UNIVERSITY OF MONTENEGRO ELECTRICAL ENGINEERING DEPARTMENT Igor DJUROVIĆ Vesna RUBEŽIĆ Time-Frequency Representations in Detection of Chaotic State in Oscillatory Circuits January, 2007

Content IChaotic systems IIChaos detection IIISignals from non-linear oscillatory circuits in frequency domain IVProposed detector VResults of simulations VIDetector accuracy VIIApplication in chaotic communications Conclusion

I Chaotic systems Deterministic chaos can be observed in numerous natural and man-made systems. Chaos is non-linear phenomena that is observed in: technique, astronomy, biology, biophysics, chemistry, geology, mathematics, meteorology, medicine, and even in social sciences. Deterministic chaos is aperiodic motion (behavior) caused by deterministic dynamic laws or some deterministic procedure.

(a) Double-spiral Chua’s chaotic attractor; (b) Lorenz’s attractor; (c) Rossler’s attractor. Complicated chaotic motion can be visualized with so called strange attractors – chaotic attractors.

Chua circuit Chua circuit can be described by state-space equations: (a) Chua circuit; (b) Non-linear v-i function of the Chua’s diode. where and is a piece-wise linear characteristic of the Chua’s diode.

II Chaos detection The most important groups of chaos detection techniques are: ITechniques based on the Lyapunov exponents IICalculation of informational and topological measures for attractors reconstructed based on available data IIIDetection of non-linear deterministic signal in noisy environment IVTitration based techniques

Comparison of existing chaos detection techniques IIIIIIIV Calculation complexity **** Can be used for short sequences * Can be applied to signals embedded in noise * Sensitive to algorithm parameter selection **** Can be applied to hyper-chaotic systems *** Can be applied to non-autonomous systems ***

III Signals from nonlinear oscillators in frequency domain a) G=530μS (period-1 limit cycle): a fundamental frequency and higher harmonics; (b) G=537μS (period-2 limit cycle): periodic signal with a discrete power spectrum with several components; (c) G=550μS (Spiral-Chua chaotic attractor): chaotic waveform with broadband spectrum; (d) G=565μS (double-scroll Chua chaotic attractor): broadband signal with numerous components.

Motivation for research In periodic regime signals from chaotic oscillators can be represented as a sum of several sinusoidal components, i.e., as a sum of Dirac pulses in frequency domain. In chaotic regime it can be observed numerous additional components in spectral domain. Circuit state can be estimated using signal spectra and by comparing appropriately designed measure that produces different results for periodic and chaotic regimes. We assumed that some of parameters of the system could vary in time and that its values (in current and previous instants) determine state of the circuit. Spectral content of the signal from these circuits is time-varying: frequency of components is varying, some components and noise-like structures could appear or disappear, etc. Then the time-frequency representations as generalization of the standard FT concept are natural tool for analysis of such signals.

Time-frequency representations Short-time Fourier transform (STFT) is applied in development of our detector as the simplest and the most commonly used TFR: where x(t) is considered signal (in electric circuit voltage or current) and w(t) is window function.

(a) STFT; (b) Logarithm of the STFT magnitude for t=76.8ms – periodic window; (c) Logarithm of the STFT magnitude for t=57.8ms - chaos; (d) Logarithm of the STFT magnitude for t=39.8ms - period doubling; (e) Logarithm of the STFT magnitude for t=20.8ms - period 1 limit cycles. IV Proposed detector Chaos in the Chua’s circuit

We can conclude that estimation of the chaotic circuit state can be performed by designing of the specific concentration measure of the TF representations for region between the direct-current and the first harmonic component. Then we decided to measure region in which the TFR has high magnitude. Concentration measure is calculated as: where Ω(t) is the algorithm parameter, f m (t) is frequency of dominant spectral component, while u Ω(t) (t; f) is given as:

Algorithm can be summarized as follows. Step I - Calculation of the STFT for a given signal. Step II – Estimation of the frequency of dominant spectral component: Step III – Selection of the parameter Ω(t) is such manner that the STFT samples with larger magnitude than Ω(t) have almost entire signal energy, i.e., that signal region below Ω(t) is small: In our experiments ε= Algorithm I - Counting to the first spectral peak

Step IV – Evaluation of the detector response function. In order to avoid influence of noise and other errors, detector response m(t) has been averaged in narrow interval around the considered instant: Step V – Decision about regime in the circuit based on m ’ (t): chaotic regime periodic regime where F w (t) is width of the main lattice of the window function in the spectral domain.

Chaos in the Colpitts circuit (a) STFT; (b) Logarithm of the STFT magnitude for t=2ms – periodic regime (c) Logarithm of the STFT magnitude for t=3ms – period 2 cycles; (d) Logarithm of the STFT magnitude for t=5ms – chaos; (e) Logarithm of the STFT magnitude for t=10ms – periodic window.

