| Project/Area Number |
11304008
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| Research Category |
Grant-in-Aid for Scientific Research (A)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
Basic analysis
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| Research Institution | Shimane University |
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Principal Investigator |
SUGIE Jitsuro Shimane Univ., Dept. of Math. Professor, 総合理工学部, 教授 (40196720)
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| Co-Investigator(Kenkyū-buntansha) |
MIZUTA Yoshihiro Hiroshima Univ., Dept. of Math. Professor, 総合科学部, 教授 (00093815)
YAMASAKI Maretsugu Shimane Univ., Dept. of Math. Professor, 総合理工学部, 教授 (70032935)
AIKAWA Hiroaki Shimane Univ., Dept. of Math. Professor, 総合理工学部, 教授 (20137889)
HARA Tadayuki Osaka Pref. Univ., Dept. of Math. Professor, 工学部, 教授 (20029565)
MURATA Minoru Tokyo Inst. Tech., Dept. of Math. Professor, 理学部, 教授 (50087079)
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| Project Period (FY) |
1999 – 2002
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| Keywords | separatrix / limit cycle / homoclinic orbit / Lie^^'nard system / Euler equation / self-adjoint equation / elliptic equation / positive solution |
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| Research Abstract |
Van der Pol's equation was formulated to describe relaxation oscillations in electrical circuits, and played an important role in development of the theory of nonlinear oscillations and the theory of Hopf bifurcation. It is well-known that this equation has exactly one limit cycle with two unbounded separatrices. Although the separatrices are closely related to the limit cycle, little is known about the position of separatrices. In this research, we estimate the position by use of phase plane analysis and some Liapunov functions. Also, we consider the Lie^^'nard system which is a generalization of van del Pol's equation and give some conditions under which the Lie^^'nard system has homoclinic orbits.
We deal with the oscillation problem for various differential equations of Euler type and nonlinear self-adjoint differential equations, and present necessary and sufficient conditions for all nontrivial solutions to be oscillatory. The obtained theorems extend many previous results on this problem. We also discuss whether all solutions of nonlinear differential equations with time delay (or with decaying coefficients) oscillate or not. Changing variables, we can rewrite those equations into systems of Lie^^'nard type. For this reason, by means of phase plane analysis of the systems, we can examine the asymptotic behaviour of solutions in detail.
Combining the above results and the so-called "supersolution-subsolution method", we obtain sufficient conditions for quasilinear elliptic equations (or Schro^^<..>dinger equations) to have a positive solution which decays at infinity.
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