**6. Fermi-Pasta-Ulam recurrence**

#### **6.1. Historical reviews**

In 1952, Fermi et al. [17] led a digital experience on a nonlinear constituted 64 particle system point of identical mass related to their neighbors by springs weakly nonlinear. They expected that the introduced nonlinear coupling between neighboring oscillators would allow a transfer of energy between successive vibration modes, thus causing an equipartition of energy over a wide spectrum. Against all odds, the system introduced a quasi-periodic behavior of the most complex. They found that all the energy which was initially excited almost returned to fundamental mode. This phenomenon was later called recurrence Fermi-Pasta-Ulam (FPU). This experience was important for two reasons—first, it highlighted the complexity of nonlinear systems; second, it demonstrated the power of the complex systems.

## **6.2. Experimental observation of the Fermi-Pasta-Ulam recurrence in a transmission electrical nonlinear bi-inductance line**

The electrical nonlinear bimodal transmission line presents a level of complexity compared to its counterpart mono inductance as far as it has two modes of propagations, the HF and BF mode. We propose to conduct a study of the recurrence of FPU in each mode.

#### *6.2.1. The FPU recurrence in the BF mode*

**Figure 13.** Observation of modulation instability in HF mode in the transmission of nonlinear bi-inductance line for a

amplitude VM = 3.6 V. We observe in **Figure 11**, the signals are collected at the level of the

Zooming on the recording of the signal shows that plane wave, which is injected at the

Modulation instability was also observed in HF mode. In this case, we have chosen a signal whose frequency is in the region of modulation instability such as provided for by the calculations in HF mode and which has a magnitude large enough to initiate the disruption that will trigger the MI of the wave. To do this, we have chosen a sinusoidal signal whose frequency is *f* = 897 KHz and amplitude VM = 3.6 V. We observe in **Figure 13**, the signals collected at the level of the 130th cell.

In 1952, Fermi et al. [17] led a digital experience on a nonlinear constituted 64 particle system point of identical mass related to their neighbors by springs weakly nonlinear. They expected that the introduced nonlinear coupling between neighboring oscillators would allow a transfer of energy between successive vibration modes, thus causing an equipartition of energy over a wide spectrum. Against all odds, the system introduced a quasi-periodic behavior of

entrance to the line, shows the MI in amplitude whose rate is 0.18 (**Figure 12**).

signal of frequency *f* = 897 KHz and amplitude VM = 3.6 V observed at 130th cell.

172 Advanced Electronic Circuits - Principles, Architectures and Applications on Emerging Technologies

**6. Fermi-Pasta-Ulam recurrence**

**6.1. Historical reviews**

138th cell.

In BF mode, we introduce a sine wave of frequency *f* = 475 KHz in the line with an amplitude Vsignal = 1.5 V, the polarization of the line voltage is V0 = 1.5 V, and we collect the signal level of the inductance of each cell *L*<sup>2</sup> (*L*<sup>2</sup> inductors are located on cells of even order). We see that the collected signal presents deformations of stochastic appearance that initially will grow as it sinks into the line to finally find the sinusoidal shape of the signal of departure to the 22nd cell (**Figure 14**). This phenomenon observed in the electrical nonlinear bimodal transmission line is known as the Fermi-Pasta-Ulam (FPU) recurrence.

**Figure 14.** FPU Recurrence observed in electrical nonlinear transmission line in BF mode on inductance L2, for a signal of frequency *f* = 475 KHz and Vsignal = 1.5 V and V0 = 1.5 V as voltage polarization of the line.

We continue to find the signal in the following cells (44th, 66th, 88th, and 110th), which enabled us to confirm the return period of the line to 22nd cells (**Figure 15**).

Again, we see that the amplitude of the signals decreases by increasing order from recurrence. This is due to the joule effect because let us not forget that our line is dissipative for the presence of inductor *L*<sup>1</sup> , and *r* <sup>1</sup> <sup>=</sup> <sup>5</sup>Ω his resistor and the inductor *L*<sup>2</sup> and its *r* <sup>2</sup> <sup>=</sup> <sup>8</sup>Ω.

#### *6.2.1.1. Comparison of the recurrence at the level of cells of inductance L1 and L2*

We also conducted the study of signals in cells of type *L*<sup>1</sup> (chokes on the cells of odd order). We find that the period of recurrence for the inductances of *L*<sup>1</sup> type is the same as that measured for inductors of type *L*<sup>2</sup> . We present the first recurrence observed for *L*<sup>2</sup> and *L*<sup>1</sup> cells at the level of cell 22 and 23, respectively, in **Figure 16**. Later, we see that successive recurrences on the inductances of *L*<sup>1</sup> type intervene at the level of cells 45, 67, 89, 111, and so on. For inductor *L*<sup>1</sup> , we found recurrence period also at 22nd cell.

