The topological invariance of Lyapunov exponents in embedded dynamics

There are two contributions of this paper. One is to provide a theoretical ground for the Jacobian methods by showing that n of the Lyapunov exponents of the estimated Jacobian are the Lyapunov exponents of an n dimensional unknown dynamical system. The second is to show that it is necessary that the Lyapunov exponents be calculated in the tangent space of the attractor in the embedding. The method of reconstruction is to form vectors of m consecutive observations, which for m > 2n is generically an embedding. This is Takens' result. A function of rn variables is then fit to the data and the Jacobian matrix is constructed at each point in the orbit of the data. When embedding occurs at dimension m = n, then the Lyapunov exponents of the reconstructed dynamics are the Lyapunov exponents of the original dynamics, This is the case for the Henon, the Lorenz and the Mackey-Glass systems with a first component observer. However, if embedding only occurs for an m > n, then the Jacobian method yields m Lyapunov exponents, only n of which are the Lyapunov exponents of the original system, The problem is that as currently used, the Jacobian method is applied to the full m-dimensional space of the reconstruction, and not just to the n-dimensional manifold that is the image of the embedding map. Our examples show that it is possible to get 'spurious' Lyapunov exponents that are even larger than the largest Lyapunov exponent of the original system.