, we firstly define some cross-constrained invariant sets for the inhomogeneous nonlinear Schrödinger equation, then we obtain some sharp conditions for global existence and blow up of solutions. As a consequence it is shown that the solution is globally well-posed in with the -norm of the initial data which is dominated by the minimal value of the constrained variational problem. [Copyright &y& Elsevier]

Abstract: Some limit-point criteria are obtained for higher-dimensional semi-degenerate singular Hamiltonian differential systems with perturbation potential terms by using -theory. Results in this paper cover many previous results of Hartman, Levinson, Titchmarsh and Read. [Copyright &y& Elsevier]

Abstract: We prove existence and nonexistence of positive solutions for the quasilinear system where , , , Ω is a bounded domain in , and the coefficients and are allowed to change sign. [Copyright &y& Elsevier]

Abstract: In this paper, we consider a Holling–Tanner system with ratio-dependence. First, we establish the sufficient conditions for the global stability of positive equilibrium by constructing Lyapunov function. Second, through a simple change of variables, we transform the ratio-dependent Holling–Tanner model into a better studied Liénard equation. As a result, the uniqueness of limit cycle can be solved. [Copyright &y& Elsevier]

Abstract: In this paper, a class of second-order servo plants under relay feedback is studied. Complete results on the uniqueness of solutions, existence and stability of the limit cycles are established using the point transformation method. And a numerical method is developed for determining the amplitude and period of a stable limit cycle from the plant parameters. [Copyright &y& Elsevier]

Abstract: In this paper, as in [E.M. Bonotto, M. Federson, Topological conjugation and asymptotic stability in impulsive semidynamical systems, J. Math. Anal. Appl. (2006), doi:10.1016/j.jmaa.2006.03.042], we continue to study the dynamics of flows defined in impulsive semidynamical systems , where X is a metric space, is a semidynamical system, M denotes an impulsive set and I is an impulsive operator. We generalize some results of non-impulsive flows of characteristic () for systems with impulses. In particular, we state conditions so that the limit set of an impulsive system of is either a periodic orbit or a single rest point. We also give conditions for a subset H in to be globally asymptotically stable in the impulsive system, provided the flow is of . [Copyright &y& Elsevier]

Abstract: We study quadratic perturbations of the integrable system , where . We prove that the first three Melnikov functions associated to the perturbed system give rise at most to three limit cycles. [Copyright &y& Elsevier]

Abstract: Let be a one-sided symbolic space (with two symbols), and let σ be the shift on Σ. In this paper, we prove that there exists a minimal set such that is Wiggins chaotic, Martelli chaotic, distributionally chaotic, strictly ergodic, topologically weakly mixing and has zero topological entropy. [Copyright &y& Elsevier]

Abstract: In this paper, we consider a time-delayed periodic system which describes the competition among mature populations. By appealing to theories of monotone dynamical systems, periodic semiflows and uniform persistence, we analyze the evolutionary behavior of the system and establish sufficient conditions for competitive coexistence and exclusion. [Copyright &y& Elsevier]

Abstract: We use sampling techniques to reconstruct the characteristic function associated with the eigenvalues of two linked Sturm–Liouville operators by a transmission condition. The particular case when the transmission matrix is the identity yields the well-known miss-distance function which is used in the computation of eigenvalues by shooting methods. [Copyright &y& Elsevier]

Abstract: Let M be a smooth compact manifold (possibly with boundary) or compact manifold with or . We prove that generic homeomorphism on M has the periodic shadowing property. [Copyright &y& Elsevier]

Abstract: In this work, we prove that the exact controllability of linear autonomous systems are conserved with “small” Desch–Schappacher perturbations arising, e.g., from the perturbations of dynamic operator''s domain. Our results are illustrated by an application to controlled systems with dynamic and boundary perturbations. [Copyright &y& Elsevier]

Abstract: We consider planar cubic systems with a unique rest point of center-focus type and constant angular velocity. For such systems we obtain an affine classification in three families, and, for two of them, their corresponding phase portraits on the Poincaré sphere. We also prove that for two of these families there is uniqueness of limit cycle. With respect the third family, we give the bifurcation diagram and phase portraits on the Poincaré sphere of a one-parameter sub-family exhibiting at least two limit cycles. [Copyright &y& Elsevier]

In this paper, criteria for limit-point (n) case of a singular discrete Hamiltonian system are established. Furthermore, the lower bound of the essential spectrum is obtained and the present of pure point spectrum is discussed for such system by using the spectral theory of self-adjoint operators in a Hilbert space. [Copyright &y& Elsevier]

Abstract: In this article, we show that if f has a snap-back repeller then any small perturbation of f has a snap-back repeller, and hence has Li–Yorke chaos and positive topological entropy, by simply using the implicit function theorem. We also give some examples. [Copyright &y& Elsevier]