428
11 Nonlinear Dynamics of Circuits with Mem-Elements
q M (t) + q MC (t) = Q 0 ,
f M (q M (t)) − ϕ L (t) − f MC (σ MC (t)) = Φ 0 }
where (Φ 0 , Q 0 ) T ∈ R 2 .
We leave to the reader the problem of inserting two independent sources in the
circuit in order to programme the manifolds and reduced-order circuit dynamics.
11.8 Discussion
Nonlinear dynamic memory elements, as memristors, memcapacitors, and
meminductors (a.k.a. mem-elements), are of paramount importance in conceiving
neural networks, mem-computing machines, and reservoir computing systems
with advanced computational primitives. This chapter has developed a systematic
methodology for analyzing complex dynamics in nonlinear circuits with such
emerging nanoscale mem-elements. The technique extends FCAM for nonlinear
circuits with memristors to a broader class of nonlinear circuits N e containing also
memcapacitors and meminductors. FCAM has also been extended to deal with
circuits containing mem-elements in combination with nonlinear capacitors and
inductors.
After deriving the CR and equivalent circuit in the (ϕ, q)-domain of each twoterminal element in N e , the chapter has focused on relevant subclasses of N e for
which an SE description can be obtained. On this basis, salient features of the
dynamics are highlighted and studied analytically: (1) the presence of invariant
manifolds in the autonomous case; (2) the coexistence of infinitely many different
reduced-order dynamics on manifolds; (3) the presence of bifurcations due to
changing the ICs for a fixed set of parameters (bifurcations without parameters).
Analytic formulas are also given to design non-autonomous circuits subject to
pulses that drive trajectories through different manifolds and nonlinear reducedorder dynamics.
The results in the chapter provide a method for a comprehensive understanding
of complex dynamical features and computational capabilities in nonlinear circuits
with mem-elements, which is fundamental for a holistic approach in neuromorphic
systems with such emerging nanoscale devices. The richness of the dynamical
behavior highlighted via FCAM in mem-circuits and the possibility to tune via
pulses the different dynamics make the circuits studied in the chapter valid
candidates as sources of controllable complex dynamics to be incorporated in future
neuromorphic architectures. Such architectures, which operate in the flux-charge
domain, may be implemented at nanoscale exploiting the full variety of emerging
mem-elements including memcapacitors and meminductors.
11 Nonlinear Dynamics of Circuits with Mem-Elements
q M (t) + q MC (t) = Q 0 ,
f M (q M (t)) − ϕ L (t) − f MC (σ MC (t)) = Φ 0 }
where (Φ 0 , Q 0 ) T ∈ R 2 .
We leave to the reader the problem of inserting two independent sources in the
circuit in order to programme the manifolds and reduced-order circuit dynamics.
11.8 Discussion
Nonlinear dynamic memory elements, as memristors, memcapacitors, and
meminductors (a.k.a. mem-elements), are of paramount importance in conceiving
neural networks, mem-computing machines, and reservoir computing systems
with advanced computational primitives. This chapter has developed a systematic
methodology for analyzing complex dynamics in nonlinear circuits with such
emerging nanoscale mem-elements. The technique extends FCAM for nonlinear
circuits with memristors to a broader class of nonlinear circuits N e containing also
memcapacitors and meminductors. FCAM has also been extended to deal with
circuits containing mem-elements in combination with nonlinear capacitors and
inductors.
After deriving the CR and equivalent circuit in the (ϕ, q)-domain of each twoterminal element in N e , the chapter has focused on relevant subclasses of N e for
which an SE description can be obtained. On this basis, salient features of the
dynamics are highlighted and studied analytically: (1) the presence of invariant
manifolds in the autonomous case; (2) the coexistence of infinitely many different
reduced-order dynamics on manifolds; (3) the presence of bifurcations due to
changing the ICs for a fixed set of parameters (bifurcations without parameters).
Analytic formulas are also given to design non-autonomous circuits subject to
pulses that drive trajectories through different manifolds and nonlinear reducedorder dynamics.
The results in the chapter provide a method for a comprehensive understanding
of complex dynamical features and computational capabilities in nonlinear circuits
with mem-elements, which is fundamental for a holistic approach in neuromorphic
systems with such emerging nanoscale devices. The richness of the dynamical
behavior highlighted via FCAM in mem-circuits and the possibility to tune via
pulses the different dynamics make the circuits studied in the chapter valid
candidates as sources of controllable complex dynamics to be incorporated in future
neuromorphic architectures. Such architectures, which operate in the flux-charge
domain, may be implemented at nanoscale exploiting the full variety of emerging
mem-elements including memcapacitors and meminductors.