Block diagram of the electronic oscillator with delayed feedback disturbed by a pulse signal. 

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... delay time ␶ 1 is always less than T a / 2 regardless of the nonlinear function of system ͑ 1 ͒ . For the higher-order modes, which can take place for very small ␧ 1 , we have ␶ 1 Ͻ nT a / 2, where n is the mode order. Furthermore, the range of values of T ensuring an accurate recovery of ␶ 1 can be different for different nonlinear functions. For a first approximation we suggest to take T close to 0.8 T a and then vary it, if necessary. As a criterion for a successful choice of T and other parameters of the pulse signal one can use the presence of a single pronounced minimum located just before the absolute maximum in the N ͑ ␶ ͒ plot ͓ see Fig. 2 ͑ b ͔͒ . We have also tested the method by varying the pulse duration from M = 0.05 T to M = 0.5 T . The method is still efficient but needs the amplitude A to increase for small M . With an increase of M , the value of A can be decreased. The considered impulsive disturbance can have an advantage over the system disturbance by a strong stochastic force used in Ref. ͓ 24 ͔ for the delay time estimation in periodic regime. The system is disturbed now not permanently but by a pulse signal. It is easy to control the parameters of the impulsive disturbance, choosing an appropriate variant from short but strong disturbance to long but low-amplitude one. To recover the parameter ␧ 1 and the nonlinear function f from the system ͑ 1 ͒ periodic time series one can use the method proposed in Ref. ͓ 11 ͔ and modified in Ref. ͓ 27 ͔ , where it was applied to chaotic time series of the time-delay system. Following this method, we have to project the system ͑ 1 ͒ trajectory on the plane ( x ͑ t − ␶ 1 ͒ , ␧ ̇ ͑ t ͒ + x ͑ t ͒ ) under variation of ␧ and calculate the length L ͑ ␧ ͒ of a line, connecting all points ordered with respect to the abscissa in the mentioned plane. When the parameter ␧ coincides with the true parameter ␧ 1 , the points of the projection lie on a single- valued curve, reproducing the nonlinear function f . The length L ͑ ␧ ͒ is minimal in this case. In the case of inaccurate parameter estimation a set of points in the plane, to which the trajectory of the system is projected, becomes more dis- persed. As a result, the polygon line connecting these points has a greater length than in the previous case. Figure 3 ͑ a ͒ shows the L ͑ ␧ ͒ plot constructed at the recovered delay time ␶ 1 = 300 and the step of ␧ variation equal to 0.01. The time derivative ̇ ͑ t ͒ is estimated from the time series by applying a local parabolic approximation. The minimum of L ͑ ␧ ͒ is observed at ␧ = ␧ 1 = 10.00. The nonlinear function, reconstructed for ␶ = 300 and ␧ = 10 from the sys- tem 1 periodic RCV time t = − series, V t + is f presented V t − 1 in + F Fig. 3 t , b . Such 3 a technique where V ͑ t ͒ allows and V us ͑ t − to ␶ 1 ͒ recover are the only delay a fragment line input of and the output func- voltages, tion f , since respectively. the oscillations In the take absence place of only an impulsive in a small distur- region of phase space because of their periodicity. For a more ex- tended recovery of the nonlinear function one should exploit the time series of the perturbed system ͑ 2 ͒ . In this case only the points of the time series corresponding to the intrinsic dynamics of the time-delay system should be used for plot- ting ␧ 1 ̇ ͑ t ͒ + x ͑ t ͒ versus x ͑ t − ␶ 1 ͒ . It means that we have to exploit the points from the intervals between the successive pulses of the external signal. The nonlinear function recovered in this manner is depicted in Fig. 3 ͑ c ͒ . It coincides well with the true quadratic function of the system ͑ 1 ͒ . To test the method efficiency in the presence of noise we apply it to the data produced by adding a zero-mean Gaussian white noise to the time series of Eq. ͑ 2 ͒ . For the cases where the additive noise has a standard deviation of up to 10% of the standard deviation of the data without noise the N ͑ ␶ ͒ plot still shows the minimum accurately at the delay time. As for the parameter ␧ 1 and the nonlinear function, they are recovered with a good accuracy using the time series of the perturbed system—i.e., exploiting only the points from the intervals between the