THE MELNIKOV CRITERION OF AN INFINITE SEPARATRIX LOOP AND THE DISTRIBUTION OF LIMIT CYCLES IN A QUADRATIC SYSTEM

FENG BEI-YE

Acta Mathematicae Applicatae Sinica ›› 1993, Vol. 16 ›› Issue (4) : 482-492.

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Acta Mathematicae Applicatae Sinica ›› 1993, Vol. 16 ›› Issue (4) : 482-492. DOI: 10.12387/C1993058

THE MELNIKOV CRITERION OF AN INFINITE SEPARATRIX LOOP AND THE DISTRIBUTION OF LIMIT CYCLES IN A QUADRATIC SYSTEM

  • FENG BEI-YE
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Abstract

This paper gives an exact condition of applying Melnikovs criterion by defining the concept of suppressing condition. For the case where the separatrix tends to an infinite saddle point this paper gives the Melnikov criterion which is similar to the Melnikov criterion in the case of a finite saddle point, and points out the difference between the two cases. By applying the new criterion given in this paper the author discusses the bifurcation diagram of a quadratic system which has a 3-order fine focus. We prove the impossibility of generating a distribution of limit cycles of the type (0,4) by bifurcation from the fine focus and the infinite separatrix at the same parameter values. This paper determines the approximate bifurcation value by thoeretical analysis. The bifurcation value thus determined precisely coincides with the value as determined on a computer.

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FENG BEI-YE. THE MELNIKOV CRITERION OF AN INFINITE SEPARATRIX LOOP AND THE DISTRIBUTION OF LIMIT CYCLES IN A QUADRATIC SYSTEM. Acta Mathematicae Applicatae Sinica, 1993, 16(4): 482-492 https://doi.org/10.12387/C1993058

References

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