Yet the study of the evolution of the waveforms between two recurrences often shows a sensitive form between the signals taken on *L*<sup>2</sup> and those taken on *L*<sup>1</sup> . We see the signals observed at the level of cells 10 and 11 for the inductors *L*<sup>1</sup> and *L*<sup>2</sup> , respectively, are quite dissimilar in shape in **Figure 17**.

In **Figure 18**, we observe the spectral decomposition of signals collected on inductance *L*<sup>2</sup> and *L*2 at cells 10 and 11, respectively. We see that the two signals contain the fundamental term

**Figure 16.** Observation of the first recurrence at the 22nd and 23rd cells on inductors *L*<sup>1</sup>

**Figure 17.** Observation of the signals measured on the inductance *L*<sup>2</sup>

and *L*<sup>1</sup>

the BF mode amplitude Vsignal = 1.5 V, the voltage polarization of the line V0 = 1.5 V, and frequency *f* = 475 kHz.

in the BF mode of frequency *f* = 475 kHz and Vsignal = 1.5 V, the voltage polarization of line is V0 = 1.5 V.

and *L*<sup>2</sup>

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at cells 10 and 11, respectively, for a signal in

, respectively, for a signal

**Figure 15.** Observation of the first five recurrences for the electrical nonlinear bimodal transmission line in the BF for inductance *L* 2, with *f* = 475 kHz, Vsignal = 1.5 V, and polarization voltage of the line V0 = 1.5 V.

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**Figure 16.** Observation of the first recurrence at the 22nd and 23rd cells on inductors *L*<sup>1</sup> and *L*<sup>2</sup> , respectively, for a signal in the BF mode of frequency *f* = 475 kHz and Vsignal = 1.5 V, the voltage polarization of line is V0 = 1.5 V.

**Figure 17.** Observation of the signals measured on the inductance *L*<sup>2</sup> and *L*<sup>1</sup> at cells 10 and 11, respectively, for a signal in the BF mode amplitude Vsignal = 1.5 V, the voltage polarization of the line V0 = 1.5 V, and frequency *f* = 475 kHz.

**Figure 15.** Observation of the first five recurrences for the electrical nonlinear bimodal transmission line in the BF for

We continue to find the signal in the following cells (44th, 66th, 88th, and 110th), which

Again, we see that the amplitude of the signals decreases by increasing order from recurrence. This is due to the joule effect because let us not forget that our line is dissipative for the pres-

. We present the first recurrence observed for *L*<sup>2</sup>

of cell 22 and 23, respectively, in **Figure 16**. Later, we see that successive recurrences on the

Yet the study of the evolution of the waveforms between two recurrences often shows a sensi-

In **Figure 18**, we observe the spectral decomposition of signals collected on inductance *L*<sup>2</sup>

at cells 10 and 11, respectively. We see that the two signals contain the fundamental term

type intervene at the level of cells 45, 67, 89, 111, and so on. For inductor *L*<sup>1</sup>

and those taken on *L*<sup>1</sup>

and *L*<sup>2</sup>

and its *r*

<sup>2</sup> <sup>=</sup> <sup>8</sup>Ω.

(chokes on the cells of odd order). We

and *L*<sup>1</sup>

type is the same as that measured

. We see the signals observed

, respectively, are quite dissimilar in

cells at the level

,

and

 *and L2*

<sup>1</sup> <sup>=</sup> <sup>5</sup>Ω his resistor and the inductor *L*<sup>2</sup>

enabled us to confirm the return period of the line to 22nd cells (**Figure 15**).

174 Advanced Electronic Circuits - Principles, Architectures and Applications on Emerging Technologies

*6.2.1.1. Comparison of the recurrence at the level of cells of inductance L1*

We also conducted the study of signals in cells of type *L*<sup>1</sup>

find that the period of recurrence for the inductances of *L*<sup>1</sup>

ence of inductor *L*<sup>1</sup>

for inductors of type *L*<sup>2</sup>

inductances of *L*<sup>1</sup>

shape in **Figure 17**.

*L*2

, and *r*

we found recurrence period also at 22nd cell.

at the level of cells 10 and 11 for the inductors *L*<sup>1</sup>

tive form between the signals taken on *L*<sup>2</sup>

inductance *L* 2, with *f* = 475 kHz, Vsignal = 1.5 V, and polarization voltage of the line V0 = 1.5 V.

