By V. A. Tkachenko
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Additional info for A problem in the spectral theory of an ordinary differential operator in a complex domain
1 A General Study In this section we assume for simplicity that the considered system is a continuous dynamical system, so that one can write: dX (t) = F(X ,t), dt (2) where t is the time, X is the variable of space (it is often a vector), F is a C1 -function (in the general case) in X. Further, in this study, the system is assumed to be autonomous. Therefore the evolution in time of an infinitesimal difference in initial conditions δ X0 can be evaluated through the formula: dδ X (t) = J(X0 )δ X (t), (3) dt where δ X (t) is the divergence at time t, J(X0 ) is the Jacobian matrix of the transformation F evaluated at initial condition X0 : [J(X0 )]i j = ∂ Fi (X = X0 ).
On the Numerical Value of Finite-Time Pseudo-Lyapunov Exponents 23 References 1. A. M. Lyapunov. The general problem of the stability of motion. International Journal of Control, 55(3):531–773, 1992. 2. V. I. Oseledec. A multiplicative ergodic theorem: Lyapunov characteristic numbers for dynamical systems. Transactions of the Moscow Mathematical Society, 19:197–231, 1968. 3. A. Wolf, J. B. Swift, H. L. Swinney, and J. A. Vastano. Determining Lyapunov exponents from a time series. Physica D: Nonlinear Phenomena, 16(3):285–317, 1985.
Therefore we will use the following multi-scale expansion: √ Ψ j = ε (ψ j + εψ j, 1 + ε 2 ψ j, 2 + . I. V. Smirnov ε 3/2 : 1 γ i∂τ0 ψ j,1 + i∂τ1 ψ j + ψ j + ∂ξ2 (ψ j − ψ¯ j ) − (ψ3− j − ψ¯ 3− j ) 2 2 3 − 4β (ψ j − ψ¯ j ) = 0, ψ j,1 = χ j,1 eiτ0 , (21) γ 1 i∂τ0 χ j,1 + i∂τ1 χ j + ∂ξ2 (χ j − χ¯ j e−2iτ0 ) − (χ3− j − χ¯ 3− j e−2iτ0 ) 2 2 − 4β (χ j eiτ0 − χ¯ j eiτ0 )3 e−iτ0 = 0. Integrating last equations (21) with respect to “fast” time τ0 , we get two coupled equations: γ 1 i∂τ1 χ j + ∂ξ 2 χ j − χ3− j + 12β |χ j |2 χ j = 0.