Weiss proved that Devaney chaos does not imply topological chaos and Oprocha pointed out that Devaney chaos does not imply distributional chaos. In this paper, by constructing a simple example which is Devaney chaotic...Weiss proved that Devaney chaos does not imply topological chaos and Oprocha pointed out that Devaney chaos does not imply distributional chaos. In this paper, by constructing a simple example which is Devaney chaotic but neither distributively nor topologically chaotic, we give a unified proof for the results of Weiss and Oprocha.展开更多
The relation among transitivity, indecomposability and Z-transitivity is discussed. It is shown that for a non-wandering system （each point is non-wandering）, indecomposability is equivalent to transitivity, and for...The relation among transitivity, indecomposability and Z-transitivity is discussed. It is shown that for a non-wandering system （each point is non-wandering）, indecomposability is equivalent to transitivity, and for the dynamical systems without isolated points, Z-transitivity and transitivity are equivalent. Besides, a new transitive level as weak transitivity is introduced and some equivalent conditions of Devaney＇s chaos are given by weak transitivity. Moreover, it is proved that both d- shadowing property and d-shadowing property imply weak transitivity.展开更多
Recently, C. Tain and G. Chen introduced a new concept of sequence of time invariant function. In this paper we try to investigate the chaotic behavior of the uniform limit function f ： X →X of a sequence of continu...Recently, C. Tain and G. Chen introduced a new concept of sequence of time invariant function. In this paper we try to investigate the chaotic behavior of the uniform limit function f ： X →X of a sequence of continuous topologically transitive （in strongly successive way） functions fn ： X →X, where X is a compact interval. Surprisingly, we find that the uniform limit function is chaotic in the sense of Devaney. Lastly, we give an example to show that the denseness property of Devaney＇s definition is lost on the limit function.展开更多
We give a summary on the recent development of chaos theory in topological dynamics, focusing on Li-Yorke chaos, Devaney chaos, distributional chaos, positive topological entropy, weakly mixing sets and so on, and the...We give a summary on the recent development of chaos theory in topological dynamics, focusing on Li-Yorke chaos, Devaney chaos, distributional chaos, positive topological entropy, weakly mixing sets and so on, and their relationships.展开更多
In this note,it is proved that for the annihilation operator B of the unforced quantum harmonic oscillator,B^n is mixing and generically 5-chaotic with any 0 < δ < 2 for each positive integer n.Besides,by using...In this note,it is proved that for the annihilation operator B of the unforced quantum harmonic oscillator,B^n is mixing and generically 5-chaotic with any 0 < δ < 2 for each positive integer n.Besides,by using the result in[Wu X and Zhu P,J.Phys.A:Math.Theor.,2011,44:505101],the authors obtain that the principal measure of B^n is equal to 1 for each positive integer n.展开更多
In this paper, we explain why the chaotic mutation （CM） model of J. M. Bahi and C. Michel （2008） simulates the genes mutations over time with good accuracy. It is firstly shown that the CM model is a truly chaotic...In this paper, we explain why the chaotic mutation （CM） model of J. M. Bahi and C. Michel （2008） simulates the genes mutations over time with good accuracy. It is firstly shown that the CM model is a truly chaotic one, as it is defined by Devaney. Then, it is established that mutations occurring in genes mutations have indeed a same chaotic dynamic, thus making relevant the use of chaotic models for genomes evolution. Transposition and inversion dynamics are finally investigated.展开更多
In this paper, we show that a delayed discrete Hopfield neural network of two nonidentical neurons with no self-connections can demonstrate chaotic behavior in a region away from the origin. To this end, we first tran...In this paper, we show that a delayed discrete Hopfield neural network of two nonidentical neurons with no self-connections can demonstrate chaotic behavior in a region away from the origin. To this end, we first transform the model, by a novel way, into an equivalent system which enjoys some nice properties. Then, we identify a chaotic invariant set for this system and show that the system within this set is topologically conjugate to the full shift map on two symbols. This confirms chaos in the sense of Devaney. Our main result is complementary to the results in Kaslik and Balint (2008) and Huang and Zou (2005), where it was shown that chaos may occur in neighborhoods of the origin for the same system. We also present some numeric simulations to demonstrate our theoretical results.展开更多
基金2013 Jilin's universities science and technology project during the 12th five-year planthe financial special funds for projects of higher education of Jilin province
文摘Weiss proved that Devaney chaos does not imply topological chaos and Oprocha pointed out that Devaney chaos does not imply distributional chaos. In this paper, by constructing a simple example which is Devaney chaotic but neither distributively nor topologically chaotic, we give a unified proof for the results of Weiss and Oprocha.
