A study of the bi-shadowing property on topologically conjugate dynamical systems on group space

Introduction

Shadowing theory is a major theory in dynamical systems. It has an important role in studying asymptotic behavior and stability in systems, see also [1,2] . With it, computer calculations of the dynamic system can be confirmed and the existence of the true-orbit of the system near the pseudo-orbit, see [3,4] . The first to develop the concept of shadowing was Walters P, [5] , see more [6][7][8] and the concept of inverse shadowing and the concept of bi-shadowing were developed by other researchers, see [9][10][11] . In [12,13] the research are studied the topologically conjugate systems of the shadowing, inverse shadowing and bi-shadowing in the dynamical system. Later, these and other properties of dynamical systems in group space were studied and good results were obtained. See [14][15][16][17][18][19] . As in paper [19] , we will mention in section two of this paper the principles of the metric group space and some basic concepts of the shadowing theory such as the definitions of (𝛼, 𝛽)-bi-shadowing, bi-shadowing, the concept of topological conjugate and topologically conjugate actions. Later in this paper, we will prove some theorems that show the transfer of the bi-shadowing property in its two types and under some conditions from action in dynamical system to another action in different dynamical system between which there is a topological conjugate.

Fundamental Definitions

Let 𝔾 be a group, 𝜒 be a Hausdorff topological space, we called the map Φ: 𝔾 × 𝜒 → 𝜒 which satisfies the following conditions is an action:

For any 𝕘 1 , 𝕘 2 ∈ 𝔾, and any 𝑥 ∈ 𝜒, and 𝑒 is the identity of the group 𝔾.

In the definitions, theories and remarks that will be mentioned in this paper we will suppose that the action Φ: 𝔾 × 𝜒 → 𝜒 of a finitely generated group 𝔾 with respect to the finite symmetric generating set 𝕊, (𝕊 is any finite symmetric generating set of 𝔾) on a metric group space (𝜒, 𝕕 𝜒 ). And the image of 𝑥 by Φ is Φ(𝕤, 𝑥), and the inverse image is Φ -1 (𝕤, 𝑥) = Φ(𝕤 -1 , 𝑥) for 𝕤 ∈ 𝕊.

We need to introduce the true-orbit and pseudo-orbit definitions Definition 2.1. [18] We called the sequence {𝑥 𝕘: 𝕘 ∈ 𝔾} is true-orbit of the action Φ if Φ(𝕤, 𝑥 𝕘 ) = 𝑥 𝕘𝕤 for 𝕤 ∈ 𝕊, 𝕘 ∈ 𝔾.

For 𝜌 > 0, we called the sequence {𝑦 𝕘: 𝕘 ∈ 𝔾} is 𝜌-pseudo-orbit of an action Φ if 𝕕 𝜒 (Φ(𝕤, 𝑦 𝕘 ), 𝑦 𝕘𝕤 ) < 𝜌, for 𝕤 ∈ 𝕊, 𝕘 ∈ 𝔾.

Below we mention the bi-shadowing definition, therefore, we need to provide the definition of distance between two actions. For Φ and Ψ are actions, the distance between them is given by:

Definition 2.2. [17] Let (𝜒, 𝕕 𝜒 ) be metric group space, we called the action Φ: 𝔾 × 𝜒 → 𝜒 is bi-shadowing with positive parameters 𝛼 and 𝛽 if there exists 0 < 𝜌 ≤ 𝛽 such that for any 𝜌-pseudo-orbit {𝑦 𝕘: 𝕘 ∈ 𝔾} of Φ and any action 𝜓: 𝔾 × 𝜒 → 𝜒 satisfying

Then there exists a true-orbit {𝒙 𝕘: 𝕘 ∈ 𝔾} of 𝝍 such that 𝕕 𝜒 (𝑥 𝕘 , 𝑦 𝕘 ) ≤ 𝛼 (𝜌 + 𝕕 𝜒 0 (Φ, Ψ)) ≤ 𝛼𝛽, for all 𝕘 ∈ 𝔾.

We called the set of all actions {𝜓 1 , 𝜓 2 , … } which satisfies (1) is the comparison class 𝔸(𝜒), and if the action Φ is bi-shadowing with positive parameters 𝛼 and 𝛽 with respect to {𝜓 1 , 𝜓 2 , … } then for simply we called the action Φ is (𝛼, 𝛽, 𝔸(𝜒))-bi-shadowing.

