Gyllenberg and Yan(Discrete Contin Dyn Syst Ser B 11(2):347–352,2009)presented a system in Zeeman’s class 30 of 3-dimensional Lotka-Volterra(3D LV)competitive systems to admit at least two limit cycles,one of which ...Gyllenberg and Yan(Discrete Contin Dyn Syst Ser B 11(2):347–352,2009)presented a system in Zeeman’s class 30 of 3-dimensional Lotka-Volterra(3D LV)competitive systems to admit at least two limit cycles,one of which is generated by the Hopf bifurcation and the other is obtained by the Poincaré-Bendixson theorem.Yu et al.(J Math Anal Appl 436:521–555,2016,Sect.3.4)recalculated the first Liapunov coefficient of Gyllenberg and Yan’s system to be positive,rather than negative as in Gyllenberg and Yan(2009),and pointed out that the Poincaré-Bendixson theorem is not applicable for that system.Jiang et al.(J Differ Equ 284:183–218,2021,p.213)proposed an open question:“whether Zeeman’s class 30 can be rigorously proved to admit at least two limit cycles by the Hopf theorem and the Poincaré-Bendixson theorem?”This paper provides four systems in Zeeman’s class 30 to admit at least two limit cycles by the Hopf theorem and the Poincaré-Bendixson theorem and gives an answer to the above question.展开更多
The Arun and Tista Rivers,which flow across the Himalayas,are commonly known as antecedent valleys that overcame the rapid uplift of the Higher Himalayan ranges.To clarify whether the idea of antecedent rivers is acce...The Arun and Tista Rivers,which flow across the Himalayas,are commonly known as antecedent valleys that overcame the rapid uplift of the Higher Himalayan ranges.To clarify whether the idea of antecedent rivers is acceptable,we investigated the geomorphology of the Himalayas between eastern Nepal and Bhutan Himalayas.The southern part of Tibetan Plateau,extending across the Himalayas as tectonically un-deformed glaciated terrain named as'Tibetan Corridor,'does not suggest the regional uplift of the Higher Himalayas.The 8,000-m class mountains of Everest,Makalu,and Kanchenjunga are isolated residual peaks on the glaciated terrain composed of mountain peaks of 4,000–6,000 m high.The Tibetan glaciers commonly beheaded by Himalayan glaciers along the great watershed of the Himalayas suggest the expansion of Himalayan river drainage by glaciation.For the narrow upstream regions of the Arun and Tista Rivers with less precipitation behind the range,it is hard to collect enough water for the power of down-cutting their channels against the uplifting Himalayas.The fission track ages of the Higher Himalayan Crystalline Nappe suggest that the Himalayas attained their present altitude by 11–10 Ma,and the Arun and Tista Rivers formed deep gorges across the Himalayas by headward erosion.展开更多
基金the National Natural Science Foundation of China(NSFC)under Grant No.12171321.
文摘Gyllenberg and Yan(Discrete Contin Dyn Syst Ser B 11(2):347–352,2009)presented a system in Zeeman’s class 30 of 3-dimensional Lotka-Volterra(3D LV)competitive systems to admit at least two limit cycles,one of which is generated by the Hopf bifurcation and the other is obtained by the Poincaré-Bendixson theorem.Yu et al.(J Math Anal Appl 436:521–555,2016,Sect.3.4)recalculated the first Liapunov coefficient of Gyllenberg and Yan’s system to be positive,rather than negative as in Gyllenberg and Yan(2009),and pointed out that the Poincaré-Bendixson theorem is not applicable for that system.Jiang et al.(J Differ Equ 284:183–218,2021,p.213)proposed an open question:“whether Zeeman’s class 30 can be rigorously proved to admit at least two limit cycles by the Hopf theorem and the Poincaré-Bendixson theorem?”This paper provides four systems in Zeeman’s class 30 to admit at least two limit cycles by the Hopf theorem and the Poincaré-Bendixson theorem and gives an answer to the above question.
基金This work was supported by Grants-in-Aid for Scientific Research of the Japanese Society for the Promotion of Science(JSPS KAKENHI)Grant Number 18H00766(principal investigator:Takashi Nakata)Grant Number 18KK0027(principal investigator:Yasuhiro Kumahara).
文摘The Arun and Tista Rivers,which flow across the Himalayas,are commonly known as antecedent valleys that overcame the rapid uplift of the Higher Himalayan ranges.To clarify whether the idea of antecedent rivers is acceptable,we investigated the geomorphology of the Himalayas between eastern Nepal and Bhutan Himalayas.The southern part of Tibetan Plateau,extending across the Himalayas as tectonically un-deformed glaciated terrain named as'Tibetan Corridor,'does not suggest the regional uplift of the Higher Himalayas.The 8,000-m class mountains of Everest,Makalu,and Kanchenjunga are isolated residual peaks on the glaciated terrain composed of mountain peaks of 4,000–6,000 m high.The Tibetan glaciers commonly beheaded by Himalayan glaciers along the great watershed of the Himalayas suggest the expansion of Himalayan river drainage by glaciation.For the narrow upstream regions of the Arun and Tista Rivers with less precipitation behind the range,it is hard to collect enough water for the power of down-cutting their channels against the uplifting Himalayas.The fission track ages of the Higher Himalayan Crystalline Nappe suggest that the Himalayas attained their present altitude by 11–10 Ma,and the Arun and Tista Rivers formed deep gorges across the Himalayas by headward erosion.
