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͑ a ͒ Block diagram of the electronic oscillator with delayed feedback. ͑ b ͒ 2000 points of a realization of Eq. ͑ 8 ͒ with the nonlinear function ͑ 9 ͒ for ϭ 1.9, 0 ϭ 1000, RC ϭ 10. ͑ c ͒ Number
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We propose a method that allows one to estimate the parameters of model scalar time-delay differential equations from time series. The method is based on a statistical analysis of time intervals between extrema in the time series. We verify our method by using it for the reconstruction of time-delay differential equations from their chaotic solutio...
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... Fig. 6a the block diagram of the electronic oscillator with delayed feedback is sketched for the case when the filter is a low-frequency first-order RC filter. We used several ver- sions of this scheme with various combinations of analogue and digital elements that were connected with the help of analog-to-digital and digital-to-analog ...
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... V(t) and V(t 0 ) are the delay line input and output voltages, respectively; R and C are the resistance and capaci- tance, respectively. Eq. 8 is of form 1 with RC. Figure 6b shows part of a realization of Eq. 8 with the following nonlinear function ...
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... is a nonlinearity parameter. The time series is sampled in such a way that 1000 points in time series cover a period of time equal to the delay time 0 1000. The time series exhibits about 400 extrema. N() presented in Fig. 6c allows us to define 0 accurately. The true nonlinear function and the recovered function are compared in Fig. 6d. The estimated from the time series RC9.9. It is possible to estimate RC from the magnitude s m 0 , where m is the value, at which the absolute maximum of N() is observed. By varying the values of RC, , and 0 within a wide ...
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... is a nonlinearity parameter. The time series is sampled in such a way that 1000 points in time series cover a period of time equal to the delay time 0 1000. The time series exhibits about 400 extrema. N() presented in Fig. 6c allows us to define 0 accurately. The true nonlinear function and the recovered function are compared in Fig. 6d. The estimated from the time series RC9.9. It is possible to estimate RC from the magnitude s m 0 , where m is the value, at which the absolute maximum of N() is observed. By varying the values of RC, , and 0 within a wide range, we obtained the following empirical relationship: s RC/2/2. Thus, one can approximately estimate RC ...
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... linear function is not single valued and has to be averaged. Since the maximum of N() is clearly defined for two to three times higher noise levels than the N() minimum, the value of m can be used as an upper estimate of 0 from the data heavily corrupted by noise. In Fig. 7 we apply the method to two experimental time series produced by a setup Fig. 6a with radiophysical RC filter. In Figs. 7a and 7b the nonlinear device and the delay line are simulated on the computer and in Figs. 7c and 7d these elements are the radiophysical ones as well as the filter. The delay time is accurately identified in Fig. 7a from the time series sampled with a time step, which is ten times smaller than ...
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... verify the method efficiency for a system of form 12 we have applied it to a time series gained from an electronic oscillator with delayed feedback that is similar to that shown in Fig. 6a, but contains two identical in-series RC filters. The model equation for this oscillator with a two-section filter derived from Kirchhoff's laws has the form of second- order delay-differential ...
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... However, the delay time also leads to the so-called time delay signature (TDS) [6]. Once the TDS is known, the chaotic dynamics of OFSLs have the potential to be identified and modeled [9]. Consequently, the TDS is a crucial secure key from the perspective of communication security and thus the problem of TDS extraction is still of great interest [6], [10]. ...
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... An excellent review can be found in Sozou, Lane, Addis, and Gobet (2017). Much work used clever analysis of time series with assumed form of laws to recover particular systems (Bezruchko, Karavaev, Ponomarenko, & Prokhorov, 2001;Bü nner, Meyer, Kittel, & Parisi, 1997;Crutchfield & McNamara, 1987;Smith, 1992). ...
Discovering the governing equations for a measured system is the gold standard for modeling, predicting, and understanding complex dynamic systems. Very complex systems, such as human minds, pose stark challenges to this mode of explanation, especially in ecological tasks. Finding such “equations of mind” is sometimes difficult, if impossible. We introduce recent directions in data science to infer differential equations directly from data. To illustrate this approach, the simple but elegant example of sparse identification of nonlinear dynamics (SINDy; Brunton, Proctor, & Kutz, 2016) is used. We showcase this method on known systems: the logistic map, the Lorenz system, and a bistable attractor model of human choice behavior. We describe some of SINDy’s limitations, and offer future directions for this data science approach to cognitive dynamics, including how such methods may be used to explore social dynamics.
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Model validation from experimental data is an important and not trivial topic which is too often reduced to a simple visual inspection of the state portrait spanned by the variables of the system. Synchronization was suggested as a possible technique for model validation. By means of a topological analysis, we revisited this concept with the help of an abstract chemical reaction system and data from two electrodissolution experiments conducted by Jack Hudson's group. The fact that it was possible to synchronize topologically different global models led us to conclude that synchronization is not a recommendable technique for model validation. A short historical preamble evokes Jack Hudson's early career in interaction with Otto E. Rössler.
