6.3 Third-Order Memristor Chaotic Circuits
259
In the next sections, bifurcations on a fixed manifold and bifurcations without
parameters for the SEs (6.48) are presented. The quantity X 0 is varied by changing
the initial flux in the memristor ϕ M 0 , whereas the remaining initial conditions are set
to q C 1 0 = C 1 v C 1 0 = 0, q C 2 0 = C 2 v C 2 0 = 1, and ϕ L 0 = Li L 0 = 0 in all numerical
simulations.
6.3.1.1 Bifurcations on a Fixed Manifold
It is apparent that the nonlinear dynamics of the SEs (6.48) on the fixed manifold
M(0) present the same bifurcations (exactly for the same values of α) of the
canonical Chua’s oscillator. Let us then consider the fixed manifold M(X 0 )
in (6.49b) with ϕ M 0 = 0.02, hence (6.49a) gives X 0 = −0.0029. The aim is
to show bifurcation phenomena on M(−0.0029) due to changes of the circuit
parameter α. Figure 6.27a, b, and c shows that a period-doubling cascade takes
place by increasing α from the value 8.8 to 8.9 and then to 9.0. A further increase
of α leads to chaotic behavior. The same period-doubling cascade occurs in the
canonical Chua’s oscillator (i.e., for X 0 = 0), but for slightly different values of α.
Numerical simulations show that the canonical Chua’s oscillator exhibits a similar
form for the projection of limit cycles as in Fig. 6.27a, b, and c approximately for
α = 8.79, α = 9.0, and α = 9.01, respectively. The only difference between the
projection of the limit cycles of the canonical Chua’s oscillator and those reported
in Fig. 6.27a, b, and c is the position in the phase-space because the limit cycles
are embedded into different invariant manifolds (M(0) for the canonical Chua’s
oscillator and M(−0.0029) for the memristor-based chaotic circuit in Fig. 6.25
described by (6.48)). We can conclude that: on the fixed invariant manifold M(0),
and on manifolds M(X 0 ) with X 0 close to 0, the nonlinear dynamics in the
memristor-based chaotic circuit of Fig. 6.25 are analogous to those of the canonical
Chua’s oscillator, i.e., nonlinear attractors and bifurcations are of a very similar
type, but they are displayed for slightly different circuit parameters.
6.3.1.2 Period-Doubling Bifurcations Without Parameters
Let us pick α = 8.7 in the SEs (6.48), in which case the MCC in Fig. 6.25 displays
periodic oscillations. The aim is to show that by changing X 0 , for example by means
of ϕ M 0 , gives rise to bifurcation phenomena. Figure 6.28 presents the projection on
the (x, y) plane of one of the limit cycles—in the circuit described by (6.48)—
when X 0 = 0, X 0 = 0.0103, and X 0 = 0.0281, i.e., according to (6.49b), ϕ M 0 = 0,
ϕ M 0 = −0.0725, and ϕ M 0 = −0.2, respectively. It turns out that the period-doubling
bifurcations of the limit cycle take place without changing the circuit parameters,
but only the initial condition ϕ M 0 .
Similar results can be obtained for α = 9.5 in (6.48), in which case the MCC in
Fig. 6.25 displays chaotic attractors. Figure 6.29a, b, and c show how the double-
259
In the next sections, bifurcations on a fixed manifold and bifurcations without
parameters for the SEs (6.48) are presented. The quantity X 0 is varied by changing
the initial flux in the memristor ϕ M 0 , whereas the remaining initial conditions are set
to q C 1 0 = C 1 v C 1 0 = 0, q C 2 0 = C 2 v C 2 0 = 1, and ϕ L 0 = Li L 0 = 0 in all numerical
simulations.
6.3.1.1 Bifurcations on a Fixed Manifold
It is apparent that the nonlinear dynamics of the SEs (6.48) on the fixed manifold
M(0) present the same bifurcations (exactly for the same values of α) of the
canonical Chua’s oscillator. Let us then consider the fixed manifold M(X 0 )
in (6.49b) with ϕ M 0 = 0.02, hence (6.49a) gives X 0 = −0.0029. The aim is
to show bifurcation phenomena on M(−0.0029) due to changes of the circuit
parameter α. Figure 6.27a, b, and c shows that a period-doubling cascade takes
place by increasing α from the value 8.8 to 8.9 and then to 9.0. A further increase
of α leads to chaotic behavior. The same period-doubling cascade occurs in the
canonical Chua’s oscillator (i.e., for X 0 = 0), but for slightly different values of α.
Numerical simulations show that the canonical Chua’s oscillator exhibits a similar
form for the projection of limit cycles as in Fig. 6.27a, b, and c approximately for
α = 8.79, α = 9.0, and α = 9.01, respectively. The only difference between the
projection of the limit cycles of the canonical Chua’s oscillator and those reported
in Fig. 6.27a, b, and c is the position in the phase-space because the limit cycles
are embedded into different invariant manifolds (M(0) for the canonical Chua’s
oscillator and M(−0.0029) for the memristor-based chaotic circuit in Fig. 6.25
described by (6.48)). We can conclude that: on the fixed invariant manifold M(0),
and on manifolds M(X 0 ) with X 0 close to 0, the nonlinear dynamics in the
memristor-based chaotic circuit of Fig. 6.25 are analogous to those of the canonical
Chua’s oscillator, i.e., nonlinear attractors and bifurcations are of a very similar
type, but they are displayed for slightly different circuit parameters.
6.3.1.2 Period-Doubling Bifurcations Without Parameters
Let us pick α = 8.7 in the SEs (6.48), in which case the MCC in Fig. 6.25 displays
periodic oscillations. The aim is to show that by changing X 0 , for example by means
of ϕ M 0 , gives rise to bifurcation phenomena. Figure 6.28 presents the projection on
the (x, y) plane of one of the limit cycles—in the circuit described by (6.48)—
when X 0 = 0, X 0 = 0.0103, and X 0 = 0.0281, i.e., according to (6.49b), ϕ M 0 = 0,
ϕ M 0 = −0.0725, and ϕ M 0 = −0.2, respectively. It turns out that the period-doubling
bifurcations of the limit cycle take place without changing the circuit parameters,
but only the initial condition ϕ M 0 .
Similar results can be obtained for α = 9.5 in (6.48), in which case the MCC in
Fig. 6.25 displays chaotic attractors. Figure 6.29a, b, and c show how the double-