Experimental realization of strange nonchaotic attractors in a quasiperiodically forced electronic circuit

We have identified the three prominent routes, namely Heagy-Hammel, fractalization, and intermittency routes, and their mechanisms for the birth of strange nonchaotic attractors (SNAs) in a quasiperiodically forced electronic system constructed using a negative conductance series $LCR$ circuit with a diode both numerically and experimentally. The birth of SNAs by these three routes is verified from both experimental and their corresponding numerical data by maximal Lyapunov exponents, and their variance, Poincaré maps, Fourier amplitude spectrum, spectral distribution function, and finite-time Lyapunov exponents. Although these three routes have been identified numerically in different dynamical systems, the experimental observation of all these mechanisms is reported here in a single second order electronic circuit.

Figure 1

(a) Circuit realization of a simple nonautonomous circuit. Here, $D$ is the $p\text{−}n$ junction diode, and $g_N$ is negative conductance. The parameter values of the other elements are fixed as $L=50.0\mathrm{mH}$, $C=10.32\mathrm{nF}$. The external emf $f_1\left(t\right)=E_{f1}\sin {\omega }_{f1}t$ and $f_2\left(t\right)=E_{f2}\sin {\omega }_{f2}t$ are the function generators (HP $33120\mathrm{A}$). The values of ${\omega }_{f1}$ and ${\omega }_{f2}$ are chosen as $5982.0\mathrm{Hz}$ and $13533.0\mathrm{Hz}$, respectively. The forcing amplitude $E_{f2}$ is fixed as $0.15\mathrm{V}$. The other forcing amplitude $E_{f1}$ and the resistance $R$ are taken as control parameters which are being varied in our analysis, (b) $i\text{−}v$ characteristics of the $p\text{−}n$ junction diode and (c) two segment piecewise-linear function.

Figure 2

(Color online) (a) Phase diagram in the $\left(a\text{−}{E}_1\right)$ plane for the circuit given in Fig. 1, represented by Eq. (2) and obtained from numerical data. 3T and 6T correspond to torus of period-3 and period-6 attractors, respectively. F, HH, and INT denote the formation of SNAs through gradual fractalization, Heagy-Hammel, and intermittency routes, respectively. C corresponds to chaotic attractor. (b) An enlarged version of the intermittent region is indicated in (a).

Figure 3

Projection of the numerically simulated attractors of Eqs. (2) in the $\left(\varphi \text{−}x\right)$ plane for fixed $E_1=0.44$ and various values of $a$ indicating the transition from quasiperiodic attractor to SNA through the Heagy-Hammel mechanism: (a) period-3 torus (3T) for $a=0.95632$, (b) period-6 torus (6T) for $a=0.95593$, and (c) SNA at $a=0.95592$.

Figure 4

Projection of the numerically simulated attractors of Eqs. (2) in the $(x,y)$ plane for fixed $E_1=0.44$ and various values of $a$ indicating the transition from a quasiperiodic attractor to an SNA through the Heagy-Hammel route: (a) period-3 torus (3T) for $a=0.95632$, (b) period-6 torus (6T) for $a=0.95593$, and (c) SNA at $a=0.95592$: (i) phase trajectory in the $\left(x\text{−}y\right)$ space; (ii) power spectrum.

Figure 5

Transition from three doubled torus to SNA through Heagy-Hammel route in region HH obtained from numerical data: (a) the behavior of the maximal Lyapunov exponent $\left(\Lambda \right)$ and (b) the variance $\left(\sigma \right)$ for $E_1=0.44$.

Figure 6

(Color online) Attractors obtained experimentally from the circuit given in Fig. 1 corresponding to Figs. 4. (a) period-3 torus (3T) for $R=2109\Omega$, (b) period-6 torus (6T) for $R=2106\Omega$, and (c) SNA at $R=2104\Omega$ for fixed value of $E_{f1}=0.22\mathrm{V}$: (i) phase trajectory in the $\left(v\text{−}{i}_L\right)$ space; (ii) power spectrum.

Figure 7

Spectral distribution function for the quasiperiodic attractors and SNAs created through the Heagy-Hammel route: (a) quasiperiodic attractor, (b) strange nonchaotic attractor. Here the numerical study is indicated by filled circles, and experimental result is denoted by filled triangles.