Definition 2.3. [19] Let (𝜒, 𝕕 𝜒 ) be metric group space, we called the action Φ: 𝔾 × 𝜒 → 𝜒 is bi-shadowing if for all 𝛽 > 0 there exists 𝛼 > 0 and 0 < 𝜌 ≤ 𝛽 𝛼 ⁄ such that for any

Topologically Conjugate Systems

For the invariance of (𝛼, 𝛽)-bi-shadowing for topologically conjugate systems, we conclude the following: Theorem 3.1. Let (𝜒, 𝕕 𝜒 ) and (Υ, 𝕕 Υ ) be metric group spaces and let Φ: 𝔾 × 𝜒 → 𝜒 and Ψ: 𝔾 × Υ → Υ be actions that are topologically conjugate by 𝕋: 𝔾 × 𝜒 → 𝛶. Assume also that 𝔸(𝜒) and 𝔸(Υ) are (𝔸(𝜒), 𝔸(Υ))-topologically conjugate. If there exists 𝜆 ≥ 1 such that

Proof. a) Let {y 𝕘: 𝕘 ∈ 𝔾} be a ρ-pseudo-orbit of an action Ψ, with 0 ≤ ρ ≤ β, which implies that 𝕕 Υ (Ψ(𝕤, y 𝕘 ), y 𝕘𝕤 ) < ρ, and let

Note that condition (2) is equivalent to the following condition:

For 𝕤 ∈ 𝕊 and for all 𝑦 ′ , 𝑦 ′′ ∈ Υ.

𝕕 𝜒 (Φ (𝕤, (𝕋(𝕤 -1 , 𝑦 𝕘 ))) , 𝕋(𝕤 -1 , 𝑦 𝕘𝕤 )) = 𝕕 𝜒 (𝕋 (𝕤 -1 , Ψ(𝑦 𝕘 )) , 𝕋(𝕤 -1 , 𝑦 𝕘𝕤 )) ≤ 𝕕 Υ (Ψ(𝑦 𝕘 ), 𝑦 𝕘𝕤 ) < 𝜌.

Hence {𝑥 𝕘: 𝕘 ∈ 𝔾} = {𝕋(𝕤 -1 , 𝑦 𝕘 ): 𝕘 ∈ 𝔾} is a 𝜌-pseudo-orbit of Φ. Let 𝑥 ∈ 𝜒 and 𝜓 ∈ 𝔸(𝜒), then using (2) and the conjugacy 𝕋, we obtain

From ( 3) and ( 5) and for any 𝜓 ∈ 𝔸(𝜒), where 𝜓 = 𝕋

From ( 6) and ( 7) we get for any 𝜑 ∈ 𝔸(Υ), where 𝜑 = 𝕋 ∘ 𝜓 ∘ 𝕋 -

Note that condition (2) is equivalent to the following condition

𝕕 𝜒 (𝕋(𝕤 -1 , 𝑦 ′ ), 𝕋(𝕤 -1 , 𝑦 ′′ )) ≤ 𝕕 Υ (𝑦 ′ , 𝑦 ′′ ) ≤ 𝜆𝕕 𝜒 (𝕋(𝕤 -1 , 𝑦 ′ ), 𝕋(𝕤 -1 , 𝑦 ′′ )), for 𝕤 ∈ 𝕊 and for all 𝑦 ′ , 𝑦 ′′ ∈ Υ.

Now, 𝕕 𝜒 (Φ (𝕤, (𝕋(𝕤 -1 , 𝑦 𝕘 ))) , 𝕋(𝕤 -1 , 𝑦 𝕘𝕤 )) = 𝕕 𝜒 (𝕋 (𝕤 -1 , Ψ(𝑦 𝕘 )) , 𝕋(𝕤 -1 , 𝑦 𝕘𝕤 )) ≤ 𝕕 Υ (Ψ(𝑦 𝕘 ), 𝑦 𝕘𝕤 ) < 𝜌′.

Hence {𝑥 𝕘: 𝕘 ∈ 𝔾} = {𝕋(𝕤 -1 , 𝑦 𝕘 ): 𝕘 ∈ 𝔾} is a 𝜌-pseudo-orbit of Φ. Let 𝑥 ∈ 𝜒 and 𝜓 ∈ 𝔸(𝜒), Then using (2) and the conjugacy 𝕋, we obtain

From ( 10) and ( 12) and ( 8) and for any 𝜓 ∈ 𝔸(𝜒), where 𝜓 = 𝕋 -1 ∘ 𝜑 ∘ 𝕋, we have 𝜌′ + sup 𝑥∈𝜒 𝕕 𝜒 (Ψ(𝕤, 𝑥), 𝜓(𝕤, 𝑥)) ≤ 𝛽′ ≤ 𝛽.