successive external pulses. The proposed method can be applied for reconstruction of time-delay systems of high order and multiple delays in periodic regimes. In these cases for estimating the parameters of the disturbed system one can use the methods developed in Ref. ͓ 28 ͔ for reconstruction of high-order time-delay systems and systems with several delays from chaotic time series. However, the method is not valid for such time-delay systems as those considered in Ref. ͓ 29 ͔ , containing the nonlinear term, which is a function of the nondelayed signal. Let us consider application of the method to experimental time series gained from an electronic oscillator with delayed feedback perturbed by an external pulse signal. A block diagram of the experimental setup is shown in Fig. 4. The delay of the signal V ͑ t ͒ for time ␶ 1 is provided by a delay line constructed using digital elements. The role of the nonlinear device is played by an amplifier constructed using bipolar transistors and having a quadratic transfer function. The inertial properties of the oscillator are defined by a low- frequency first-order RC filter, which resistance R and ca- pacitance C specify ␧ 1 = RC . The analog and digital elements of the scheme are connected with the help of analog-to- digital and digital-to-analog converters. The signal from a pulse generator is applied to the oscillator using the summa- tor ⌺ . The considered oscillator is governed by the first-order delay-differential equation RCV t = − V t + f V t − 1 + F t , 3 where V ͑ t ͒ and V ͑ t − ␶ 1 ͒ are the delay line input and output voltages, respectively. In the absence of an impulsive distur- bance the oscillator shows at 1 = 4.16 ms and RC = 0.32 ms periodic self-sustained oscillations of period T a = 8.88 ms. Using an analog-to-digital converter with sampling frequency f s = 50 kHz we record the signal V ͑ t ͒ at the pulse signal parameters A = 1.6 V, T = 7.5 ms, and M = 1.5 ms. Part of the experimental time series is shown in Fig. 5 ͑ a ͒ . For various ␶ values we count the number N of situations when V ̇ ͑ t ͒ and V ̇ ͑ t − ␶ ͒ are simultaneously equal to zero and construct the N ͑ ␶ ͒ plot ͓ Fig. 5 ͑ b ͔͒ . The step of ␶ variation in Fig. 5 ͑ b ͒ is equal to the sampling time T s = 0.02 ms. The absolute minimum of N ͑ ␶ ͒ takes place exactly at the delay time ␶ = ␶ 1 = 4.16 ms. The L ͑ ␧ ͒ plot, constructed with the recovered ␶ 1 = 4.16 ms and the step of ␧ variation equal to 0.01 ms, exhibits the minimum at ␧ = ␧ 1 = 0.32 ms ͓ Fig. 5 ͑ c ͔͒ . In Fig. 5 ͑ d ͒ the nonlinear function recovered from the time series of the disturbed system is presented. This function coincides closely with the true transfer function f of the amplifier. The use of strong disturbance leading to the appearance of a transient process in a time-delay system performing periodic oscillations is not always possible. Because of the pe- culiarities of the system dynamics, the strong disturbance can result in undesirable qualitative change of the system behavior or even cause a destruction of the system. In these cases it is preferable to use small disturbances for estimating the system parameters. In this section we propose an original method for the recovery of time-delay systems in periodic regimes based on the analysis of the system response to perturbation by a small periodic signal. Let us consider a ring time-delayed feedback system composed of a delay line, nonlinear device, and filter ͓ Fig. 6 ͑ a ͔͒ , performing periodic self-sustained oscillations x ͑ t ͒ with pe- riod T a Fig. 6 b . We disturb the system by an external signal y ͑ t ͒ having the form of rectangular radio pulses with linearly increasing filling frequency ͓ Fig. 6 ͑ c ͔͒ . The filling period is always less than the period of oscillations, T a . The pulse period T greatly exceeds the preliminary estimation of the delay time, which is usually less than T a / 2. The form of the model equation for this system is determined by the filter parameters and the point of the external signal injection into the ring time-delay system. In the case where the filter is composed of three identical in-series low-frequency RC filters and the signal y ͑ t ͒ is added to the system between the filter and the delay line ͓ Fig. 6 ͑ a ͔͒ , the considered system is governed by the third-order delay-differential ...