394
11 Nonlinear Dynamics of Circuits with Mem-Elements
To clarify the implications of the presence of invariant manifolds M(Q 0 , Φ 0 ) in
the state-space (σ MC (t), q MC (t), ϕ M (t)) T ∈ R 3 , let us focus on a specific MC − M
circuit with nonlinearities
f MC (σ MC ) = σ MC +
1
3
σ
3
MC
and
f M (ϕ M ) = −ϕ M +
1
3
ϕ
3
M .
By changing the ICs in the (v, i)-domain in such a way that Φ 0 and Q 0 are
varied, different dynamical behaviors and bifurcations occur in the MC − M circuit
described by (11.4), even if circuit parameters and f MC (·) and f M (·) are held fixed.
For simplicity, assume Φ 0 = 0, but Q 0 can change. In this case, each manifold
M(Q 0 ) = M(Q 0 , 0) can be represented by a curve in the (σ MC , q MC ) plane given
by
M(Q 0 ) = {(σ MC , ˙
σ MC = q MC )
τ
∈ R
2
: q MC = −f M (f MC (σ MC )) + Q 0 }.
Three of these manifolds (for Q 0 = 0, Q 0 = 0.3, and Q 0 = 0.7) are represented
in Fig. 11.2. The first-order dynamics on each manifold can be studied via the
Dynamic Route Maps (DRMs) reported in the same figure (cf. Chap. 4), where the
arrowheads denote the direction of motion on each invariant manifold.
The DRM analysis allows us to draw the following scenario:
• any solution of (11.4) is bounded and since (11.4) is a first-order autonomous
system, any solution converges to an EP (cf. Chap. 4);
• the MC − M circuit exhibits a bistable behavior on the invariant manifold
M(Q 0 = 0), named zero-manifold, due to one unstable EP at the origin and
two asymptotically stable EPs;
• the MC − M circuit is bistable also on the invariant manifold M(Q 0 = 0.3),
however, there is a different location of the three EPs;
• the MC − M circuit has a globally convergent dynamics, i.e., every solution
of (11.4) approaches the unique EP, on the invariant manifold M(Q 0 = 0.7).
The results illustrated in Fig. 11.2 make clear the presence of infinitely many
different dynamics for fixed circuit parameters, that can be tuned by the ICs in the
(v, i)-domain. In particular, increasing (or decreasing) Q 0 causes the disappearance
of a pair of EPs of (11.4) via a saddle-node bifurcation without parameters (Chap. 6).
11 Nonlinear Dynamics of Circuits with Mem-Elements
To clarify the implications of the presence of invariant manifolds M(Q 0 , Φ 0 ) in
the state-space (σ MC (t), q MC (t), ϕ M (t)) T ∈ R 3 , let us focus on a specific MC − M
circuit with nonlinearities
f MC (σ MC ) = σ MC +
1
3
σ
3
MC
and
f M (ϕ M ) = −ϕ M +
1
3
ϕ
3
M .
By changing the ICs in the (v, i)-domain in such a way that Φ 0 and Q 0 are
varied, different dynamical behaviors and bifurcations occur in the MC − M circuit
described by (11.4), even if circuit parameters and f MC (·) and f M (·) are held fixed.
For simplicity, assume Φ 0 = 0, but Q 0 can change. In this case, each manifold
M(Q 0 ) = M(Q 0 , 0) can be represented by a curve in the (σ MC , q MC ) plane given
by
M(Q 0 ) = {(σ MC , ˙
σ MC = q MC )
τ
∈ R
2
: q MC = −f M (f MC (σ MC )) + Q 0 }.
Three of these manifolds (for Q 0 = 0, Q 0 = 0.3, and Q 0 = 0.7) are represented
in Fig. 11.2. The first-order dynamics on each manifold can be studied via the
Dynamic Route Maps (DRMs) reported in the same figure (cf. Chap. 4), where the
arrowheads denote the direction of motion on each invariant manifold.
The DRM analysis allows us to draw the following scenario:
• any solution of (11.4) is bounded and since (11.4) is a first-order autonomous
system, any solution converges to an EP (cf. Chap. 4);
• the MC − M circuit exhibits a bistable behavior on the invariant manifold
M(Q 0 = 0), named zero-manifold, due to one unstable EP at the origin and
two asymptotically stable EPs;
• the MC − M circuit is bistable also on the invariant manifold M(Q 0 = 0.3),
however, there is a different location of the three EPs;
• the MC − M circuit has a globally convergent dynamics, i.e., every solution
of (11.4) approaches the unique EP, on the invariant manifold M(Q 0 = 0.7).
The results illustrated in Fig. 11.2 make clear the presence of infinitely many
different dynamics for fixed circuit parameters, that can be tuned by the ICs in the
(v, i)-domain. In particular, increasing (or decreasing) Q 0 causes the disappearance
of a pair of EPs of (11.4) via a saddle-node bifurcation without parameters (Chap. 6).