Since Φ is (𝛼, 𝛽)-bi-shadowing that is (there exists 0 < 𝜌 = 𝜌′ ≤ 𝛽 such that for any This ends the proof that Ψ is (𝜆𝛼 ′ , 𝛽 ′ , 𝔸(Υ))-bi-shadowing. ∎

Proof. b) Let {x 𝕘: 𝕘 ∈ 𝔾} be a ρ-pseudo-orbit of an action Φ with 0 ≤ ρ ≤ β, which implies that 𝕕 χ (Φ(𝕤, x 𝕘 ), x 𝕘𝕤 ) < ρ, and let ψ ∈ 𝔸(χ) satisfy 𝕕 χ 0 (Φ, 𝜓) = sup 𝑥∈𝜒 𝕕 𝜒 (Φ(𝑥), 𝜓(𝑥)) ≤ 𝛽 -𝜌.

Now, 𝕕 Υ (Ψ (𝕤, 𝕋(𝕤, 𝑥 𝕘 )) , 𝕋(𝕤, 𝑥 𝕘𝕤 )) = 𝕕 Υ (h (𝕤, Ψ(𝕤, 𝑥 𝕘 )) , 𝕋(𝕤, 𝑥 𝕘𝕤 )) ≤ 𝜆𝕕 𝜒 (Φ(𝕤, 𝑥 𝕘 ), 𝑥 𝕘𝕤 ),

Hence 𝕕 Υ (Ψ (𝕤, 𝕋(𝕤, 𝑥 𝕘 )) , 𝕋(𝕤, 𝑥 𝕘𝕤 )) < 𝜆𝜌.

It follows that {𝑦 𝕘: 𝕘 ∈ 𝔾} = {𝕋(𝕤, 𝑥 𝕘 ): 𝕘 ∈ 𝔾} is a 𝜆𝜌-pseudo-orbit of Ψ. Now let 𝑦 ∈ Υ and 𝜑 ∈ 𝔸(Υ), then using (11) we have 𝕕 Υ (Ψ(𝕤, 𝑦), 𝜑(𝕤, 𝑦)) ≤ 𝜆𝕕 𝜒 (𝕋(𝕤 -1 , Ψ(𝕤, 𝑦)), 𝕋(𝕤 -1 , 𝜑(𝕤, 𝑦))) = 𝜆𝕕 𝜒 (Φ(𝕤, 𝕋(𝕤 -1 , 𝑦)), 𝜓(𝕤, 𝕋(𝕤 -1 , 𝑦))) = 𝜆𝕕 𝜒 (Φ(𝕤, 𝑥), 𝜓(𝕤, 𝑥)),

Where 𝑥 = 𝕋(𝕤 -1 , 𝑦). Therefore 𝕕 Υ (Ψ(𝕤, 𝑦), 𝜑(𝕤, 𝑦)) ≤ 𝜆𝕕 𝜒 (Φ(𝕤, 𝑥), 𝜓(𝕤, 𝑥)) for every 𝑦 ∈ Υ and 𝑥 = 𝕋(𝕤 -1 , 𝑦), hence sup 𝑦∈Υ 𝕕 Υ (Ψ(𝕤, 𝑦), 𝜑(𝕤, 𝑦)) ≤ 𝜆 sup 𝑥∈𝜒 𝕕 𝜒 (Φ(𝕤, 𝑥), 𝜓(𝕤, 𝑥)) . (

From ( 13) and ( 14) and ( 9) we get for any 𝜑 ∈ 𝔸(Υ), where 𝜑 = 𝕋 ∘ 𝜓 ∘ 𝕋 -1 , that 𝜆𝜌 + sup 𝑦∈Υ 𝕕 Υ (Ψ(𝕤, 𝑦), 𝜑(𝕤, 𝑦)) ≤ 𝜆𝛽 ≤ 𝜆𝛽′.

