## Abstract

We consider shadowing properties for vector fields corresponding to different type of reparametrizations. We give an example of a vector field which has the oriented shadowing properties, but does not have the standard shadowing property.

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## Acknowledgments

The work is partially supported by Chebyshev Laboratory (Department of Mathematics and Mechanics, St. Petersburg State University) under RF Government grant 11.G34.31.0026, JSC “Gazprom neft,” by the St. Petersburg State University in the framework of project 6.38.223.2014, Russian Foundation of Basic Research 15-01-03797a, and by the German-Russian Interdisciplinary Science Center (G-RISC) funded by the German Federal Foreign Office via the German Academic Exchange Service (DAAD).

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## Appendix

### Appendix

### Proof of Lemma 1

Note that

Item (i). Let us show that *Y* ∈ **C**
^{1}(*ℝ*
^{2}). Since *b*(*r*)∈**C**
^{1}(0, + *∞*), it is enough to prove continuity of *D*
*Y*(*x*) at *x* = 0. Assume that \(\sqrt {{x_{1}^{2}} + {x_{2}^{2}}} < 2l\). The following holds:

Since

and *r*
*b*
^{′}(*r*) → 0 as *r* → 0, the following holds

Arguing similarly for \(\frac {\partial Y_{2}}{\partial x_{1}}, \frac {\partial Y_{2}}{\partial x_{2}}\), we conclude that

Note that

which implies that

and completes the proof of item (i).

Item (ii). By the equality (10), it is enough to show that for *r* > 0, *T*
_{0} < 0 holds the inequality

Without loss of generality, we can assume that *r* < 2*l*. The following holds

Item (ii) is proved.

Item (iii). Note that

Fix *a* > 0. Choose small enough *l*, satisfying the following inequalities

Let *x*
_{0} = (*r*
_{0}, *φ*
_{0}), *x*
_{1} = (*r*
_{1}, *φ*
_{1}) and *h*(*t*)∈*R*
*e*
*p*(*l*) satisfy assumptions of the lemma. The following holds

Let us consider *T* > 0 and Δ∈*ℝ* such that

Consider points *x*
_{2} = *ψ*(*T*, *x*
_{0}) = (*r*
_{2}, *φ*
_{2}) *x*
_{3} = *ψ*(*h*(*T*),*x*
_{1}) = (*r*
_{3}, *φ*
_{3}). Note that *r*
_{2} = *K*
*l*. Inequality (2) implies

Using inequalities (11), we conclude that

Equality (10) implies that

Relations (2) and (15) implies

The following holds

Relations (2) and (15) imply that *e*
^{a(h(T)+Δ)}
*r*
_{0} = *e*
^{ah(T)}
*r*
_{1} > (*K*−1)*l* and hence

Relations (18) imply inequalities

and hence

Since *h*(*t*)∈Rep(*l*) and *T* = (ln(*K*
*l*)− ln*r*
_{0})/*a* using inequalities (11), (12), we conclude that

Inequalities (14) imply that for *τ* ∈ [0,Δ] holds the inequality *e*
^{aτ}
*r*
_{0} < 2*l*, hence *b*(*e*
^{aτ}
*r*
_{0}) = 1/ ln(*e*
^{aτ}
*r*
_{0}). Inequalities (12), (13), (21) imply that *a*|Δ|<−(ln*r*
_{0})/2, which implies |*b*(*e*
^{aτ}
*r*
_{0})| < 2*b*(*r*
_{0}) = −2/ ln*r*
_{0} and

Combining this with relations (19), (20), we conclude that

and hence, (17) implies |*φ*
_{0}−*φ*
_{1}|<*π*/4. Item (iii) is proved.

### Proof of Lemma 2

Let us fix a linear map *Q* and a number *D* > 0. Consider the lines *l*
_{1}⊂*S*
_{1}, *l*
_{2}⊂*S*
_{2} defined by *x*
_{2} = *x*
_{3} = 0, *y*
_{2} = *y*
_{3} = 0, respectively. Note that *Q*
*l*
_{2}≠*l*
_{1}. Let us consider plane *V*⊂*S*
_{1} containing *l*
_{1} and *Q*
*l*
_{2}. Consider a parralelogram *P*⊂*V*, symmetric with respect to 0 with sides parralel to *l*
_{1} and *Q*
*l*
_{2}, satisfying the relation

Let us choose *R* > 0, such that the following inclusions hold

Let *z*
_{1} be a point of intersection Sp_{1} and the line *V*∩{*x*
_{1} = 0}. Condition (23) implies that *z*
_{1} ∈ *P*. Consider the line *k*
_{1}, containing *z*
_{1} and parallel to *l*
_{1}. Inclusion (22) implies that *k*
_{1}∩*P*⊂Cyl_{1}.

Similarly, let *z*
_{2} be a point of intersection of Sp_{2} and *V*∩{*y*
_{1} = 0}. Condition (23) implies the inclusion *Q*
*z*
_{2} ∈ *P*. Let *k*
_{2} be the line containing *Q*
*z*
_{2} and parallel to *Q*
*l*
_{2}. Inclusion (22) implies that *Q*
^{−1}(*k*
_{2}∩*V*)⊂Cyl_{2}.

Since \(k_{1} \nparallel k_{2}\), there exists a point *z* ∈ *k*
_{1}∩*k*
_{2}. Inclusions *z*
_{1}, *z*
_{2} ∈ *P* imply that *z* ∈ *P*. Hence, *z* ∈ Cyl_{1}∩*Q*Cyl_{2}. Lemma 2 is proved.

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### Cite this article

Tikhomirov, S. An Example of a Vector Field with the Oriented Shadowing Property.
*J Dyn Control Syst* **21, **643–654 (2015). https://doi.org/10.1007/s10883-015-9272-9

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### Keywords

- Shadowing
- Vector field
- Reparametrization
- Structural stability

### Mathematics Subject Classification (2010)

- 37C50
- 37C10