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6 Memristor Circuits: Invariant Manifolds, Coexisting Attractors, Extreme. . .
2. Foliation of state space, invariant manifolds, reduction of order, and smoother
dynamics: in the chapter we used FCAM to analyze the dynamics of a M–C
circuit, a M–L–C circuit, and a Memristor-based Chaotic Circuit (MCC) in the
class LM. In all cases FCAM has enabled to obtain a foliation of the phasespace in the (v, i)-domain in invariant manifolds where the circuit dynamics is
described by a lower-order system of SEs in the (ϕ, q)-domain. Dealing with
a lower-order system is of course advantageous and also reveals dynamical
aspects that are difficult to grasp when dealing with a higher-order system. For
instance, the M − L − C circuit is described in the (v, i)-domain by a thirdorder system whose dynamics is not easy to analyze directly. Instead, in the
flux-charge domain it is described on each manifold by a second-order system
that can be brought back to a forced Van Der Pol oscillator and analyzed via
quite standard techniques. Analogous considerations apply to the MCC, which
is equivalent in the (ϕ, q)-domain to a forced Chua’s oscillator. It is worth to
remark that a further advantage of FCAM is that the vector fields defining the SEs
in the (ϕ, q)-domain contain the memristor nonlinearity f (·) (or h(·)), which is
smoother than the nonlinearity f (·) (or h (·)) in the vector fields defining the SEs
in the (v, i)-domain. As a consequence, numerical problems in the simulations
(cf. for instance [29]) of memristor circuits are expected to be less relevant in the
(ϕ, q)-domain than in the (v, i)-domain.
3. Coexisting dynamics: the principle of foliation of the state space naturally
explains the coexistence of different dynamics and attractors for a memristor
circuit. We have shown in particular that for the M–C circuit we may have
coexistence of monostable and bistable dynamics, while for the M–L–C circuit
there coexist both convergent and oscillatory dynamics. The MCC circuit
displays a much reacher scenario with the coexistence of convergent dynamics,
periodic dynamics, and complex (chaotic) dynamics.
4. Bifurcations without parameters: Given the initial conditions for the state
variables in the (v, i)-domain, we can explicitly find via FCAM the corresponding invariant manifold where the memristor circuit dynamics evolve. As a
consequence, bifurcations induced by varying these initial conditions for fixed
circuit parameters (i.e., bifurcations without parameters) can be analytically
investigated. For the M–L–C circuit we were able to thoroughly analyze Hopf
bifurcations without parameters originating nonlinear oscillations whereas for
the MCC we investigated period-doubling cascades induced by varying the initial
conditions that lead to the birth or disappearance of a chaotic attractor. To the best
of authors’ knowledge, the M–L–C oscillatory circuit in Fig. 6.18 and the MCC
in Fig. 6.25 represent the first examples in the literature where a wide gamut of
bifurcations without parameters have been rigorously shown.
5. Overall, via FCAM we have shown that the presence of a continuum of EPs, the
existence of invariants of motion, and invariant manifolds are structurally stable
properties for a memristor circuit. Namely, they hold for any value of the circuit
parameters and are due only to the presence of the special element memristor.
The same holds for the feature of coexisting attractors and extreme multistability.
6 Memristor Circuits: Invariant Manifolds, Coexisting Attractors, Extreme. . .
2. Foliation of state space, invariant manifolds, reduction of order, and smoother
dynamics: in the chapter we used FCAM to analyze the dynamics of a M–C
circuit, a M–L–C circuit, and a Memristor-based Chaotic Circuit (MCC) in the
class LM. In all cases FCAM has enabled to obtain a foliation of the phasespace in the (v, i)-domain in invariant manifolds where the circuit dynamics is
described by a lower-order system of SEs in the (ϕ, q)-domain. Dealing with
a lower-order system is of course advantageous and also reveals dynamical
aspects that are difficult to grasp when dealing with a higher-order system. For
instance, the M − L − C circuit is described in the (v, i)-domain by a thirdorder system whose dynamics is not easy to analyze directly. Instead, in the
flux-charge domain it is described on each manifold by a second-order system
that can be brought back to a forced Van Der Pol oscillator and analyzed via
quite standard techniques. Analogous considerations apply to the MCC, which
is equivalent in the (ϕ, q)-domain to a forced Chua’s oscillator. It is worth to
remark that a further advantage of FCAM is that the vector fields defining the SEs
in the (ϕ, q)-domain contain the memristor nonlinearity f (·) (or h(·)), which is
smoother than the nonlinearity f (·) (or h (·)) in the vector fields defining the SEs
in the (v, i)-domain. As a consequence, numerical problems in the simulations
(cf. for instance [29]) of memristor circuits are expected to be less relevant in the
(ϕ, q)-domain than in the (v, i)-domain.
3. Coexisting dynamics: the principle of foliation of the state space naturally
explains the coexistence of different dynamics and attractors for a memristor
circuit. We have shown in particular that for the M–C circuit we may have
coexistence of monostable and bistable dynamics, while for the M–L–C circuit
there coexist both convergent and oscillatory dynamics. The MCC circuit
displays a much reacher scenario with the coexistence of convergent dynamics,
periodic dynamics, and complex (chaotic) dynamics.
4. Bifurcations without parameters: Given the initial conditions for the state
variables in the (v, i)-domain, we can explicitly find via FCAM the corresponding invariant manifold where the memristor circuit dynamics evolve. As a
consequence, bifurcations induced by varying these initial conditions for fixed
circuit parameters (i.e., bifurcations without parameters) can be analytically
investigated. For the M–L–C circuit we were able to thoroughly analyze Hopf
bifurcations without parameters originating nonlinear oscillations whereas for
the MCC we investigated period-doubling cascades induced by varying the initial
conditions that lead to the birth or disappearance of a chaotic attractor. To the best
of authors’ knowledge, the M–L–C oscillatory circuit in Fig. 6.18 and the MCC
in Fig. 6.25 represent the first examples in the literature where a wide gamut of
bifurcations without parameters have been rigorously shown.
5. Overall, via FCAM we have shown that the presence of a continuum of EPs, the
existence of invariants of motion, and invariant manifolds are structurally stable
properties for a memristor circuit. Namely, they hold for any value of the circuit
parameters and are due only to the presence of the special element memristor.
The same holds for the feature of coexisting attractors and extreme multistability.