Georg Börner, Hauke Haehne, Jose Casadiego, Marc Timme

Chaos 33, 073136 (2023)

The dynamics of a complex system is fundamentally governed by the number of its active dynamical variables, the system’s state space dimension. However, identifying state space dimension constitutes a difficult task, in particular if the dimension is much larger than the number of variables observed. Here, we show that it is mathematically possible in principle to infer the dimension of the state space using time series observations of just one variable, for arbitrarily high state space dimensions. We discuss how in practice the success of this inference depends on numerical constraints of data evaluation and experimental choices, such as the sampling intervals and total duration of observations. We illustrate how the approach may be applied to high-dimensional systems, e.g., with 100 variables, and provide general rules of thumb for performing and evaluating measurements of a given system. Our results provide a novel approach for inferring the dimension of complex and networked dynamical systems from scalar time series data and may help to develop alternative methods, e.g., for the reconstruction of the dimensions of system attractors.

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