• Previous Article
    Spreading speed in a non-monotonic Ricker competitive integrodifference system
  • DCDS-B Home
  • This Issue
  • Next Article
    Canards and homoclinic orbits in a slow-fast modified May-Holling-Tanner predator-prey model with weak multiple Allee effect
doi: 10.3934/dcdsb.2022053
Online First

Online First articles are published articles within a journal that have not yet been assigned to a formal issue. This means they do not yet have a volume number, issue number, or page numbers assigned to them, however, they can still be found and cited using their DOI (Digital Object Identifier). Online First publication benefits the research community by making new scientific discoveries known as quickly as possible.

Readers can access Online First articles via the “Online First” tab for the selected journal.

Inheritance of $ {\mathscr F}- $chaos and $ {\mathscr F}- $sensitivities under an iteration for non-autonomous discrete systems

Faculty of Mathematics, Kim Il Sung University, Pyongyang, Democratic People's Republic of Korea

*Corresponding author: JinHyon Kim

Received  July 2021 Revised  December 2021 Early access March 2022

This paper is concerned with chaos and sensitivity via Furstenberg families in a non-autonomous discrete system defined by a sequence of continuous self-maps on a compact metric space $ (X, \; d) $. First we consider the properties $ P(k) $ and $ Q(k) $ introduced in the literature. We show that if $ {\mathscr F} $ is a Furstenberg family with the property $ P(k) $ then its dual family $ k{\mathscr F} $ has the property $ Q(k) $ and that if $ {\mathscr F} $ is a filter with the property $ Q(k) $ then its dual family $ k{\mathscr F} $ has the property $ P(k) $. Next, for a given positive integer $ k $, it is shown that $ ({\mathscr F}_{1} , \; {\mathscr F}_{2} )- $chaos, generically $ {\mathscr F}- $chaos, dense $ {\mathscr F}- $chaos and $ {\mathscr F}- $sensitivities are inherited under the $ k $th iteration when $ \{ f_{n} \} _{n = 1}^{\infty } $ is equicontinuous on $ X $ and, $ {\mathscr F}_{1} , \; {\mathscr F}_{2} $ and $ {\mathscr F} $ are translation invariant Furstenberg families with the properties $ P(k) $ and $ Q(k) $. It is to weaken the condition in the literature that $ \{ f_{n} \} _{n = 1}^{\infty } $ uniformly converges on a compact metric space $ X $.

Citation: JinHyon Kim, HyonHui Ju, WiJong An. Inheritance of $ {\mathscr F}- $chaos and $ {\mathscr F}- $sensitivities under an iteration for non-autonomous discrete systems. Discrete and Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2022053
References:
[1] E. Akin, Recurrence in Topological Dynamics, Furstenberg Families and Ellis Actions, Plenum Press, New York, 1997. 
[2]

W. HuangD. KhilkoS. Kolyada and G. Zhang, Dynamical compactness and sensitivity, J. Differ. Equ., 260 (2016), 6800-6827.  doi: 10.1016/j.jde.2016.01.011.

[3]

J. Kim and H. Ju, About two definitions of ${\mathscr F}-$sensitivity, Topol. Appl., 269 (2020), 106927, 10pp. doi: 10.1016/j.topol.2019.106927.

[4]

S. Kolyada and L. Snoha, Topological entropy of nonautonomous dynamical systems, Random. Comput. Dyn., 4 (1996), 205-233. 

[5]

R. LiY. ZhaoH. WangR. Jiang and H. Liang, ${\mathscr F}-$sensitivity and $({\mathscr F}_{1}, \; {\mathscr F}_{2})-$sensitivity between dynamical systems and their induced hyperspace dynamical systems, J. Nonlinear Sci. Appl., 10 (2017), 1640-1651.  doi: 10.22436/jnsa.010.04.28.

[6]

Z. LiH. Wang and J. Xiong, Some remarks on $({\mathscr F}_{1}, \; {\mathscr F}_{2})-$scrambled sets, ACTA Math. Sin. (Chin. Ser.), 53 (2010), 727-732. 

[7]

T. LuP. Zhu and X. Wu, Distributional chaos in non-autonomous discrete systems, Acta Math. Sci. (Chin. Ser.), 35 (2015), 558-566. 

