Reconstructing the state space of continuous time chaotic systems using power spectra

Lipton, J. M.; Dabke, K. P.

doi:10.1016/0375-9601(95)00876-4

Cited by 13 publications

(3 citation statements)

References 31 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The power spectra of various types of time series possess different characteristics [24][25][26]. For the white noise, the power spectrum is a flat line.…”

Section: Power Spectrummentioning

confidence: 99%

The nonlinear nature of friction coefficient in lubricated sliding friction

Zhou

Zhu

Zuo

et al. 2015

Tribology International

View full text Add to dashboard Cite

“…The power spectra of various types of time series possess different characteristics [24][25][26]. For the white noise, the power spectrum is a flat line.…”

Section: Power Spectrummentioning

confidence: 99%

The nonlinear nature of friction coefficient in lubricated sliding friction

Zhou

Zhu

Zuo

et al. 2015

Tribology International

View full text Add to dashboard Cite

“…When the Fourier transform is computed for such a signal, the singularities must be avoided in the complex plane through an adequate integration path and in this way exponential terms appear on its associated Fourier transform (Sigeti, 1995). In the presence of noise, the exponential frequency falloff relationship will be noticeable up to a given frequency and afterwards it will decay as a power of f -n where f is the frequency and n a natural number (Lipton & Dabke, 1996). These phenomena are also observable in chaotic systems as well, independently of the appearance of attractors or not in their dynamic behaviour (van Wyk &.…”

Section: Performance Evaluationmentioning

confidence: 99%

Models of Continuous-Time Linear Time-Varying Systems with Fully Adaptable System Modes

Anda¹,

Sarmiento-Reyes²,

Kaszyński³

et al. 2008

New Approaches in Automation and Robotics

View full text Add to dashboard Cite

“…Fourier analysis has also been employed to approximate the transfer function to obtain an intermediate discrete-time reduced order model with stability guarantees for very large scale linear systems [67,30]. For general phase space reconstruction, asymptotic decay rates from Fourier analysis have been leveraged to infer appropriate sampling intervals and number of delays [40]. We thus leverage a Fourier basis representation to uncover the structure of time delay embeddings in linear models of non-linear dynamical systems.…”

mentioning

confidence: 99%

On the Structure of Time-delay Embedding in Linear Models of Non-linear Dynamical Systems

Pan,

Duraisamy

2019

Preprint

View full text Add to dashboard Cite

It is known that for a non-linear dynamical system, periodic and quasi-periodic attractors can be reconstructed in a discrete sense using time-delay embedding. Following this argument, it has been shown that even chaotic non-linear systems can be represented as a linear system with intermittent forcing. Although it is known that linear models such as those generated by the Hankel Dynamic Mode Decomposition can -in principle -reconstruct any ergodic dynamical system, quantitative details such as the required sampling rate and the number of delays remain unknown. This work addresses fundamental issues related to the structure and conditioning of linear time delayed models of non-linear dynamics on an attractor. First, we prove that, for scalar systems, the minimal number of time delays required for perfect signal recovery is solely determined by the sparsity in the Fourier spectrum. For the vector case, we devise a rank test and provide a geometric interpretation of the necessary and sufficient conditions for the existence of an accurate linear time delayed model. Further, we prove that the output controllability index of a certain associated linear system serves as a tight upper bound on the minimal number of time delays required. An explicit expression for the exact representation of the linear model in the spectral domain is also provided. From a computational implementation perspective, the effect of the sampling rate on the numerical conditioning of the time delayed model is examined. As a natural extension of Bazán's work, an upper bound on the 2-norm condition number is derived, with the implication that conditioning can be improved with additional time delays and/or decreasing sampling rates. Finally, it is explicitly shown that the underlying dynamics can be accurately recovered using only a partial period of trajectory data.

show abstract

Reconstructing the state space of continuous time chaotic systems using power spectra

Cited by 13 publications

References 31 publications

The nonlinear nature of friction coefficient in lubricated sliding friction

The nonlinear nature of friction coefficient in lubricated sliding friction

Models of Continuous-Time Linear Time-Varying Systems with Fully Adaptable System Modes

On the Structure of Time-delay Embedding in Linear Models of Non-linear Dynamical Systems

Contact Info

Product

Resources

About