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Tosiro KOGA, Masaharu SHINAGAWA, Satoshi HASAKO, "Synthesis of Linard's Equations Having More than One Periodic Solution" in IEICE TRANSACTIONS on Fundamentals,
vol. E76-A, no. 6, pp. 848-857, June 1993, doi: .
Abstract: As is well known, Linard's equation +µf (χ)+g(χ)=0 represents a wide class of oscillatory circuits as an extension of van der Pol's equation, and Linard's theorem guarantees the existence of a unique periodic solution which is orbitally stable. However, we sometimes meet such cases in engineering applications that the symmetry of the equation is violated, for instance, by a constant bias force. While, it has been known that asymmetric Linard's equation can have more than one periodic solution. The problem of finding the maximum number of such solutions, known as a special case of Hilbert's sixteenth problem, has recently been solved by T. Koga, one of the present authors. This paper first describes fundamental theorems due to T. Koga, and presents a solution to the synthesis problem of asymmetric Linard's systems, which generates an arbitrarily prescribed number of limit cycles, and which is considered to be important in relation to the stability of Linard's systems. Then, as application of this result, we give a method of determining parameters included in Linard's systems which may produce two limit cycles depending on the parameters. We also give a Linard's system which have three limit cycles. In addition, a new result on the parameter dependency of the number of limit cycles is presented.
URL: https://global.ieice.org/en_transactions/fundamentals/10.1587/e76-a_6_848/_p
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@ARTICLE{e76-a_6_848,
author={Tosiro KOGA, Masaharu SHINAGAWA, Satoshi HASAKO, },
journal={IEICE TRANSACTIONS on Fundamentals},
title={Synthesis of Linard's Equations Having More than One Periodic Solution},
year={1993},
volume={E76-A},
number={6},
pages={848-857},
abstract={As is well known, Linard's equation +µf (χ)+g(χ)=0 represents a wide class of oscillatory circuits as an extension of van der Pol's equation, and Linard's theorem guarantees the existence of a unique periodic solution which is orbitally stable. However, we sometimes meet such cases in engineering applications that the symmetry of the equation is violated, for instance, by a constant bias force. While, it has been known that asymmetric Linard's equation can have more than one periodic solution. The problem of finding the maximum number of such solutions, known as a special case of Hilbert's sixteenth problem, has recently been solved by T. Koga, one of the present authors. This paper first describes fundamental theorems due to T. Koga, and presents a solution to the synthesis problem of asymmetric Linard's systems, which generates an arbitrarily prescribed number of limit cycles, and which is considered to be important in relation to the stability of Linard's systems. Then, as application of this result, we give a method of determining parameters included in Linard's systems which may produce two limit cycles depending on the parameters. We also give a Linard's system which have three limit cycles. In addition, a new result on the parameter dependency of the number of limit cycles is presented.},
keywords={},
doi={},
ISSN={},
month={June},}
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TY - JOUR
TI - Synthesis of Linard's Equations Having More than One Periodic Solution
T2 - IEICE TRANSACTIONS on Fundamentals
SP - 848
EP - 857
AU - Tosiro KOGA
AU - Masaharu SHINAGAWA
AU - Satoshi HASAKO
PY - 1993
DO -
JO - IEICE TRANSACTIONS on Fundamentals
SN -
VL - E76-A
IS - 6
JA - IEICE TRANSACTIONS on Fundamentals
Y1 - June 1993
AB - As is well known, Linard's equation +µf (χ)+g(χ)=0 represents a wide class of oscillatory circuits as an extension of van der Pol's equation, and Linard's theorem guarantees the existence of a unique periodic solution which is orbitally stable. However, we sometimes meet such cases in engineering applications that the symmetry of the equation is violated, for instance, by a constant bias force. While, it has been known that asymmetric Linard's equation can have more than one periodic solution. The problem of finding the maximum number of such solutions, known as a special case of Hilbert's sixteenth problem, has recently been solved by T. Koga, one of the present authors. This paper first describes fundamental theorems due to T. Koga, and presents a solution to the synthesis problem of asymmetric Linard's systems, which generates an arbitrarily prescribed number of limit cycles, and which is considered to be important in relation to the stability of Linard's systems. Then, as application of this result, we give a method of determining parameters included in Linard's systems which may produce two limit cycles depending on the parameters. We also give a Linard's system which have three limit cycles. In addition, a new result on the parameter dependency of the number of limit cycles is presented.
ER -
nard's Equations Having More than One Periodic Solution
English