[8]

M. Salman and R. Das, Multi-sensitivity and other stronger forms of sensitivity in non-autonomous discrete systems, Chaos Solitons Fractals, 115 (2018), 341-348.  doi: 10.1016/j.chaos.2018.07.031.

[9]

H. Shao, G. Chen and Y. Shi, Topological conjugacy between induced non-autonomous set-valued systems and subshifts of finite type, Qual. Theory Dyn. Syst., 19 (2020), Paper No. 34, 26 pp.

[10]

H. Shao and Y. Shi, Some weak versions of distributional chaos in non-autonomous discrete systems, Commun. Nonlinear Sci. Numer. Simulat., 70 (2019), 318-325.  doi: 10.1016/j.cnsns.2018.11.005.

[11]

H. ShaoY. Shi and H. Zhu, On distributional chaos in non-autonomous discrete systems, Chaos Solitons Fractals, 107 (2018), 234-243.  doi: 10.1016/j.chaos.2018.01.005.

[12]

F. Tan and J. Xiong, Chaos via Furstenberg family couple, Topol. Appl., 156 (2009), 525-532.  doi: 10.1016/j.topol.2008.08.006.

[13]

F. Tan and R. Zhang, On ${\mathscr F}-$sensitive pairs, Acta Math. Sci., 31 (2011), 1425-1435.  doi: 10.1016/S0252-9602(11)60328-7.

[14]

X. Tang, G. Chen and T. Lu, Some iterative properties of $({\mathscr F}_{1}, \; {\mathscr F}_{2})-$chaos in non-autonomous discrete systems, Entropy, 20 (2018), Paper No. 188, 9 pp. doi: 10.3390/e20030188.

[15]

C. Tian and G. Chen, Chaos of a sequence of maps in a metric space, Chaos Solitons Fractals, 28 (2006), 1067-1075.  doi: 10.1016/j.chaos.2005.08.127.

[16]

R. Vasisht and R. Das, Exploring ${\mathscr F}-$Sensitivity for Non-Autonomous Systems, preprint, arXiv: 1806.00693v1.

[17]

X. WuJ. Wang and G. Chen, ${\mathscr F}-$sensitivity and multi-sensitivity of hyperspatial dynamical systems, J. Math. Anal. Appl., 429 (2015), 16-26.  doi: 10.1016/j.jmaa.2015.04.009.

[18]

X. Wu and P. Zhu, Chaos in a class of non-autonomous discrete systems, Appl. Math. Lett., 26 (2013), 431-436.  doi: 10.1016/j.aml.2012.11.003.

show all references

References:
[1] E. Akin, Recurrence in Topological Dynamics, Furstenberg Families and Ellis Actions, Plenum Press, New York, 1997. 
[2]

W. HuangD. KhilkoS. Kolyada and G. Zhang, Dynamical compactness and sensitivity, J. Differ. Equ., 260 (2016), 6800-6827.  doi: 10.1016/j.jde.2016.01.011.

[3]

J. Kim and H. Ju, About two definitions of ${\mathscr F}-$sensitivity, Topol. Appl., 269 (2020), 106927, 10pp. doi: 10.1016/j.topol.2019.106927.

[4]

S. Kolyada and L. Snoha, Topological entropy of nonautonomous dynamical systems, Random. Comput. Dyn., 4 (1996), 205-233. 

[5]

R. LiY. ZhaoH. WangR. Jiang and H. Liang, ${\mathscr F}-$sensitivity and $({\mathscr F}_{1}, \; {\mathscr F}_{2})-$sensitivity between dynamical systems and their induced hyperspace dynamical systems, J. Nonlinear Sci. Appl., 10 (2017), 1640-1651.  doi: 10.22436/jnsa.010.04.28.

[6]

Z. LiH. Wang and J. Xiong, Some remarks on $({\mathscr F}_{1}, \; {\mathscr F}_{2})-$scrambled sets, ACTA Math. Sin. (Chin. Ser.), 53 (2010), 727-732. 

[7]

T. LuP. Zhu and X. Wu, Distributional chaos in non-autonomous discrete systems, Acta Math. Sci. (Chin. Ser.), 35 (2015), 558-566. 

