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Register a new userPredicting catastrophes in nonlinear dynamical systems by compressive sensing
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An extremely challenging problem of significant interest is to predict catastrophes in advance of their occurrences. We present a general approach to predicting catastrophes in nonlinear dynamical systems under the assumption that the system equations are completely unknown and only time series reflecting the evolution of the dynamical variables of the system are available. Our idea is to expand the vector field or map of the underlying system into a suitable function series and then to use the compressive-sensing technique to accurately estimate the various terms in the expansion. Examples using paradigmatic chaotic systems are provided to demonstrate our idea.
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An efficient technique is introduced for model inference of complex nonlinear dynamical systems driven by noise. The technique does not require extensive global optimization, provides optimal compensation for noise-induced errors and is robust in a broad range %of parameters of dynamical models. It is applied to clinically measured blood pressure signal for the simultaneous inference of the strength, directionality, and the noise intensities in the nonlinear interaction between the cardiac and respiratory oscillations.
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An effective modeling method for nonlinear distributed parameter systems (DPSs) is critical for both physical system analysis and industrial engineering. In this Rapid Communication, we propose a novel DPS modeling approach, in which a high-order nonlinear Volterra series is used to separate the time/space variables. With almost no additional computational complexity, the modeling accuracy is improved more than 20 times in average comparing with the traditional method.
Scaling regions -- intervals on a graph where the dependent variable depends linearly on the independent variable -- abound in dynamical systems, notably in calculations of invariants like the correlation dimension or a Lyapunov exponent. In these applications, scaling regions are generally selected by hand, a process that is subjective and often challenging due to noise, algorithmic effects, and confirmation bias. In this paper, we propose an automated technique for extracting and characterizing such regions. Starting with a two-dimensional plot -- e.g., the values of the correlation integral, calculated using the Grassberger-Procaccia algorithm over a range of scales -- we create an ensemble of intervals by considering all possible combinations of endpoints, generating a distribution of slopes from least-squares fits weighted by the length of the fitting line and the inverse square of the fit error. The mode of this distribution gives an estimate of the slope of the scaling region (if it exists). The endpoints of the intervals that correspond to the mode provide an estimate for the extent of that region. When there is no scaling region, the distributions will be wide and the resulting error estimates for the slope will be large. We demonstrate this method for computations of dimension and Lyapunov exponent for several dynamical systems, and show that it can be useful in selecting values for the parameters in time-delay reconstructions.
Improving the understanding of diffusive processes in networks with complex topologies is one of the main challenges of todays complexity science. Each network possesses an intrinsic diffusive potential that depends on its structural connectivity. However, the diffusion of a process depends not only on this topological potential but also on the dynamical process itself. Quantifying this potential will allow the design of more efficient systems in which it is necessary either to weaken or to enhance diffusion. Here we introduce a measure, the {em diffusion capacity}, that quantifies, through the concept of dynamical paths, the potential of an element of the system, and also, of the system itself, to propagate information. Among other examples, we study a heat diffusion model and SIR model to demonstrate the value of the proposed measure. We found, in the last case, that diffusion capacity can be used as a predictor of the evolution of the spreading process. In general, we show that the diffusion capacity provides an efficient tool to evaluate the performance of systems, and also, to identify and quantify structural modifications that could improve diffusion mechanisms.
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