Since Ψ is (𝛼 ′ , 𝜆𝛽 ′ )-bi-shadowing that is (there exists 0 < 𝜌′ = 𝜆𝜌 ≤ 𝜆𝛽′ such that for any 𝜌′-pseudo-orbit {𝑦 𝕘: 𝕘 ∈ 𝔾} of Ψ and any action 𝜑: 𝔾 × Υ → Υ satisfying 𝕕 Υ 0 (Φ, 𝜑) ≤ 𝛽′ -𝜌′, then there exists a true-orbit {𝑧 𝕘: 𝕘 ∈ 𝔾} of 𝜑 such that 𝕕 Υ (𝑦 𝕘 , 𝑧 𝕘 ) ≤ 𝛼′ (𝜌′ + 𝕕 Υ 0 (Ψ, 𝜑))) . This shows that Φ is (𝜆𝛼, 𝛽, 𝔸(𝜒))-bi-shadowing. ∎ Remark 3.4. In Theorem 3.3, if 𝜆 = 1, then we conclude that the action Φ: 𝔾 × 𝜒 → 𝜒 is (𝛼, 𝛽, 𝔸(𝜒))-bi-shadowing if and only if Ψ: 𝔾 × Υ → Υ is (𝛼 ′ , 𝛽 ′ , 𝔸(Υ))-bi-shadowing.

As for the non-parametric bi-shadowing for topologically conjugate systems, we conclude the following: Theorem 3.5. Let (χ, 𝕕 χ ) and (Υ, 𝕕 Υ ) be metric group spaces and let Φ: 𝔾 × χ → χ and Ψ: 𝔾 × Υ → Υ be actions that are topologically conjugate by 𝕋: 𝔾 × χ → Υ. Assume also that 𝔸(χ) and 𝔸(Υ) are (𝔸(χ), 𝔸(Υ))-topologically conjugate. If there exists λ ≥ 1 such that equation ( 2) is valid then we have

If Φ is bi-shadowing with to the comparison class 𝔸(χ), then Ψ is 𝔸(Υ)-bi-shadowing.

Proof. Assume that for any β′ > 0 there exists α′ > 0 and 0 ≤ ρ′ ≤ β′ α′ ⁄ , let {y 𝕘: 𝕘 ∈ 𝔾} be a ρ′-pseudo-orbit of an action Ψ, which implies that 𝕕 Υ (Ψ(𝕤, y 𝕘 ), y 𝕘𝕤 ) < ρ′, and let φ ∈ 𝔸(Υ) satisfy𝕕 Υ 0 (Ψ, φ) = sup y∈Υ 𝕕 Υ (Ψ(y), φ(y)) ≤ β′ -ρ′.

ρ′ + sup y∈Υ 𝕕 Υ (Ψ(y), φ(y)) ≤ β′.

Note that condition (2) is equivalent to the following condition:

𝕕 χ (𝕋(𝕤 -1 , y ′ ), 𝕋(𝕤 -1 , y ′′ )) ≤ 𝕕 Υ (y ′ , y ′′ ) ≤ λ𝕕 χ (𝕋(𝕤 -1 , y ′ ), 𝕋(𝕤 -1 , y ′′ )), for 𝕤 ∈ 𝕊 and for all y ′ , y ′′ ∈ Υ.

Now,

𝕕 χ (Φ (𝕤, (𝕋(𝕤 -1 , y 𝕘 ))) , 𝕋(𝕤 -1 , y 𝕘𝕤 )) = 𝕕 χ (𝕋 (𝕤 -1 , Ψ(y 𝕘 )) , 𝕋(𝕤 -1 , y 𝕘𝕤 )) ≤ 𝕕 Υ (Ψ(y 𝕘 ), y 𝕘𝕤 ) < ρ′.

Hence {x 𝕘: 𝕘 ∈ 𝔾} = {𝕋(𝕤 -1 , y 𝕘 ): 𝕘 ∈ 𝔾} is a ρ′-pseudo-orbit of Φ. Let x ∈ χ and ψ ∈ 𝔸(χ), then using (2) and the conjugacy 𝕋, we obtain 𝕕 χ (Φ(𝕤, x), ψ(𝕤, x)) ≤ 𝕕 Υ (𝕋(𝕤, Φ(𝕤, x)), 𝕋(𝕤, ψ(𝕤, x))) = 𝕕 Υ (Ψ(𝕤, 𝕋(𝕤, x)), φ(𝕤, 𝕋(𝕤, x))) = 𝕕 Υ (Ψ(𝕤, y), φ(𝕤, y)),

Where y = 𝕋(𝕤, x). This implies that sup x∈χ 𝕕 χ (Φ(𝕤, x), ψ(𝕤, x)) ≤ sup y∈Υ 𝕕 Υ (Ψ(𝕤, y), φ(𝕤, y)).

From ( 15) and ( 16) and for any ψ ∈ 𝔸(χ), where ψ = 𝕋 -1 ∘ φ ∘ 𝕋, we have ρ′ + sup x∈χ 𝕕 χ (Ψ(𝕤, x), ψ(𝕤, x)) ≤ β′.