[8]

M. Salman and R. Das, Multi-sensitivity and other stronger forms of sensitivity in non-autonomous discrete systems, Chaos Solitons Fractals, 115 (2018), 341-348.  doi: 10.1016/j.chaos.2018.07.031.

[9]

H. Shao, G. Chen and Y. Shi, Topological conjugacy between induced non-autonomous set-valued systems and subshifts of finite type, Qual. Theory Dyn. Syst., 19 (2020), Paper No. 34, 26 pp.

[10]

H. Shao and Y. Shi, Some weak versions of distributional chaos in non-autonomous discrete systems, Commun. Nonlinear Sci. Numer. Simulat., 70 (2019), 318-325.  doi: 10.1016/j.cnsns.2018.11.005.

[11]

H. ShaoY. Shi and H. Zhu, On distributional chaos in non-autonomous discrete systems, Chaos Solitons Fractals, 107 (2018), 234-243.  doi: 10.1016/j.chaos.2018.01.005.

[12]

F. Tan and J. Xiong, Chaos via Furstenberg family couple, Topol. Appl., 156 (2009), 525-532.  doi: 10.1016/j.topol.2008.08.006.

[13]

F. Tan and R. Zhang, On ${\mathscr F}-$sensitive pairs, Acta Math. Sci., 31 (2011), 1425-1435.  doi: 10.1016/S0252-9602(11)60328-7.

[14]

X. Tang, G. Chen and T. Lu, Some iterative properties of $({\mathscr F}_{1}, \; {\mathscr F}_{2})-$chaos in non-autonomous discrete systems, Entropy, 20 (2018), Paper No. 188, 9 pp. doi: 10.3390/e20030188.

[15]

C. Tian and G. Chen, Chaos of a sequence of maps in a metric space, Chaos Solitons Fractals, 28 (2006), 1067-1075.  doi: 10.1016/j.chaos.2005.08.127.

[16]

R. Vasisht and R. Das, Exploring ${\mathscr F}-$Sensitivity for Non-Autonomous Systems, preprint, arXiv: 1806.00693v1.

[17]

X. WuJ. Wang and G. Chen, ${\mathscr F}-$sensitivity and multi-sensitivity of hyperspatial dynamical systems, J. Math. Anal. Appl., 429 (2015), 16-26.  doi: 10.1016/j.jmaa.2015.04.009.

[18]

X. Wu and P. Zhu, Chaos in a class of non-autonomous discrete systems, Appl. Math. Lett., 26 (2013), 431-436.  doi: 10.1016/j.aml.2012.11.003.

[1]

Aurelia Dymek. Proximality of multidimensional $ \mathscr{B} $-free systems. Discrete and Continuous Dynamical Systems, 2021, 41 (8) : 3709-3724. doi: 10.3934/dcds.2021013

[2]

Karim Samei, Arezoo Soufi. Quadratic residue codes over $\mathbb{F}_{p^r}+{u_1}\mathbb{F}_{p^r}+{u_2}\mathbb{F}_{p^r}+...+{u_t}\mathbb{F}_ {p^r}$. Advances in Mathematics of Communications, 2017, 11 (4) : 791-804. doi: 10.3934/amc.2017058

[3]

Renato Manfrin. On the boundedness of solutions of the equation $u''+(1+f(t))u=0$. Discrete and Continuous Dynamical Systems, 2009, 23 (3) : 991-1008. doi: 10.3934/dcds.2009.23.991

[4]

Sara D. Cardell, Joan-Josep Climent, Daniel Panario, Brett Stevens. A construction of $ \mathbb{F}_2 $-linear cyclic, MDS codes. Advances in Mathematics of Communications, 2020, 14 (3) : 437-453. doi: 10.3934/amc.2020047

[5]

Fanghui Ma, Jian Gao, Fang-Wei Fu. New non-binary quantum codes from constacyclic codes over $ \mathbb{F}_q[u,v]/\langle u^{2}-1, v^{2}-v, uv-vu\rangle $. Advances in Mathematics of Communications, 2019, 13 (3) : 421-434. doi: 10.3934/amc.2019027

[6]

