224
6 Memristor Circuits: Invariant Manifolds, Coexisting Attractors, Extreme. . .
Remark 6.2 (Structural Stability of Non-isolated EPs, Invariants of Motion and
Invariant Manifolds) The examples discussed so far feature the peculiar property
that there exists a continuum of non-isolated EPs. Such property is structurally
stable in the considered circuits, i.e., it persists under perturbations of the circuit
parameters. Indeed, we leave to the reader the verification that any of the two circuits
still displays a line of non-isolated equilibria for any value of C 1 and C 2 and the
other parameters. We refer the reader to [2] and [3] for further considerations on the
existence of non-isolated equilibrium points for RLC circuits and their structural
stability or instability properties.
However, we stress that it is a special property for an RLCcircuit to possess a
continuum of equilibria. For example, according to Theorem 14 in [2], it is seen
that the existence of non-isolated EPs is related to the presence of particular circuit
structures as cut-sets made of capacitors only (cf. Example 6.2) or loops of inductors
only. Such structures also imply the existence of invariants of motion and invariant
manifolds [3], that are again observable only in special classes of RLC circuits.
Different from RLC circuits, in memristor circuits, due to the presence of
the new element memristor, we are in a situation where the circuit displays a
continuum of EPs in the (v, i)-domain (cf. Theorem 2.2 in Chap. 2). Moreover, this
is accompanied by the presence of invariants of motion and invariant manifolds.
Such properties are structurally stable, in the sense that they hold for any value of
the circuit parameters. These issues will be discussed and illustrated in detail with
the memristor circuits studied in the remaining part of the book.
Remark 6.3 (Linear Systems with an Invariant of Motion) It is quite a special
situation for a linear system to admit an invariant of motion. For instance, the system
˙
x = −x
˙
y = −y
does not have a continuous non-constant invariant of motion (cf. [4]). Note that such
a system has a unique EP.
Instead, the linear system
˙
x = y
˙
y = −x
which can be associated with a simple conservative L − C circuit, has a unique EP,
and has a non-constant invariant of motion w(x, y) = x 2 + y 2 − γ , where γ ∈ R.
6 Memristor Circuits: Invariant Manifolds, Coexisting Attractors, Extreme. . .
Remark 6.2 (Structural Stability of Non-isolated EPs, Invariants of Motion and
Invariant Manifolds) The examples discussed so far feature the peculiar property
that there exists a continuum of non-isolated EPs. Such property is structurally
stable in the considered circuits, i.e., it persists under perturbations of the circuit
parameters. Indeed, we leave to the reader the verification that any of the two circuits
still displays a line of non-isolated equilibria for any value of C 1 and C 2 and the
other parameters. We refer the reader to [2] and [3] for further considerations on the
existence of non-isolated equilibrium points for RLC circuits and their structural
stability or instability properties.
However, we stress that it is a special property for an RLCcircuit to possess a
continuum of equilibria. For example, according to Theorem 14 in [2], it is seen
that the existence of non-isolated EPs is related to the presence of particular circuit
structures as cut-sets made of capacitors only (cf. Example 6.2) or loops of inductors
only. Such structures also imply the existence of invariants of motion and invariant
manifolds [3], that are again observable only in special classes of RLC circuits.
Different from RLC circuits, in memristor circuits, due to the presence of
the new element memristor, we are in a situation where the circuit displays a
continuum of EPs in the (v, i)-domain (cf. Theorem 2.2 in Chap. 2). Moreover, this
is accompanied by the presence of invariants of motion and invariant manifolds.
Such properties are structurally stable, in the sense that they hold for any value of
the circuit parameters. These issues will be discussed and illustrated in detail with
the memristor circuits studied in the remaining part of the book.
Remark 6.3 (Linear Systems with an Invariant of Motion) It is quite a special
situation for a linear system to admit an invariant of motion. For instance, the system
˙
x = −x
˙
y = −y
does not have a continuous non-constant invariant of motion (cf. [4]). Note that such
a system has a unique EP.
Instead, the linear system
˙
x = y
˙
y = −x
which can be associated with a simple conservative L − C circuit, has a unique EP,
and has a non-constant invariant of motion w(x, y) = x 2 + y 2 − γ , where γ ∈ R.