7.7 Examples on Memristor Circuit Programming
303
Let us assume k 0 = 0 and show that the nonlinear dynamics of MCC can be
programmed by means of suitable pulses a(·) with finite time duration as those in
Fig. 7.8. Due to the pulse, the dynamics evolves from the manifold M(k 0 ) onto a
different manifold M(k 1 ) for τ > T 1 , where k 1 is given by (7.42), i.e., k 1 = k u =
AΔ.
Suppose we first choose A = −0.00281. In this case, as seen in Fig. 6.29c of
Chap. 6, the dynamics for τ > T 1 on the manifold M(−0.0281) is characterized
by a spiral chaotic attractor. Figure 7.9 shows how the pulse with A = −0.00281
in Fig. 7.8 induces a bifurcation of the chaotic attractor in MCC with fixed circuit
parameters α = 9.5 and β = 15. Such results are also shown in Fig. 7.9b where the
whole waveforms of X(τ ), Y 1 (τ ) and Y 2 (τ ) are reported for τ ∈ [0, 1000]. It is seen
that the double-scroll chaotic attractor (blue curve) for k 0 = 0 (i.e., for 0 ≤ τ ≤ T 0 )
turns into a spiral chaotic attractor (green curve) for k 1 = −0.0281 (i.e., for τ ≥ T 1 ).
When the pulse in Fig. 7.8 is applied (i.e., T 0 ≤ τ ≤ T 1 ) the dynamics (red curve
in Fig. 7.9a) evolves from the manifold M(k 0 = 0) to M(k 1 = 0.0281), i.e., from
the double-scroll to the spiral chaotic attractor. Similar results have been obtained
if the rectangular pulses in Fig. 7.8 are replaced by triangular pulses or pulses with
arbitrary waveforms, provided their momentum is equal to −0.0281.
Figure 7.10 shows how the pulse with A = 0.00781 in Fig. 7.8 induces a different
bifurcation. With reference to Fig. 7.10, the blue curve is the dynamics for 0 ≤ τ ≤
300, (i.e., when the pulse has not yet been applied), the red curve is the dynamics
for 300 ≤ τ ≤ 310 (i.e., during the pulse application) and the green curve is the
dynamics for τ ≥ 310 (i.e., when the pulse is over). In this case the double-scroll
chaotic attractor turns into a period-three limit cycle (black curve in Fig. 7.10) due
to the pulse.
Remark 7.3 It is worth noting that a single impulsive source a(t) is sufficient to set
the desired manifolds and reduced-order dynamics on manifolds in the MCC. This
is true although the MCC has four state variables in the (v, i)-domain.
Remark 7.4 In the MCC there coexists convergent, periodic, and complex chaotic
dynamics for the same set of circuit parameters and nonlinearity. We have seen
that via pulses we can easily control and set the desired regime. According to the
viewpoint in [16], MCC can be thought of as being a potential source of controllable
complex dynamics to be used in future neuromorphic architectures.
Remark 7.5 Several other pulse-induced bifurcations have been observed in other
memristor circuits N = N R
N D . This allows us to draw the conclusion that the
nonlinear dynamics of memristor circuits can be programmed via pulse with finite
time duration (i.e., constant momentum sources in the (ϕ, q)-domain). The dynamic
analysis through manifolds in the (ϕ, q)-domain permits to effectively design the
required pulse parameters (i.e., duration and amplitude).
303
Let us assume k 0 = 0 and show that the nonlinear dynamics of MCC can be
programmed by means of suitable pulses a(·) with finite time duration as those in
Fig. 7.8. Due to the pulse, the dynamics evolves from the manifold M(k 0 ) onto a
different manifold M(k 1 ) for τ > T 1 , where k 1 is given by (7.42), i.e., k 1 = k u =
AΔ.
Suppose we first choose A = −0.00281. In this case, as seen in Fig. 6.29c of
Chap. 6, the dynamics for τ > T 1 on the manifold M(−0.0281) is characterized
by a spiral chaotic attractor. Figure 7.9 shows how the pulse with A = −0.00281
in Fig. 7.8 induces a bifurcation of the chaotic attractor in MCC with fixed circuit
parameters α = 9.5 and β = 15. Such results are also shown in Fig. 7.9b where the
whole waveforms of X(τ ), Y 1 (τ ) and Y 2 (τ ) are reported for τ ∈ [0, 1000]. It is seen
that the double-scroll chaotic attractor (blue curve) for k 0 = 0 (i.e., for 0 ≤ τ ≤ T 0 )
turns into a spiral chaotic attractor (green curve) for k 1 = −0.0281 (i.e., for τ ≥ T 1 ).
When the pulse in Fig. 7.8 is applied (i.e., T 0 ≤ τ ≤ T 1 ) the dynamics (red curve
in Fig. 7.9a) evolves from the manifold M(k 0 = 0) to M(k 1 = 0.0281), i.e., from
the double-scroll to the spiral chaotic attractor. Similar results have been obtained
if the rectangular pulses in Fig. 7.8 are replaced by triangular pulses or pulses with
arbitrary waveforms, provided their momentum is equal to −0.0281.
Figure 7.10 shows how the pulse with A = 0.00781 in Fig. 7.8 induces a different
bifurcation. With reference to Fig. 7.10, the blue curve is the dynamics for 0 ≤ τ ≤
300, (i.e., when the pulse has not yet been applied), the red curve is the dynamics
for 300 ≤ τ ≤ 310 (i.e., during the pulse application) and the green curve is the
dynamics for τ ≥ 310 (i.e., when the pulse is over). In this case the double-scroll
chaotic attractor turns into a period-three limit cycle (black curve in Fig. 7.10) due
to the pulse.
Remark 7.3 It is worth noting that a single impulsive source a(t) is sufficient to set
the desired manifolds and reduced-order dynamics on manifolds in the MCC. This
is true although the MCC has four state variables in the (v, i)-domain.
Remark 7.4 In the MCC there coexists convergent, periodic, and complex chaotic
dynamics for the same set of circuit parameters and nonlinearity. We have seen
that via pulses we can easily control and set the desired regime. According to the
viewpoint in [16], MCC can be thought of as being a potential source of controllable
complex dynamics to be used in future neuromorphic architectures.
Remark 7.5 Several other pulse-induced bifurcations have been observed in other
memristor circuits N = N R
N D . This allows us to draw the conclusion that the
nonlinear dynamics of memristor circuits can be programmed via pulse with finite
time duration (i.e., constant momentum sources in the (ϕ, q)-domain). The dynamic
analysis through manifolds in the (ϕ, q)-domain permits to effectively design the
required pulse parameters (i.e., duration and amplitude).