TAILIEUCHUNG - Báo cáo toán học: " Irreducible Quadratic Perturbation of Spatial Oscillator"

báo cáo này, chúng tôi xem xét các nhiễu loạn bất khả quy bậc hai dao động tuyến tính thứ ba chiều. Sử dụng phương pháp Poincare, chúng tôi điều tra điều kiện đảm bảo sự tồn tại (thiếu) các giải pháp định kỳ. | Vietnam Journal of Mathematics 35 1 2007 61-72 Viet n a m J 0 u r n a I of MATHEMATICS VAST 2007 Irreducible Quadratic Perturbation of Spatial Oscillator O. RabieiMotlagh1 and Z. Afsharnejad2 Dept. of Math. University of Birjand Birjand Iran 2 Dept. of Math. Ferdowsi University of Mashhad Mashhad Iran Received April 05 2006 Revised September 26 2006 Abstract. In this paper we consider the irreducible quadratic perturbation for the third dimensional linear oscillator. Using the Poincare method we investigate conditions guaranteeing existence lack of periodic solutions. Also we study the role of the iterative derivatives of the displacement function on constitution of periodic solutions the type of the stability and global bifurcation of the system. 2000 Mathematics Subject Classification 65Lxx. Keywords Periodic Solution Poincare map Bifurcation 1. Introduction Third order differential equations are recently the subject of much research specially because of their role in modeling of natural phenomena spatial oscillatory systems are of great importance. These kinds of equations arise in biology 9 12 and physical behaviour of a fluid 2 3 10 17 . Although there are a few papers for the persistence of the periodic solutions 4- 8 19 but for almost all of them the existence of a family of periodic solutions for a primary system is assumed. Therefore the major problem is still finding periodic solutions for the primary system. Because of the topological characteristics of the three-dimensional space the investigation of periodic solutions for the nonlinear third-order differential equations is a difficult problem. The three dimensional linear oscillator appears in some phenomena such as turbulent fluid dynamics 1 16 . The concept of this oscillatory system is adopted from linear oscillation in a plane which is modeled 62 O. RabieiMotlagh and Z. Afsharnejad by the equation X a2x 0. This introduces an object moving on an ellipse in xy-plane. Differentiating the above .

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