Discovery of causal relations is fundamental for understanding the dynamics of complex systems. While causal interactions are well defined for acyclic systems that can be separated into causally effective subsystems, a mathematical definition of gradual causal interaction is still lacking for nonseparable dynamical systems. The solution proposed here is analytically tractable for time discrete chaotic maps and is shown to fulfill basic requirements for causality measures. It implies a method for determination of directed effective influences using pairs of measurements from dynamical systems. Applications to time series from systems of coupled differential equations and linear stochastic systems demonstrate its general utility.

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