**Figure 18.** Observation of the Fourier transform of signals measured on the inductance *L*<sup>2</sup> and *L*<sup>1</sup> at cells 10 and 11, respectively, for a signal taken in mode BF having a frequency *f* = 475 kHz, and amplitude Vsignal = 1.5 V, with a voltage polarization V0 = 1.5 V.

resorted the spectral analysis of the transform of Fourier to ensure that the wave has covered its sinusoidal shape. In precedent works [18, 19], we present the evolution of the signal and Fourier transformation (signal frequency *f* = 910 kHz, Vsignal = 2.2 V and polarized voltage V0 = 1.5 V). In these works, we see that the amplitude of the signal of the positive half-wave is higher than that of the wave for the negative alternation. This comes from the fact that the nonlinearity of the capacity C(V) does not similarly affect the two alternations. The minimum of the negative alternation seems more flattened than the positive alternation. On the other hand, Fourier analysis reveals the presence of a harmonic with twice the fundamental fre-

23, respectively) for a signal in the BF mode *f* = 475 kHz frequency and amplitude Vsignal = 1.5 V for a voltage polarization V0 = 1.5 V. There is a return to the ground state (*f* = 475 kHz) contribution of the harmonics of orders 1, 2, and 3 is marginal.

order, the recurrences occur in cells 4th, 10th, 16th, 22nd, and so on. The return period of the

We also studied the evolution of the return period depending on the amplitude of the signal in the BF mode and frequency *f* = 475 KHz. By varying the amplitude of the input signal offered in the line, we see a linear dependence of the period of recurrence with the inverse of the square root of the amplitude of the voltage of the signal [19] (**Figure 20**). Toda [20], in the case of the electric transmission line mono-modal,determined theoretically this dependence

, located in the cells of odd order, recurrences occur to the

and *L*<sup>2</sup>

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, located on the cells of even

in the first recurrence (cell 22 and

quency. We find for the inductors *L*<sup>1</sup>

cells 5th, 11th, 17th, 23rd, and so on. With regard to inductance *L*<sup>2</sup>

**Figure 19.** Observation of the Fourier transform of signals measured on inductors *L*<sup>1</sup>

line for the HF mode considered is therefore an average of 6th cell.

of the return period with the amplitude of the applied signal.

**6.3. Evolution of the period of recurrence based on the amplitude of the signal**

that corresponds to the frequency of the sine wave introduced (*f* = 475 KHz) and located at frequency harmonics 940 KHz, 1.41 MHz, 1.88 MHz, and so on. We see, for that concerning the cell 10, the signal on the inductor *L*<sup>2</sup> , the contribution of the first three harmonics prevail over others in addition to the fundamental. However, with regard to the collected cell on the 11th cell, inductance *L*<sup>1</sup> , the contribution of the following harmonics are predominant in addition to the fundamental. We can deduce that signals of inductors *L*<sup>1</sup> and *L*<sup>2</sup> are in fact identical in spectral terms and that is the relative contribution of the different harmonics, which explains the difference in shape between these signals.

On the other hand, the occurrence of recurrences marks the return to the ground state of the signals observed at the level of the inductance *L*<sup>1</sup> as *L*<sup>2</sup> . In **Figure 19**, we see the fundamental term becomes predominant, the contribution of the various harmonics being then quite marginal.

#### *6.2.2. The FPU recurrence in the HF mode*

Observation of recurrent FPU in HF mode may prove to be more challenging than the BF mode. This is primarily because, in this mode, HF signal amplitude is much lower than in BF mode because of the preponderance of the dissipative effects on nonlinear effects. Let us add to it in high frequencies, the periods of recurrence are much smaller and we count the periods by units of cell. However, we observed the FPU recurrence in HF mode by substantially increasing the amplitude of the input signals. Unlike the BF mode, we have sometimes

**Figure 19.** Observation of the Fourier transform of signals measured on inductors *L*<sup>1</sup> and *L*<sup>2</sup> in the first recurrence (cell 22 and 23, respectively) for a signal in the BF mode *f* = 475 kHz frequency and amplitude Vsignal = 1.5 V for a voltage polarization V0 = 1.5 V. There is a return to the ground state (*f* = 475 kHz) contribution of the harmonics of orders 1, 2, and 3 is marginal.

resorted the spectral analysis of the transform of Fourier to ensure that the wave has covered its sinusoidal shape. In precedent works [18, 19], we present the evolution of the signal and Fourier transformation (signal frequency *f* = 910 kHz, Vsignal = 2.2 V and polarized voltage V0 = 1.5 V). In these works, we see that the amplitude of the signal of the positive half-wave is higher than that of the wave for the negative alternation. This comes from the fact that the nonlinearity of the capacity C(V) does not similarly affect the two alternations. The minimum of the negative alternation seems more flattened than the positive alternation. On the other hand, Fourier analysis reveals the presence of a harmonic with twice the fundamental frequency. We find for the inductors *L*<sup>1</sup> , located in the cells of odd order, recurrences occur to the cells 5th, 11th, 17th, 23rd, and so on. With regard to inductance *L*<sup>2</sup> , located on the cells of even order, the recurrences occur in cells 4th, 10th, 16th, 22nd, and so on. The return period of the line for the HF mode considered is therefore an average of 6th cell.

