Observability is a very useful concept for determining whether the dynamics of complicated systems can be correctly reconstructed from a single (univariate or multivariate) time series. When the governing equations of dynamical systems are high-dimensional and/or rational, analytical computations of observability coefficients produce large polynomial functions with a number of terms that become exponentially large with the dimension and the nature of the system. In order to overcome this difficulty, we introduce here a symbolic observability coefficient based on a symbolic computation of the determinant of the observability matrix. The computation of such coefficients is straightforward and can be easily analytically carried out, as demonstrated in this paper for a five-dimensional rational system.

The inference of an underlying network topology from local observations of a complex system composed of interacting units is usually attempted by using statistical similarity measures, such as cross-correlation (CC) and mutual information (MI). The possible existence of a direct link between different units is, however, hindered within the time-series measurements. Here we show that, for the class of systems studied, when an abrupt change in the ordered set of CC or MI values exists, it is possible to infer, without errors, the underlying network topology from the time-series measurements, even in the presence of observational noise, non-identical units, and coupling heterogeneity. We find that a necessary condition for the discontinuity to occur is that the dynamics of the coupled units is partially coherent, i.e., neither complete disorder nor globally synchronous patterns are present. We critically compare the inference methods based on CC and MI, in terms of how effective, robust, and reliable they are, and conclude that, in general, MI outperforms CC in robustness and reliability. Our findings could be relevant for the construction and interpretation of functional networks, such as those constructed from brain or climate data.Inferring the underlying topology of a complex system from observed data is currently the object of intense research. However, the limits for the exact inference of direct links in realworld systems composed by interacting dynamical units are still not fully understood. Understanding this limitations is often crucial in many applications in social and natural sciences. In order to infer the underlying network, usually, the observed data comes from timeseries recorded at the different units. Then, a direct link between units is assumed depending on how interdependent these observations are. For example, by recording the activity of different brain regions, one wishes to infer which are the relevant structural or functional brain connections by comparing similarity patterns [1][2][3]. In general, the outcome is a complex network [4,5] that interconnects the individual units and allows for a better understanding of the overall system behavior.The main statistical tools used to determine the interdependence of the units have been the cross-correlation (CC) and the mutual information (MI) between their dynamical trajectories [6][7][8][9][10][11][12][13][14][15][16]. Depending on the field of application, the choice of similarity estimators is wider and includes partial correlations, graphical models, and adapted estimators, such as event synchronization [17] (recently used to analyze the summer monsoon rainfall over the Indian peninsula [18]) or response dynamics [19,20]. However, any similarity measure used to compare two time-series usually results in a non-zero value [21][22][23][24][25]. A reason for this is that, in finite data sets, the presence of persistent trends and/or deterministic recurrent oscillations results in spurious correlations [26][27][28]. Therefore, network reconstruct...

When the state of the whole reaction network can be inferred by just measuring the dynamics of a limited set of nodes the system is said to be fully observable. However, as the number of all possible combinations of measured variables and time derivatives spanning the reconstructed state of the system exponentially increases with its dimension, the observability becomes a computationally prohibitive task. Our approach consists in computing the observability coefficients from a symbolic Jacobian matrix whose elements encode the linear, nonlinear polynomial or rational nature of the interaction among the variables. The novelty we introduce in this paper, required for treating large-dimensional systems, is to identify from the symbolic Jacobian matrix the minimal set of variables (together with their time derivatives) candidate to be measured for completing the state space reconstruction. Then symbolic observability coefficients are computed from the symbolic observability matrix. Our results are in agreement with the analytical computations, evidencing the correctness of our approach. Its application to efficiently exploring the dynamics of real world complex systems such as power grids, socioeconomic networks or biological networks is quite promising.

The existence of a special periodic window in the two-dimensional parameter space of an experimental Chua's circuit is reported. One of the main reasons that makes such a window special is that the observation of one implies that other similar periodic windows must exist for other parameter values. However, such a window has never been experimentally observed, since its size in parameter space decreases exponentially with the period of the periodic attractor. This property imposes clear limitations for its experimental detection.

Concepts from Ergodic Theory are used to describe the existence of special nontransitive maps in attractors of phase synchronous chaotic oscillators. In particular, it is shown that for a class of phase-coherent oscillators, these special maps imply phase synchronization. We illustrate these ideas in the sinusoidally forced Chua's circuit and two coupled Rössler oscillators. Furthermore, these results are extended to other coupled chaotic systems. In addition, a phase for a chaotic attractor is defined from the tangent vector of the flow. Finally, it is discussed how these maps can be used to a real-time detection of phase synchronization in experimental systems.