Shadowing for differential equations with grow-up

We consider the problem of shadowing for differential equations with grow-up. We introduce so-called nonuniform shadowing properties (in which size of the error depends on the point of the phase space) and prove for them analogs of shadowing lemma. Besides, we prove a theorem about weighted shadowing for flows. We compactify the system (using Poincare compactification, for example), apply the results about nonuniform or weighted shadowing to the compactified system, and then transfer the results back to the initial system using the decompactification procedure.Comment: preprint, 19 pages, 3 figure

Shadowable Points for flows

A shadowable point for a flow is a point where the shadowing lemma holds for pseudo-orbits passing through it. We prove that this concept satisfies the following properties: the set of shadowable points is invariant and a $G_{\delta}$ set. A flow has the pseudo-orbit tracing property if and only if every point is shadowable. The chain recurrent and nonwandering sets coincide when every chain recurrent point is shadowable. The chain recurrent points which are shadowable are exactly those that can be are approximated by periodic points when the flow is expansive. We study the relations between shadowable points of a homeomorphism and the shadowable points of its suspension flow. We characterize the set of forward shadowable points for transitive flows and chain transitive flows. We prove that the geometric Lorenz attractor does not have shadowable points. We show that in the presence of shadowable points chain transitive flows are transitive and that transitivity is a necessary condition for chain recurrent flows with shadowable points whenever the phase space is connected. Finally, as an application these results we give concise proofs of some well known theorems establishing that flows with POTP admitting some kind of recurrence are minimal. These results extends those presented in [10].Comment: 18 page

Vector Fields with the Oriented Shadowing Property

We give a description of the \Cone-interior (\Int^1(\OrientSh)) of the set of smooth vector fields on a smooth closed manifold that have the oriented shadowing property. A special class \Bb of vector fields that are not structurally stable is introduced. It is shown that the set \Int^1(\OrientSh\setminus\Bb) coincides with the set of structurally stable vector fields. An example of a field of the class \Bb belonging to \Int^1(\OrientSh) is given. Bibliography: 18 titles.Comment: 42 page