#### **6.3. Evolution of the period of recurrence based on the amplitude of the signal**

that corresponds to the frequency of the sine wave introduced (*f* = 475 KHz) and located at frequency harmonics 940 KHz, 1.41 MHz, 1.88 MHz, and so on. We see, for that concerning the

respectively, for a signal taken in mode BF having a frequency *f* = 475 kHz, and amplitude Vsignal = 1.5 V, with a voltage

others in addition to the fundamental. However, with regard to the collected cell on the 11th

spectral terms and that is the relative contribution of the different harmonics, which explains

On the other hand, the occurrence of recurrences marks the return to the ground state of the

Observation of recurrent FPU in HF mode may prove to be more challenging than the BF mode. This is primarily because, in this mode, HF signal amplitude is much lower than in BF mode because of the preponderance of the dissipative effects on nonlinear effects. Let us add to it in high frequencies, the periods of recurrence are much smaller and we count the periods by units of cell. However, we observed the FPU recurrence in HF mode by substantially increasing the amplitude of the input signals. Unlike the BF mode, we have sometimes

becomes predominant, the contribution of the various harmonics being then quite marginal.

as *L*<sup>2</sup>

, the contribution of the following harmonics are predominant in addition

, the contribution of the first three harmonics prevail over

and *L*<sup>2</sup>

. In **Figure 19**, we see the fundamental term

and *L*<sup>1</sup>

are in fact identical in

at cells 10 and 11,

cell 10, the signal on the inductor *L*<sup>2</sup>

the difference in shape between these signals.

signals observed at the level of the inductance *L*<sup>1</sup>

*6.2.2. The FPU recurrence in the HF mode*

to the fundamental. We can deduce that signals of inductors *L*<sup>1</sup>

**Figure 18.** Observation of the Fourier transform of signals measured on the inductance *L*<sup>2</sup>

176 Advanced Electronic Circuits - Principles, Architectures and Applications on Emerging Technologies

cell, inductance *L*<sup>1</sup>

polarization V0 = 1.5 V.

We also studied the evolution of the return period depending on the amplitude of the signal in the BF mode and frequency *f* = 475 KHz. By varying the amplitude of the input signal offered in the line, we see a linear dependence of the period of recurrence with the inverse of the square root of the amplitude of the voltage of the signal [19] (**Figure 20**). Toda [20], in the case of the electric transmission line mono-modal,determined theoretically this dependence of the return period with the amplitude of the applied signal.

*V*(*t*) = *A* sec*h*<sup>2</sup>(*t*) (4)

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We inject into the bi-inductance line a signal in sech2 profile, and we follow the evolution of the signal in the line. We observe, in **Figure 22**, the evolution of signal at input cell 72 and input cell 144. We note that the shape of the signal remains intact during its spread in the line. However, we are seeing a weakening of the amplitude that is bound to the dissipative nature of the line. On the other hand, just like Remoissenet [2], we observe an oscillatory tail that

As we previously announced in [18], the LTNL can be used to modify the forms of signals. One example is in the case of the generation of radar abrupt front signals to obtain systems that are more accurate. Even our experimental line is not designed and optimized to produce that effect. We have been able to observe some modification in waveforms by

To illustrate the phenomenon of signal compression, we inject a sinusoidal wave amplitude V = 1.75 V for a voltage polarization of the line V0 = 1.5 V and frequency *f* = 169.4 KHz, at the entrance of the line. We observe at the 101th cell, the signal compared to the starting signal as

**Figure 21.** Observation of a train of soliton produced by the fragmentation of a plane wave subject to modulation instability for the BF of 491 kHz frequency and amplitude 4 V. The positions of the C1 and C2 sliders allow seeing the

where *A* and *α* are adjustable parameters and *t* time.

accompanies the solitary wave in its spread.

a factor of compression order 2 (**Figure 23**).

evolution of a pair of soliton in the line at cell 74 and cell 86.

**7.3. Shape modification of the signal**

nonlinearity.

*7.3.1. Signal compression*

**Figure 20.** Variation of the return period expressed in number of cell based on the inverse of the square root of the amplitude of the signal applied to a frequency *f* = 475 KHz wave and voltage polarization V0 = 1.5 V.