Olof Heden, Faina I. Solov’eva. Partitions of $\mathbb F$n into non-parallel Hamming codes. Advances in Mathematics of Communications, 2009, 3 (4) : 385-397. doi: 10.3934/amc.2009.3.385

[7]

W. Cary Huffman. On the theory of $\mathbb{F}_q$-linear $\mathbb{F}_{q^t}$-codes. Advances in Mathematics of Communications, 2013, 7 (3) : 349-378. doi: 10.3934/amc.2013.7.349

[8]

Fernando Alcalde Cuesta, Ana Rechtman. Minimal Følner foliations are amenable. Discrete and Continuous Dynamical Systems, 2011, 31 (3) : 685-707. doi: 10.3934/dcds.2011.31.685

[9]

William E. Fitzgibbon. The work of Glenn F. Webb. Mathematical Biosciences & Engineering, 2015, 12 (4) : v-xvi. doi: 10.3934/mbe.2015.12.4v

[10]

Roghayeh Mohammadi Hesari, Mahboubeh Hosseinabadi, Rashid Rezaei, Karim Samei. $\mathbb{F}_{p^{m}}\mathbb{F}_{p^{m}}{[u^2]}$-additive skew cyclic codes of length $2p^s $. Advances in Mathematics of Communications, 2022  doi: 10.3934/amc.2022023

[11]

Keisuke Hakuta, Hisayoshi Sato, Tsuyoshi Takagi. On tameness of Matsumoto-Imai central maps in three variables over the finite field $\mathbb F_2$. Advances in Mathematics of Communications, 2016, 10 (2) : 221-228. doi: 10.3934/amc.2016002

[12]

Laurent Imbert, Michael J. Jacobson, Jr.. Empirical optimization of divisor arithmetic on hyperelliptic curves over $\mathbb{F}_{2^m}$. Advances in Mathematics of Communications, 2013, 7 (4) : 485-502. doi: 10.3934/amc.2013.7.485

[13]

Delphine Boucher. Construction and number of self-dual skew codes over $\mathbb{F}_{p^2}$. Advances in Mathematics of Communications, 2016, 10 (4) : 765-795. doi: 10.3934/amc.2016040

[14]

Orit Lavi, Doron Ginsberg, Yoram Louzoun. Regulation of modular Cyclin and CDK feedback loops by an E2F transcription oscillator in the mammalian cell cycle. Mathematical Biosciences & Engineering, 2011, 8 (2) : 445-461. doi: 10.3934/mbe.2011.8.445

[15]

Guangkui Xu, Longjiang Qu. Two classes of differentially 4-uniform permutations over $ \mathbb{F}_{2^{n}} $ with $ n $ even. Advances in Mathematics of Communications, 2020, 14 (1) : 97-110. doi: 10.3934/amc.2020008

[16]

Xiaofang Xu, Lisha Li, Bing Chen, Xiangyong Zeng. Maximal complete permutations over $ \mathbb{F}_2^n $. Advances in Mathematics of Communications, 2022  doi: 10.3934/amc.2022058

[17]

Byung-Soo Lee. A convergence theorem of common fixed points of a countably infinite family of asymptotically quasi-$f_i$-expansive mappings in convex metric spaces. Numerical Algebra, Control and Optimization, 2013, 3 (3) : 557-565. doi: 10.3934/naco.2013.3.557

[18]

Martin J. Blaser. Studying microbiology with Glenn F. Webb. Mathematical Biosciences & Engineering, 2015, 12 (4) : xvii-xxii. doi: 10.3934/mbe.2015.12.4xvii

[19]

David Henry, Hung-Chu Hsu. Instability of equatorial water waves in the $f-$plane. Discrete and Continuous Dynamical Systems, 2015, 35 (3) : 909-916. doi: 10.3934/dcds.2015.35.909

[20]

Xiuhong Chen, Zhihua Li. On optimality conditions and duality for non-differentiable interval-valued programming problems with the generalized (F, ρ)-convexity. Journal of Industrial and Management Optimization, 2018, 14 (3) : 895-912. doi: 10.3934/jimo.2017081

2021 Impact Factor: 1.497

Metrics

  • PDF downloads (185)
  • HTML views (84)
  • Cited by (0)

Other articles
by authors

[Back to Top]