222
6 Memristor Circuits: Invariant Manifolds, Coexisting Attractors, Extreme. . .
w(v C 1 , v C 2 ) = av C 2 + v C 1 − γ
(6.4)
is a (non-constant) invariant of motion for (6.1). In the mathematical literature, (6.4)
is also called a first integral of (6.1). This property implies that there exist ∞ 1
invariant sets for (6.1) given by
M(γ ) = {(v C 1 , v C 2 )
T
∈ R
2
: av C 2 + v C 1 = γ }
for any γ ∈ R. These are simply straight lines with slope −a in the phase space (cf.
Fig. 6.2).
Remark 6.1 We have shown that the state space (v C 1 , v C 2 ) ∈ R 2 of (6.1) can be
foliated in ∞ 1 1D invariant sets (or manifolds), where each manifold is in this
case simply a straight line. In addition, the following considerations can be easily
drawn:
• the SE (6.1) is second-order, while (6.2) is first-order. However, the right-hand
side of the latter system depends upon the initial conditions v C 1 0 and v C 2 0 for the
state variables of (6.1)
• the previous results state that the dynamics of v C 2 as a solution of (6.1) is
essentially first-order, since it satisfies the reduced-order system (6.2). However,
since (6.2) depends upon the scalar term av C 2 0 + v C 1 0 , there are actually ∞ 1
different coexisting first-order dynamics for (6.1)
• the reduced-order dynamics are all of the same type and share the same stability
properties. On each manifold there is a unique isolated EP v C 2 0 + v C 1 0 /a which
is asymptotically stable for the reduced-order system (6.2). Each solution starting
on a manifold converges toward the corresponding EP as t → ∞.
In the example considered, the existence of an invariant of motion is related to the
existence of infinitely many (a continuum of) EPs for (6.1). The reduced-order
system (6.2) has instead a unique EP depending however on the manifold.
Example 6.2 Consider the linear circuit with two capacitors and a resistor in
Fig. 6.3. Note that the two capacitors form a cut-set. The SEs describing its dynamics
are, assuming C 1 = C 2 = 1 and R = 1
Fig. 6.3 Linear circuit with
two capacitors
C 1
C 2
R
−
+
v R
i R
−
+
v C 1
i C 1
−
+
v C 2
i C 2
6 Memristor Circuits: Invariant Manifolds, Coexisting Attractors, Extreme. . .
w(v C 1 , v C 2 ) = av C 2 + v C 1 − γ
(6.4)
is a (non-constant) invariant of motion for (6.1). In the mathematical literature, (6.4)
is also called a first integral of (6.1). This property implies that there exist ∞ 1
invariant sets for (6.1) given by
M(γ ) = {(v C 1 , v C 2 )
T
∈ R
2
: av C 2 + v C 1 = γ }
for any γ ∈ R. These are simply straight lines with slope −a in the phase space (cf.
Fig. 6.2).
Remark 6.1 We have shown that the state space (v C 1 , v C 2 ) ∈ R 2 of (6.1) can be
foliated in ∞ 1 1D invariant sets (or manifolds), where each manifold is in this
case simply a straight line. In addition, the following considerations can be easily
drawn:
• the SE (6.1) is second-order, while (6.2) is first-order. However, the right-hand
side of the latter system depends upon the initial conditions v C 1 0 and v C 2 0 for the
state variables of (6.1)
• the previous results state that the dynamics of v C 2 as a solution of (6.1) is
essentially first-order, since it satisfies the reduced-order system (6.2). However,
since (6.2) depends upon the scalar term av C 2 0 + v C 1 0 , there are actually ∞ 1
different coexisting first-order dynamics for (6.1)
• the reduced-order dynamics are all of the same type and share the same stability
properties. On each manifold there is a unique isolated EP v C 2 0 + v C 1 0 /a which
is asymptotically stable for the reduced-order system (6.2). Each solution starting
on a manifold converges toward the corresponding EP as t → ∞.
In the example considered, the existence of an invariant of motion is related to the
existence of infinitely many (a continuum of) EPs for (6.1). The reduced-order
system (6.2) has instead a unique EP depending however on the manifold.
Example 6.2 Consider the linear circuit with two capacitors and a resistor in
Fig. 6.3. Note that the two capacitors form a cut-set. The SEs describing its dynamics
are, assuming C 1 = C 2 = 1 and R = 1
Fig. 6.3 Linear circuit with
two capacitors
C 1
C 2
R
−
+
v R
i R
−
+
v C 1
i C 1
−
+
v C 2
i C 2