基金Supported by National Natural Science Foundation of China(Grant No.11261039)National Natural Science Foundation of Jiangxi Province(Grant No.20132BAB201009)
文摘The relation among transitivity, indecomposability and Z-transitivity is discussed. It is shown that for a non-wandering system （each point is non-wandering）, indecomposability is equivalent to transitivity, and for the dynamical systems without isolated points, Z-transitivity and transitivity are equivalent. Besides, a new transitive level as weak transitivity is introduced and some equivalent conditions of Devaney＇s chaos are given by weak transitivity. Moreover, it is proved that both d- shadowing property and d-shadowing property imply weak transitivity.
基金CSIR ( project no. F.NO. 8/3(45)/2005-EMR-I)for providing financial support to carry out the research work
文摘Recently, C. Tain and G. Chen introduced a new concept of sequence of time invariant function. In this paper we try to investigate the chaotic behavior of the uniform limit function f ： X →X of a sequence of continuous topologically transitive （in strongly successive way） functions fn ： X →X, where X is a compact interval. Surprisingly, we find that the uniform limit function is chaotic in the sense of Devaney. Lastly, we give an example to show that the denseness property of Devaney＇s definition is lost on the limit function.
基金Supported by NNSF of China(Grant Nos.11371339,11431012,11401362,11471125)NSF of Guangdong province(Grant No.S2013040014084)
文摘We give a summary on the recent development of chaos theory in topological dynamics, focusing on Li-Yorke chaos, Devaney chaos, distributional chaos, positive topological entropy, weakly mixing sets and so on, and their relationships.
基金supported by YBXSZC20131046the Scientific Research Fund of Sichuan Provincial Education Department under Grant No.14ZB0007
文摘In this note,it is proved that for the annihilation operator B of the unforced quantum harmonic oscillator,B^n is mixing and generically 5-chaotic with any 0 < δ < 2 for each positive integer n.Besides,by using the result in[Wu X and Zhu P,J.Phys.A:Math.Theor.,2011,44:505101],the authors obtain that the principal measure of B^n is equal to 1 for each positive integer n.
文摘In this paper, we explain why the chaotic mutation （CM） model of J. M. Bahi and C. Michel （2008） simulates the genes mutations over time with good accuracy. It is firstly shown that the CM model is a truly chaotic one, as it is defined by Devaney. Then, it is established that mutations occurring in genes mutations have indeed a same chaotic dynamic, thus making relevant the use of chaotic models for genomes evolution. Transposition and inversion dynamics are finally investigated.
基金National Natural Science Foundation of China (Grant Nos. 11071263 and 11201504)the Natural Sciences and Engineering Research Council of Canada (Grant No. 227048-2010)
文摘In this paper, we show that a delayed discrete Hopfield neural network of two nonidentical neurons with no self-connections can demonstrate chaotic behavior in a region away from the origin. To this end, we first transform the model, by a novel way, into an equivalent system which enjoys some nice properties. Then, we identify a chaotic invariant set for this system and show that the system within this set is topologically conjugate to the full shift map on two symbols. This confirms chaos in the sense of Devaney. Our main result is complementary to the results in Kaslik and Balint (2008) and Huang and Zou (2005), where it was shown that chaos may occur in neighborhoods of the origin for the same system. We also present some numeric simulations to demonstrate our theoretical results.
