In this paper, a non-existence condition for homoclinic and heteroclinic orbits in n-dimensional, continuous-time, and smooth systems is obtained, Based on this result and an elementary example, it can be conjectured ...In this paper, a non-existence condition for homoclinic and heteroclinic orbits in n-dimensional, continuous-time, and smooth systems is obtained, Based on this result and an elementary example, it can be conjectured that there is a fourth kind of chaos in polynomial ordinary differential equation （ODE） systems characterized by the nonexistence of homoclinic and heteroclinic orbits.展开更多
A paper, ＂Non-existence of Shilnikov chaos in continuous-time systems＂ was published in the journal Applied Mathematics and Mechanics （English Edition）. The authors gave sufficient conditions for the non-existence...A paper, ＂Non-existence of Shilnikov chaos in continuous-time systems＂ was published in the journal Applied Mathematics and Mechanics （English Edition）. The authors gave sufficient conditions for the non-existence of homoclinic and heteroclinic orbits in an nth-order autonomous system. Unfortunately, we show in this comment that the proof presented is erroneous and the result is invalid. We also provide two counterexamples of the wrong criterion stated by the authors.展开更多
Comments on 'Non-existence of Shilnikov chaos in continuous-time systems' are given.An error has been found in the proof of Theorem 1 in the paper by Elhadj and Sprott(Elhadj,Z.and Sprott,J.Non-existence of Sh...Comments on 'Non-existence of Shilnikov chaos in continuous-time systems' are given.An error has been found in the proof of Theorem 1 in the paper by Elhadj and Sprott(Elhadj,Z.and Sprott,J.Non-existence of Shilnikov chaos in continuous-time systems.Applied Mathematics and Mechanics(English Edition),33(3),1-4(2012)).It makes the main conclusion of the paper incorrect,that is to say,the non-existence of Shilnikov chaos in the continuous-time systems considered cannot be ensured.Furthermore,a counter-example shows that Theorem 1 in the paper is incorrect.展开更多
文摘In this paper, a non-existence condition for homoclinic and heteroclinic orbits in n-dimensional, continuous-time, and smooth systems is obtained, Based on this result and an elementary example, it can be conjectured that there is a fourth kind of chaos in polynomial ordinary differential equation （ODE） systems characterized by the nonexistence of homoclinic and heteroclinic orbits.
基金supported by the Ministerio de Educacion y Ciencia,Plan Nacional I+D+I co-financed with FEDER Funds(No.MTM2010-20907-C02)the Consejeria de Educacion y Ciencia de la Juntade Andalucia(Nos.FQM-276,TIC-0130,and P08-FQM-03770)
文摘A paper, ＂Non-existence of Shilnikov chaos in continuous-time systems＂ was published in the journal Applied Mathematics and Mechanics （English Edition）. The authors gave sufficient conditions for the non-existence of homoclinic and heteroclinic orbits in an nth-order autonomous system. Unfortunately, we show in this comment that the proof presented is erroneous and the result is invalid. We also provide two counterexamples of the wrong criterion stated by the authors.
基金Project supported by the National Natural Science Foundation of China(No.11102156)the Northwestern Polytechnical University Foundation for Fundamental Research
文摘Comments on 'Non-existence of Shilnikov chaos in continuous-time systems' are given.An error has been found in the proof of Theorem 1 in the paper by Elhadj and Sprott(Elhadj,Z.and Sprott,J.Non-existence of Shilnikov chaos in continuous-time systems.Applied Mathematics and Mechanics(English Edition),33(3),1-4(2012)).It makes the main conclusion of the paper incorrect,that is to say,the non-existence of Shilnikov chaos in the continuous-time systems considered cannot be ensured.Furthermore,a counter-example shows that Theorem 1 in the paper is incorrect.
