6.3 Third-Order Memristor Chaotic Circuits
261
Fig. 6.28 Bifurcations without parameters of a limit cycle in the MCC of Fig. 6.25 with the circuit
parameter fixed at α = 8.7. The projection of the limit cycle for X 0 = 0 (blue curve) turns into
a different limit cycle (green curve) for X 0 = 0.0103 and then into a period-two limit cycle for
X 0 = 0.0281. The period-doubling bifurcation of the limit cycle takes place without changing the
circuit parameters, but only the initial condition ϕ M 0
scroll chaotic attractor 2 for X 0 = 0 (blue curve) turns into a different double-scroll
chaotic attractor (green curve) for X 0 = −0.0103 and then into a spiral chaotic
attractor for X 0 = −0.0281.
This shows that the MCC in Fig. 6.25 exhibits, for fixed circuit parameters,
complex bifurcation phenomena due to varying initial conditions, in particular it
displays period-doubling bifurcations without parameters leading to the birth or
disappearance of chaotic attractors.
Remark 6.22 This discussion shows that for the MCC there is coexistence of
infinitely many third-order dynamics, one for each invariant manifold. In particular,
there is coexistence of a continuum of EPs, of periodic attractors (cycles of period
one or of multiple period), and also of a continuum of different complex chaotic
attractors, for a fixed set of circuit parameters. Such an extremely rich and complex
dynamic scenario corresponds to the so-called property of extreme multistability.
Remark 6.23 We refer the reader to [16] where use is made of the harmonic balance
method (HBM) in combination with FCAM to analytically predict period-doubling
bifurcations without parameters in MCC. The article [17] instead uses HBM to study
analogous bifurcations in a class of harmonically forced second-order memristor
oscillators in the (ϕ, q)-domain.
2 The double-scroll attractor for X 0 = 0 coincides with that of the canonical Chua’s oscillator.
261
Fig. 6.28 Bifurcations without parameters of a limit cycle in the MCC of Fig. 6.25 with the circuit
parameter fixed at α = 8.7. The projection of the limit cycle for X 0 = 0 (blue curve) turns into
a different limit cycle (green curve) for X 0 = 0.0103 and then into a period-two limit cycle for
X 0 = 0.0281. The period-doubling bifurcation of the limit cycle takes place without changing the
circuit parameters, but only the initial condition ϕ M 0
scroll chaotic attractor 2 for X 0 = 0 (blue curve) turns into a different double-scroll
chaotic attractor (green curve) for X 0 = −0.0103 and then into a spiral chaotic
attractor for X 0 = −0.0281.
This shows that the MCC in Fig. 6.25 exhibits, for fixed circuit parameters,
complex bifurcation phenomena due to varying initial conditions, in particular it
displays period-doubling bifurcations without parameters leading to the birth or
disappearance of chaotic attractors.
Remark 6.22 This discussion shows that for the MCC there is coexistence of
infinitely many third-order dynamics, one for each invariant manifold. In particular,
there is coexistence of a continuum of EPs, of periodic attractors (cycles of period
one or of multiple period), and also of a continuum of different complex chaotic
attractors, for a fixed set of circuit parameters. Such an extremely rich and complex
dynamic scenario corresponds to the so-called property of extreme multistability.
Remark 6.23 We refer the reader to [16] where use is made of the harmonic balance
method (HBM) in combination with FCAM to analytically predict period-doubling
bifurcations without parameters in MCC. The article [17] instead uses HBM to study
analogous bifurcations in a class of harmonically forced second-order memristor
oscillators in the (ϕ, q)-domain.
2 The double-scroll attractor for X 0 = 0 coincides with that of the canonical Chua’s oscillator.