Figure 8

Distribution of finite-time Lyapunov exponents on SNAs created through the Heagy-Hammel route: (a) quasiperiodic attractor, (b) strange nonchaotic attractor. Finite-time Lyapunov exponents calculated from numerical data are indicated by dashed lines, and from experimental data are denoted by solid lines.

Figure 9

Projection of the numerically simulated attractors of Eqs. (2) in the $\left(\varphi \text{−}x\right)$ plane for fixed $E_1=0.34$ and various values of $a$ indicating the transition from quasiperiodic attractor to SNA through fractalization route. (a) period-3 torus (3T) for $a=0.954406$ and (b) SNA at $a=0.954351$.

Figure 10

Projection of the numerically simulated attractors of Eqs. (2) in the $(x,y)$ plane for fixed $E_1=0.34$ and various values of $a$ indicating the transition from quasiperiodic attractor to SNA through fractalization route: (a) period-3 torus (3T) for $a=0.954406$ and (b) SNA at $a=0.954351$: (i) phase trajectory in the $\left(x\text{−}y\right)$ plane; (ii) power spectrum.

Figure 11

Transition from three torus to SNA through fractalization route obtained from numerical data: (a) the behavior of the maximal Lyapunov exponent $\left(\Lambda \right)$ and (b) the variance $\left(\sigma \right)$ for $E_1=0.34$.

Figure 12

(Color online) Attractors obtained experimentally from the circuit given in Fig. 1 corresponding to Figs. 10. (a) period-3 torus (3T) for $R=2102\Omega$ and (b) SNA at $R=2101\Omega$ for fixed value of $E_{f1}=0.17\mathrm{V}$: (i) phase trajectory $\left(v\text{−}{i}_L\right)$; (ii) power spectrum.

Figure 13

Spectral distribution function for spectra of quasiperiodic attractor and SNAs created through gradual fractalization route: (a) quasiperiodic attractor, (b) strange nonchaotic attractor. Here numerical study is indicated by the filled circles and experimental study is denoted by the filled triangles.

Figure 14

Distribution of finite-time Lyapunov exponents on SNAs created through gradual fractalization: (a) quasiperiodic attractor, (b) strange nonchaotic attractor. Finite-time Lyapunov exponents calculated from numerical data are indicated by dashed lines, and from experimental data are denoted by solid lines.

Figure 15

Projection of the simulated attractors of Eqs. (2) in the $\left(\varphi \text{−}x\right)$ plane for fixed $E_1=0.635$ and various value of $a$: indicating the transition from quasiperiodic attractor to SNA through type-I intermittent route. (a) torus (3T) for $a=0.951912$ and (b) SNA at $a=0.951889$.

Figure 16

Projection of the numerically simulated attractors of Eqs. (2) for fixed $E_1=0.635$ and two different values of $a$ indicating the transition from quasiperiodic attractor to SNA through type-I intermittent route: (a) period-3 torus (3T) for $a=0.951912$ and (b) SNA at $a=0.951889$: (i) phase trajectory in the $(x-y)$ plane; (ii) power spectrum.

Figure 17

Transition from three torus to SNA through type-I intermittent mechanism obtained from numerical data: (a) the behavior of the maximal Lyapunov exponent $\left(\sigma \right)$ and (b) the variance $\left(\Lambda \right)$ for $E_1=0.635$.

Figure 18

Average laminar length $\left(⟨l⟩\right)$ vs $(a_{\mathit{\text{critic}}}-a)$ at $a_{\mathit{\text{critic}}}=0.951876$ obtained from numerical data.

Figure 19

(Color online) Attractors obtained experimentally from the circuit given in Fig. 1 corresponding to Figs. 16. (a) period-3 torus (3T) for $R=2099\Omega$ and (b) SNA at $R=2097\Omega$ for fixed value of $E_{f1}=0.318\mathrm{V}$: (i) phase trajectory $\left(v\text{−}{i}_L\right)$; (ii) power spectrum.

Figure 20

Spectral distribution function for spectrum SNAs created through type-I intermittency route (circle denotes numerical study and triangle indicates experimental study).

Figure 21

Distribution of finite-time Lyapunov exponents on SNAs created through type-I intermittency. (a) quasiperiodic attractor, (b) strange nonchaotic attractor. Finite-time Lyapunov exponents calculated from numerical data are indicated by dashed lines, and from experimental data are denoted by solid lines.