On the existence of periodic solutions for the quasi-linear third order system of O.D.Es

✍ Scribed by B. Mehri; M. Niksirat

• Publisher: John Wiley and Sons
• Year: 2002
• Weight: 82 KB
• Volume: 1
• Category: Article
• ISSN: 1617-7061
• DOI: 10.1002/1617-7061(200203)1:1<520::aid-pamm520>3.0.co;2-r

✦ Synopsis

On the existence of periodic solutions for the quasi-linear third order system of O.D.Es

In this paper we concern with the nonlinear third order quasi-linear system of ordinary differential equations as:

where X ∈ IR n and Λ is a diagonal matrix. We obtain some simple sufficient conditions for the existence of periodic solution using theorem of Brouer's degree. As we showed earlier [1], the scalar form the (1) can be treated by the Implicit Function Theorem instead of Brouer degree. Also because of the possibility of rewriting a 2n + 1 order equation into a third order system by a simple transformation [2], we can obtain useful results for such equations too. The main problem for this kind of equations is the validity of the results for the parameter free problem, i.e. when = 1. We consider it by study of dynamic of curves formed by the initial conditions that force the system be periodic when starts to increase.