Chaotic signals

In this section we consider a model of interactions between the Kerr oscillators applied by J. Fiurasek et al. [139] and Perinova and Karska [140]. Each Kerr oscillator is externally pumped and damped. If the Kerr nonlinearity is turned off, the system is linear. This enables us to perform a simple comparison of the linear and nonlinear dynamics of the system, and we have found a specific nonlinear version of linear filtering. We study numerically the possibility of synchronization of chaotic signals generated by the Kerr oscillators by employing different feedback methods. 

Flicker-noise spectroscopy — The spectral density of - flicker noise (also known as 1// noise, excess noise, semiconductor noise, low-frequency noise, contact noise, and pink noise) increases with frequency. Flicker noise spectroscopy (FNS) is a relatively new method based on the representation of a nonstationary chaotic signal as a sequence of irregularities (such as spikes, jumps, and discontinuities of derivatives of various orders) that conveys information about the time dynamics of the signal [i—iii]. This is accomplished by analysis of the power spectra and the moments of different orders of the signal. The FNS approach is based on the ideas of deterministic chaos and maybe used to identify any chaotic nonstationary signal. Thus, FNS has application to electrochemical systems (-> noise analysis). 

In order to study the oscillation frequency of the chaotic model it. is important to develop a means for decomposing a chaotic signal into its phase and amplitude components. This is non-trivial for chaotic systems where there is often no unambiguous definition of phase. In our case, the motion always shows phase coherent dynamics, so that a phase can be defined as an angle in x,y)-phase plane or via the Hilbert-transform [31]. Here, we use an alternative method which is based on counting successive maxima, that allows analysis even if the signal is spiky . In this scheme we estimate the instantaneous phase

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An exciting development has for some years been seen in the use of chaos theory in signal processing. A chaotic signal is not periodic it has random time evolution and a broadband spectrum and is produced by a deterministic nonlinear dynamical system with an irregular... 

In the next amplitude (maximum) plots, the amplitude (maximum) of the (n-l- l)th peak is plotted against the amplimde (maximum) b of the nth peak. For regular periodic oscillations, the plot will reveal a finite number of discrete points, whereas for chaotic signals such a plot will show a sharp peak. 

The following subsection details some methods to find out if a given random time series fixtm a biological source is chaotic. Further, methods to determine the dimensionality and draw the phase plot or portrait of the chaotic signal are outlined briefly. 

Frequency Analysis. The statistical analysis of chaotic signals includes spectral analysis to confirm the absence of any spectral lines, since chaotic signals do not have any periodic deterministic component. Absence of spectral lines would indicate that the signal is either chaotic or stochastic. However, chaos is a complicated nonperiodic motion distinct from stochastic processes in that the amplitude of the high-frequency spectrum shows an exponential decline. The frequency spectrum can be evaluated using FFT-ba methods outlined the earlier sections. 

Estimating Correlation Dimension of a Chaotic SignaL One of the ways to measure the dimensionality of the phase portrait of a chaotic signal is through what is known as the correlation dimension T>2- It is defined as... 

By using this measure, regions with low consistency in the estimate typically correspond to regions with chaotic signal patterns. Hence, this measure is a suitable attribute for chaotic texture. This attribute is shown in Figures 4 (d) and 5(b). The chaos texture attribute is inherently dip- and azimuth-invariant, in addition to being amplitude-invariant. This invariance is crucial and allows us to select whether to explicitly accommodate for these properties in the analysis. Dip, azimuth and amplitude may require special treatment. 

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