| Project/Area Number |
06804008
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
解析学
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| Research Institution | Shimane University (1996)
Shinshu University (1994-1995) |
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Principal Investigator |
SUGIE Jitsuro Shimane University Department of Mathematics and Computer Science Professor, 総合理工学部, 教授 (40196720)
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| Project Period (FY) |
1994 – 1996
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| Project Status |
Completed (Fiscal Year 1996)
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| Budget Amount *help |
¥2,000,000 (Direct Cost: ¥2,000,000)
Fiscal Year 1996: ¥400,000 (Direct Cost: ¥400,000)
Fiscal Year 1995: ¥400,000 (Direct Cost: ¥400,000)
Fiscal Year 1994: ¥1,200,000 (Direct Cost: ¥1,200,000)
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| Keywords | Ordinary Differential Equations / Functional Differential Equations / Qualitative Theory / Lienard Equation / Predator-Prey Systems / Periodic Solutions / Limit Cycles / Separatrix / リミット サイクル |
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| Research Abstract |
The purpose of this research is to examine 1. asymptotic behavior of trajectories of the Lienard system and 2. stability regions for systems of differential-difference equations.
1. It is very important for oscillation of solutions and other subjects to find conditions for deciding (1) whether trajectories of the Lienard system approach the origin or not and (2) whether trajectories intersect the vertical isocline. The head investigator and cooperators of this research discussed the above problems and obtained some implicit necessary and sufficient conditions, explicit necessary conditions and explicit sufficient conditions (Nonlin.Diff.Eq.Appl.). As the result, we clarified the properties of separatrices in the Lienard plane and gave existence results on periodic solutions of the lienard system with a homoclinic orbit which is a typical example of separatrix (Disc.Cont.Dynam.Syst.). We also gave a complete classification of global phase portraits of the Lienard system by the properties and the number of separatrices (J.Math.Anal.Appl.). Moreover, we discussed oscillation problems of a second order nonlinear differential equation of Euler type (Proc.Amer.Math.Soc.) and a periodically forced Lienard system (Proc.Inter.Con.Dynam.Syst.Chaos, Diff.Integ.Eq.). As an application to biological model, we considered a predator-prey system with a fairly general functional response of Holling type and presented a necessary and sufficient condition under which this system has exactly one stable limit cycle.
2. We gave some necessary and sufficient conditions for the zero solution of an n-dimensional differential-difference system to be exponential asymptotically stable (Funk.Ekvac.). The proof of our results is carried out by an application of Pontryagin criterion for quasi-polynomials to the characteristic equation of the differential-difference system. In case n=1, our results yeild a well-known necessary and sufficient condition for a scalar differential-difference equation.
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