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摘要
Let E, F subset of R be two given closed intervals, and let tau: E -> F and theta: F -> E be continuous maps. In this paper, we consider Koto's chaos, sensitivity and accessibility of a given system Psi(u,v) = (theta(v), r(u)) on a given product space E x F where u E E and v is an element of F. In particular, it is proved that for any Cournot map Psi(u, v) (theta(v), z(u)) on the product space E x F, the following hold:
(1) If 'P satisfies Kato's definition of chaos then at least one of Psi(2)vertical bar Q(1), and Psi(2)broken vertical bar Q(2), does, where Qi = {(theta(v), v) : v E F} and Q(2) = {(u, z (u)) u is an element of E}.
(2) Suppose that , Psi(2)broken vertical bar Q(1) and Psi(2)broken vertical bar Q(2), satisfy Kato's definition of chaos, and that the maps 9 and z satisfy that for any epsilon > 0, if
broken vertical bar(tau 0 theta)n (1,1) (theta 0 0)" (v(2)) broken vertical bar < epsilon and
broken vertical bar (theta 0 tau)(m)(u(1)) - (theta 0 tau)(m)(u(2)) vertical bar < epsilon
for some integers n, m > 0, then there is an integer l(n, m, s) > 0 with
broken vertical bar(tau o theta)(l(n,m,epsilon))(v(1)) (tau o theta)(l(n,m,epsilon))(v(2)) < epsilon
and
broken vertical bar(theta o tau)(l(n,m,epsilon)) - (theta o tau)(l(n,m,epsilon)) (u(2)) l < epsilon. Then 'P satisfies Kato's definition of chaos.
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单位广东海洋大学
