Search

Abstract In this paper we introduce a concept of weak stability for non-autonomous dynamical systems. We characterize the set of weak stable points and show that the set of weak stable points is residual, and investigate the relation between weak stability and shadowing property. We also discuss the relation between weak stability of a non-autonomous dynamical system and its induced set-valued system. [ABSTRACT FROM AUTHOR]

In this paper we introduce a new notion, named controlled shadowing property and we relate it to some notions in dynamical systems such as topological ergodicity, topologically mixing and specification properties. The relation between the controlled shadowing and chaos in sense of Li-Yorke is studied. At the end we give some examples to investigate the controlled shadowing property. [ABSTRACT FROM AUTHOR]

In this paper we consider shadowing and weak shadowing properties for iterated function systems IFS and give some results on these concepts. At first, a sufficient condition for shadowing property is given and by this result we present two IFS which have the shadowing property. It is proved that every uniformly expanding as well as every uniformly contracting IFS has the weak shadowing property. By an example we show that in IFS's shadowing property does not imply weak shadowing property. Finally we have the main result of the paper and prove that the weak shadowing is a generic property in the set of all IFS's. [ABSTRACT FROM AUTHOR]

We say that a dynamical system X has the Ramsey shadowing property if an arbitrary sequence of points in X can be shadowed on a set that is "large" in the sense of Ramsey theory. Our main theorem states that this property is equivalent to the existence of a dense set of minimal points. [ABSTRACT FROM AUTHOR]

It is proved that a nontrivial compact dynamical system with asymptotic average shadowing property (AASP) displays uniformly distributional chaos or distributional chaos in a sequence. Moreover, distributional chaos in a system with AASP can be uniform and dense in the measure center, that is, there is an uncountable uniformly distributionally scrambled set consisting of such points that the orbit closure of every point contains the measure center. As a corollary, the similar results hold for the system with almost specification property. [ABSTRACT FROM AUTHOR]

Using the shadowing property of the dynamical system, we prove the generalized HyersUlam stability of quintic functional equations. [ABSTRACT FROM AUTHOR]

Let ${\it\gamma}$ be a hyperbolic closed orbit of a $C^{1}$ vector field $X$ on a compact $C^{\infty }$ manifold $M$ and let $H_{X}({\it\gamma})$ be the homoclinic class of $X$ containing ${\it\gamma}$. In this paper, we prove that if a $C^{1}$-persistently expansive homoclinic class $H_{X}({\it\gamma})$ has the shadowing property, then $H_{X}({\it\gamma})$ is hyperbolic. [ABSTRACT FROM AUTHOR]

Let f be a diffeomorphism of a closed C ∞ manifold M, and let Λ ⊂ M be a closed f-invariant set. We show that if f |Λ is robustly chain transitive with orbital shadowing property, then Λ is a basic set. [ABSTRACT FROM PUBLISHER]

Numerical solutions to chaotic dynamical systems are suspect because the exponential divergence of nearby trajectories causes numerical errors to be exponentially magnified with time. To bolster confidence in the reliability of such numerical solutions, we turn to the study of shadowing: a shadow is an exact trajectory of a chaotic dynamical system that stays close to a numerical trajectory for a long time. Finding shadows of numerical trajectories is very computationally intensive, and until recently it has been infeasible to study shadows of higher-dimensional systems. This paper introduces several optimizations to previous algorithms, which collectively achieve an average speedup of almost two orders of magnitude with no measurable loss in effectiveness. We test the algorithm on systems with up to 180 dimensions. Its application to large gravitational N-body integrations has already led to a deeper understanding of the reliability of galaxy and cosmological simulations. The source code (available from the first author) provides to the experienced user a generic interface capable of searching for shadows of any set of ordinary differential equations. [ABSTRACT FROM AUTHOR]

An exact trajectory of a dynamical system lying close to a numerical trajectory is called a shadow. We present a general-purpose method for proving the existence of finite-time shadows of numerical ODE integrations of arbitrary dimension in which some measure of hyperbolicity is present and there are either 0 or 1 expanding modes, or 0 or 1 contracting modes. Much of the rigor is provided automatically by interval arithmetic and validated ODE integration software that is freely available. The method is a generalization of a previously published containment process that was applicable only to two-dimensional maps. We extend it to handle maps of arbitrary dimension with the above restrictions, and finally to ODEs. The method involves building n-cubes around each point of the discrete numerical trajectory through which the shadow is guaranteed to pass at appropriate times. The proof consists of two steps: first, the rigorous computational verification of a simple geometric property, which we call the inductive containment property, and second, a simple geometric argument showing that this property implies the existence of a shadow. The computational step is almost entirely automated and easily adaptable to any ODE problem. The method allows for the rescaling of time, which is a necessary ingredient for successfully shadowing ODEs. Finally, the method is local, in the sense that it builds the shadow inductively, requiring information only from the most recent integration step, rather than more global information typical of several other methods. The method produces shadows of comparable length and distance to all currently published results. Finally, we conjecture that the inductive containment property implies the existence of a shadow without restriction on the number of expanding and contracting modes, although proof currently eludes us. [ABSTRACT FROM AUTHOR]