Nonlinear dynamics in non-volatile locally-active memristor for periodic and chaotic oscillations

(a) Pinched hysteresis loops driven by sinusoidal voltage with amplitude A = 10 V at f = 0.5 Hz, 2 Hz, 5 Hz, and 500 Hz, with m = 0 and initial state x(0) = 0.5; (b) time domain waveforms of input voltage (red) and output current (blue) at frequency f = 2 Hz.

POP of locally-active memristor.

(a) DC V–I plot of memristor (b) zoomed DC V–I plot near negative slope region.

Small-signal equivalent circuit of memristor at operating point Q(X, V) under exciting voltage δ v.

The pole-zero diagram of the memristor in the voltage range of −10 ≤ V ≤ 10.

Frequency response and Nyquist plot of the locally-active memristor at voltage V = −8 V, showing (a) real part Re Y(iω, V) and imaginary Im Y(iω, V) part of frequency response, (b) zoomed Fig. 6(a) in range −200 rad/s ≤ ω ≤ 200 rad/s, and (c) Nyquist plot.

Oscillating circuit composed of memristor operating in locally-active region.

Trajectories of eigenvalues in range of −20 V ≤ V ≤ 20 V.

Pole-zero diagrams of Y_c(s, V) for (a) real part of pole p₁, (b) real part of pole p₂, (c) zoomed real part of poles p₁ and p₂ in the range of −10 V ≤ V ≤ 10 V, (d) imaginary part of poles p₁ and p₂, (e) the relationship between the real and imaginary parts of poles p₁ and p₂, in which the arrowheads indicate the movement direction of the pole as the voltage V increases.

Pole diagrams of admittance Y_c(s, V) with external inductance L in a range of 0 < L < 1 H, showing (a) real part of the poles, p₁ and p₂, and (b) imaginary part of the poles.

Frequency response of admittance function Y(iω, V) with respect to ω, showing (a) real part curve of frequency response of memristor in a range of −100 rad/s ≤ ω ≤ 100 rad/s, and (b) Nyquist plot of memristor frequency response in a range of −1000 rad/s ≤ ω ≤ 1000 rad/s.

Transient waveforms of state variable x and current i_L with different DC voltages, where initial states are (x(0), i_L(0)) = (0.1, −0.1) (a) DC voltage V = −8.1 V, and transient waveforms converge to stable equilibrium point Q₀; (b) DC voltage V = −7.8 V, and the system oscillates to form a limit cycle; (c) DC voltage V = −5.5 V, and the system is in unstable state, diverging from initial state; (d) DC voltage V = −5.3 V, and the system is in unstable state, diverging from initial state; (e) DC voltage V = −5.2 V, and the system oscillates to form a limit cycle; (f) DC voltage V = −4.9 V, transient waveforms converge to stable equilibrium point $Q_0^{{}^{\mathrm{\prime }}}$.

Plots of the memristor determined by Eqs. (38) and (39): (a) $V=\stackrel{\text{⌣}}{v}(X)$, (b) $I=\stackrel{\text{⌣}}{i}(X)$, and (c) DC V–I plot.

Chaotic oscillator circuit based on locally-active memristor.

Phase diagrams of this memristor-based chaotic oscillating circuit (a) x–y plane, (b) y–z plane, (c) x–z plane, and (d) chaotic pinched hysteresis loops of the memristor.

Time domain waveform of memristor state variable x, capacitor voltage v_C, and inductor current i_L.

(a) Time domain waveforms and x–y–z phase diagrams with circuit parameters fixed at (a) (x₀,y₀, z₀) = (4.5, 0.1, 0.1); (b) (x₀, y₀, z₀) = (11, 0.1, 0.1), but the initial values changed.

Nonlinear dynamic characteristics of different initial conditions.

(a) Lyapunov exponent (LE) spectrum and (b) bifurcation diagram varying with parameter a.

Phase diagram changing with parameter a on the x–y plane: (a) a = 0.8, (b) a = 1.2, (c) a = 1.793, and (d) a = 2.13.

(a) Lyapunov exponent spectrum and (b) bifurcation diagram changing with parameter b.