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On a form of lyapunov exponents (I: Establishment of the form)

Another form of Lyapunov exponents associated with certain motion that given by (1.2) is proposed. It is expressed through the variation of the disturbance direction (not through that of the disturbance norm). | Vietnam Journal of Mechanics , VAST Vol. 26 , 2005 , No. 4 (208 - 214) ON A FORM OF LYAPUNOV EXPONENTS (I: ESTABLISHMENT OF THE FORM) NGUYEN VAN DIN H Institute of Mechanics ABSTRACT. Another form of Lyapunov exponents associated with certain motion that given by (1.2) is proposed. It is expressed through the variation of the disturbance direction (not through that of the disturbance norm). 1 Introduction It is known that Lyapunov exponents associated with certain motion X(t) (equilibrium state, periodic and quasi periodic or chaotic motions) are real numbers characterizing t he behaviour of nearby motions x(t) and provide informations on the stability character of the motion under consideration. In [1] , they are defined as asymptotic quantities . 1 >(yo) = hm - ln(lly(t)ll/llY(O)ll), t->oo (1.1) t where I I denotes a vector norm, y(t) = x(t) - X(t) is the disturbance i. e. the deviation from the reference motion X(t) to its nearby ones x(t), y 0 = y(O) is the initial disturbance, ln stands for the natural logarithm. In the present paper , an attention is foccussed on another form of Lyapunov exponents, that is f t >(uo) = lim ~ t->oo t (1.2) u'(r)A(r)u(r)dr, 0 where u(t) = y(t) /i ly (t) il is the unit vector directed to t he disturbance, uo = u(O) is t he initial unit vector, A(t) is the Jacobian matrix of t he equation of variation, prime denotes the transposes operator. In this part I, the form (1.2) is established; the part II is devoted to t he verification and illustration of the results obtained. 2 The equation of variation and the expression of the disturbance Consider a system governed by the differential equation x(t) = F(x, t) or Xj(t) = Fj(X1, X2 , . ' Xn , t) , 208 (j = 1, 2, . ' n) , (2 .1) where x (x1, x2, . , Xn) is t he column vector in an n-dimensional Euclidean space with norm l xll = + + · · · + overdot denotes differentiation with respect to t ime t, F(x , t) is a column vector function (with necessary