Algorithm II – Counting samples in entire TF plane Step I – Calculation of the STFT. The STFT of the sinusoidal signal for applied Hanning window produces amplitude AN/2 on frequency of sinusoid (A is signal amplitude and N is number of samples within window) while two adjacent samples have amplitude of AN / 4. Step II – Create a function

Step III – Calculate whereis convolution in frequency domain. Step IV – Determination of threshold. Expected value of G(t,f) for frequency of sinusoid is 3AN/8. For region of the TF plane without sinusoidal component G(t,f) would be approximately 0. Then we selected mean of these two values as the threshold: it is 3AN/16, i.e., 3|STFT(t,f)|/8.

Step V – Calculation of detector response functions: where: In our experiments:

Step VII – Both functions m i (t) are averaged in order to reduce noise influence: In our experiments p=3ms. Step VII – Detection: chaotic regime periodic regime Threshold C can be selected in the wide range. We set C=2.

Period-doubling route to chaos in Chua circuit (a) TF representation; (b) solid line-detector response; dotted box-chaotic region according to theory; dashed line-threshold. V Simulation results

Quasyperiodic route to chaos in the Chua circuit (a) TF representation; (b) solid line-detector response; dotted box-chaotic region according to theory; dashed line-threshold.

Intermitent route to chaos in the Chua circuit (a) TF representation; (b) solid line-detector response; dashed line-threshold.

Period-doubling route to chaos for the Rossler system (a) TF representation; (b) solid line-detector response; dotted box-chaotic region according to theory; dashed line-threshold.

(a) TF representation; (b) solid line-detector response; dashed line-threshold. Route to chaos in the Lorenz system

Logystic map: Period-doubling route to chaos (a) TF representation; (b) solid line-detector response; dashed line-threshold. Parameter A is varying from 3.5 to 4.

Detection of higher order chaos Period-doubling route to chaos for system that contains three Chua circuits: (a) TF solid line-detector response; dashed line-threshold.

Chaos detection in non-autonomous nonlinear systems Duffing circuit route to chaos: (a) TF solid line-detector response; dashed line-threshold.

Chaos detection in Colpitts circuit Colpitts oscillator – route to chaos: (a) logarithm of |STFT(t,f) |; (b) m 1 (t); (c) m 2 (t); (d) detector response m(t) – solid line; threshold – dashed line.

Noise influence Detector response: (a) SNR=20dB; (b) SNR=10 dB. SNRAB 20dB0.4%4.70% 17dB0.95%6.31% 14dB2.04%9.51% 12dB3.37%17.74% 10dB5.98%38.09% 8dB23.68%61.51% Percentage of errors for detection of signal corrupted by the Gaussian noise. A-error in periodic regime; B- error within periodic windows. VI Detector accuracy

Window width Illustration of the STFT evaluation for instant t 0 within periodic window for two different window widths. Step I – Set of window widths where a>1, T 0 is the narrowest, a T Q is the widest window. Multiwindow detector

Step II – For each window calculate the STFT and corresponding detector. Step III – Calculate function: Step IV – Decision is made based on the following function: chaotic regime where periodic regime

Detector response for periodic region with N=311 samples: left – non-noisy signal; right –noisy signal (SNR=10 dB); (a), (e) N W =128; (b), (f) N W =180; (c), (g) N W =254; (d), (h) N W =357. (a)STFT with window width of 180 samples (zoomed region is given in (b))); STFT with window width of 357 samples (zoomed region is given in (c)).

SNRP=1P=2P=3 20dB 5.9%4.42%1.89% 18dB 4.7%4.1%1.26% 16dB 4.1%2.52%0% 14dB 3.15%0.63%0% 12dB 1.89%0% 10dB 0% 8dB 0% Percentage of errors for multiwindow algorithm (p=1, p=2, p=3) and algorithm with constant window widths within periodic regime. Percentage of errors detection for signals corrupted by Gaussian noise within chaotic regime.

VII Breaking “secure” chaotic communications (a) Binary sequence; (b) Chaotic modulated signal. ( a) TF representation of received signal; (b) solid line-detector response; dotted line- binary sequence. Percentage of errors.

Breaking “secure” chaotic communications for hyperchaotic systems (a)TF representation; (b) solid line-detector response; dotted line-binary sequence. Logarithm of |STFT| for instants t 1 =0.035s (dotted line) i t 2 =0.046s (solid line) from two different attractors. Detection threshold – dotted line.

Conclusion The proposed chaos detector:  simple for implementation  robust to moderate noise influence  robust to changes in the algorithm parameters requires relatively narrow interval (small number of samples) reasonable calculation complexity Developed for oscillatory circuits but it can be applied for some other chaotic systems.