Channels and jokers in continuous systems
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L, NONLINEAR, AND SOFT MATTER PHYSICS
Channels and Jokers in Continuous Systems O. Ya. Butkovskiœ* and M. Yu. Logunov Vladimir State University, Vladimir, 600026 Russia *e-mail: [email protected] Received October 16, 2006
Abstract—The Rössler system is used as an example to demonstrate that the reconstruction of a model nonlinear dynamical system from an observed time series reveals phase-space regions called channels and jokers [3]. The proposed method for finding such regions is shown to be robust to noise and inaccuracy (redundancy) of the models used in the reconstruction procedure. The evolution of local Lyapunov exponents of attractors is examined for the model systems, and its relation to channels and jokers is exposed. It is shown that channels and jokers can be used in predictive modeling. The quality of such models is analyzed by invoking the concept of degree of predictability. PACS numbers: 05.45.-a, 05.10.-a, 05.45.Tp DOI: 10.1134/S1063776107060143
1. CHANNELS AND JOKERS Construction of adequate mathematical models from observed time series has been the subject of numerous studies, and interest in this problem does not abate. Frequently, the reconstructed model equations describing chaotic systems do not match the upper limit for predictable time scale [1, 2], 1 σx = ------ ln -----. 2λ σ 2ξ 2
τ pred
Here, λ is the largest Lyapunov exponent; σ ξ is the variance due to noise, inaccurate modeling, etc.; and 2 σ x is the observed process variance. 2
According to Malinetsky [3], the substantial difficulties arising in modeling nonlinear processes are explained, in particular, by phase-space inhomogeneity. To describe processes and dynamical systems of this kind, the concept of joker was introduced to distinguish phase-space regions where the dynamic behavior of a system is weakly predictable because it changes and becomes more complicated or even stochastic. In opposition to jokers, channels refer to regions of stable, highly predictable motion. A more rigorous definition of channel given in [3] is as follows. Consider the n-dimensional phase space of a chaotic system F. Suppose that the system’s behavior within a phase-space region G can be described by a few-mode model F1 with phase space of dimension n1 < n. If the trajectory of the system passes through the region G sufficiently many times during the observation period, then an n1-dimensional function F1 defined on this region can be constructed and used to make a local prediction within the region. Due to simpler (few-
mode) dynamics in the region G as compared to the rest of the phase space, the system’s behavior is more accurately predicted by using the reconstructed function F1 instead of F. A region G where dynamics can be satisfactorily predicted, at least, for some modes of a chaotic system is called a channel [3]. The existence of channels admits two explanations. First, channels can be regions of few-mode dynamics where good predictability is due to the simplicity of the model. Second, channels can be stability regions in the phase space of a
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