Since Φ is bi-shadowing that is (for any β > 0 there exists α = α′ λ ⁄ > 0 and 0 < ρ = ρ′ ≤ β α ⁄ such that for any ρ-pseudo-orbit {x 𝕘: 𝕘 ∈ 𝔾} of Φ and any action ψ: 𝔾 × χ → χ satisfying 𝕕 χ 0 (Φ, ψ) ≤ β -ρ, then there exists a true-orbit {w 𝕘: 𝕘 ∈ 𝔾} of ψ such that 𝕕 χ (x 𝕘 , w 𝕘 ) ≤ α (ρ + 𝕕 χ 0 (Φ, ψ))) . This ends the proof that Ψ is 𝔸(Υ)-bi-shadowing. ∎ Remark 3.6. In the preceding theorem, if 𝜆 = 1, then we conclude that the action Φ: 𝔾 × 𝜒 → 𝜒 is 𝔸(𝜒)-bi-shadowing if and only if Ψ: 𝔾 × Υ → Υ is 𝔸(Υ)-bi-shadowing.

Conclusion

Shadowing theory plays a crucial role in understanding the behavior and stability of dynamical systems. It allows for the validation of computer-generated pseudo-orbits by confirming the presence of true orbits nearby. Walters P. first introduced the concept of shadowing, and further developments such as inverse shadowing and bi-shadowing have expanded its scope. This paper investigates the principles of bi-shadowing, topological conjugacy, and the conditions under which these properties transfer between dynamical systems. The results obtained illustrate the importance of shadowing theory in both theoretical and practical aspects of dynamical systems, highlighting its continued relevance in mathematical research and applications.

References

1. Al-Badarneh AA. On the dynamics of Zadeh extensions and set-valued induced maps. Can J Pure Appl Sci. 2015;9(3):3673-3679.
2. Aleksandrov AY, Aleksandrova EB, Platonov Voloshin MV. On the global asymptotic stability of a class of nonlinear switched systems. Nonlinear Dyn Syst Theory. 2017;17(2):107-20.
3. Al-Nayef AA. Bi-shadowing of infinite trajectories for difference equations in Banach spaces. J Differ Equ Appl. 2001;7:577-585.
4. Martynyuk AA. Analysis of a set of trajectories of generalized standard systems: Averaging technique. Nonlinear Dyn Syst Theory. 2017;17(1):29-41.
5. Al-Sharaa IMT, Al-Joboury RS. Asymptotic fitting shadowing property. Albahir J. 2018;7(13-14).
6. Ajam MHO. Some algebraic results of shadowing property in dynamical systems. J Eng Appl Sci. 2018;13(8 SI):6395-6397.
7. Al-Shara'a IMT, Al Sultani SKK. Some general properties of the inverse shadowing property. J Univ Babylon Pure Appl Sci. 2018;26(10):176-80.
8. Corless RM, Pilyugin SY. Approximate and real trajectories for generic dynamical systems. J Math Anal Appl. 1995;189(2):409-23.
9. Diamond P, Kloeden PE, Kozyakin VS, Pokrovskii AV. Robustness of the observable behavior of semihyperbolic dynamical systems. Avtomat i Telemekh. 1995;11:148159.
10. Al-Khatatneh OA, Al-Badarneh AA. Asymptotic behavior in product and conjugate dynamical systems using bi-shadowing properties. Nonlinear Dyn Syst Theory. 2020;20(5):479-489.
11. Pilyugin SY, Tikhomirov SB. Shadowing in actions of some abelian groups. Fundam Math. 2003;179(1):83-96.
12. Osipov AV, Tikhomirov SB. Shadowing for actions of some finitely generated groups. Dyn Syst. 2014;29(3):337-351.
13. Pilyugin SY. Inverse shadowing in group actions. Dyn Syst. 2017;32(2):198-210.
14. Ajam MHO, Al-Shara'a IMT. Study of chaotic behaviour with G-bi-shadowing property. J Coll Basic Educ. 2021;2(SI):110-122.
15. Ajam MHO, Al-Sharaa IMT. Some of the sufficient conditions to get the G-bi-shadowing action. Int J Nonlinear Anal Appl. 2022;13(1):1105-1112.
16. Ajam MH, Salman HM, Kadhim HI. Studying asymptotic behavior of bi-shadowing properties in multiple product dynamical systems in group space. Int J Stat Appl Math. 2024;9(3):48-56.