On the non-integrability of a generalized Darboux Halphen system

[摘要] In this paper we study a generalized Darboux Halphen system given by x(1) = x(2)x(3) - x(1)(x(2)x(3)) + tau(2)(alpha(1)alpha(2)alpha(3)x(1)x(2)x(3)), x(2) = x(3)x(1) - x(2)(x(3) + x(1)) + tau(2)(alpha(1)alpha(2)alpha(3)x(1)x(2)x(3)), x(3) = x(1)x(2) - x(3)(x(1) + x(2)) + tau(2)(alpha(1)alpha(2)alpha(3)x(1)x(2)x(3)), where x(1), x(2), x(3) are real variables, alpha(1), alpha(2), alpha(3) are real constants and tau(2)(alpha(1)alpha(2)alpha(3)x(1)x(2)x(3)) = alpha(2)(1)(x(1) - x(2))(x(3) - x(1)) + alpha(2)(2)(x(2) - x(3))(x(1) - x(2)) + alpha(2)(3)(x(3) - x(1))(x(2)-x(3)). We prove that, for any (alpha(1), alpha(2), alpha(3)) is an element of R-3 \ {(0, 0, 0)}, this system does not admit any non-constant global first integral that can be described by a formal power series. Furthermore, restricting the values of (alpha(1), alpha(2), alpha(3)) to a full Lebesgue measure set, we prove that this system does not admit any non-constant rational or Darbouxian global first integral. This is a first step toward proving that this system is chaotic. (c) 2006 Elsevier B.V. All rights reserved.

[发布日期] 2006-12-31

[关键词] generalized Halphen system;analytic and rational first integrals