On Some False Chaos Indicators When Analyzing Sampled Data

Invariance of Lyapunov exponents and Lyapunov dimension for regular and irregular linearizations

“…[60,52,1,18]) are also developed for the estimation of LEs from time series. However there are known examples in which the results of such computations differ substantially from the analytical results [59,2]. In [54] it is noted that these examples call into serious question whether the Lyapunov exponents for time series data give any sort of meaningful quantitative measurement of the original system.…”

“…Suppose that X(t, x 0 ) is a fundamental matrix of linear system (2) and all its LEs are finite. Consider a nondegenerate matrix Q (det Q = 0), and suppose X(t, x 0 ) = X(t, x 0 )Q.…”

“…But in the general case the set of LCEs of a normal fundamental matrix may not be equal to the set of LEs. Also Lyapunov showed that the so-called Lyapunov transformation x = L(t)y (non-degenerate linear transformations L(t) : R n → R n such that L(t), L −1 (t), L(t) are bounded and continuous) of coordinates of linear systems (2) preserves LCEs of this linear system. In particular, a corollary of his consideration is that the diffeomorphism y = D(z) of the phase space does not change LCEs of the bounded trajectories of nonlinear system (1) (see, e.g.…”

“…Suppose z(t, x 0 ) is a solution of system (1) with the initial data x 0 = z(0, x 0 ) uniformly bounded for t ∈ [0, +∞). Consider the linearization of system (1) along the solution z(t, x 0 ): ẋ = J(t, x 0 )x, t ∈ [0, +∞), (2) where J(t, x 0 ) = {∂F i (z)/∂z j }| z=z(t,x0) is (n×n) Jacobi matrix. Consider a fundamental matrix X(t, x 0 ) = x 1 (t, x 0 ), ..., x n (t, x 0 ) , which consists of the linearly independent solutions {x i (t, x 0 )} n 1 of linearized system (2).…”

“…Consider the linearization of system (1) along the solution z(t, x 0 ): ẋ = J(t, x 0 )x, t ∈ [0, +∞), (2) where J(t, x 0 ) = {∂F i (z)/∂z j }| z=z(t,x0) is (n×n) Jacobi matrix. Consider a fundamental matrix X(t, x 0 ) = x 1 (t, x 0 ), ..., x n (t, x 0 ) , which consists of the linearly independent solutions {x i (t, x 0 )} n 1 of linearized system (2). The fundamental matrix is often assumed to satisfy the following condition: X(0, x 0 ) = I n , where I n is a unit (n × n)-matrix.…”