科研之友
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摘要
We study in R-*(+2) the biquadratic system of two order one difference equations %26lt;br%26gt;u(n+1)u(n) = v(n)(2) - bv(n) + c, v(n+1)v(n) = u(n+1)(2) - au(n+1) + c, %26lt;br%26gt;for some values of the parameters. We show that there is an invariant function G, and so that the orbit of a point (u(0), v(0)) in some invariant open set U is on an invariant ellipse, and that the restriction on this ellipse of the associated dynamical system is conjugated to a rotation on a circle. The equilibrium is locally stable and the solutions (u(n), v(n)) are permanent. We show also that the starting points with periodic orbit are dense in U, and that every integer p %26gt;= N(a, b, c) is the minimal period of a periodic solution (u(n), v(n)). Moreover, the restriction of the dynamical system to the invariant compact %26quot;annulus%26quot; {K-1 %26lt;= G %26lt;= K-2} has global sensitivity to initial conditions, for inf(U) G %26lt; K-1 %26lt; K-2 %26lt; sup(U) G. Otherwise, outside U the solutions tend to infinity. At last we prove that the possible rational periodic solutions, when a, b, c are rational, may only be two or three-periodic, and we determine exactly the triples (a, b, c) for which such rational two or three-periodic solutions exist.
- 出版日期2012-11
