Duality of Complex Systems Built from Higher‐Order Elements

Abstract

The duality of nonlinear systems built from higher-order two-terminal Chua’s elements and independent voltage and current sources is analyzed. Two different approaches are now being generalized for circuits with higher-order elements: the classical duality principle, hitherto restricted to circuits built from R-C-L elements, and Chua’s duality of memristive circuits. The so-called storeyed structure of fundamental elements is used as an integrating platform of both approaches. It is shown that the combination of associated flip-type and shift-type transformations of the circuit elements can generate dual networks with interesting features. The regularities of the duality can be used for modeling, hardware emulation, or synthesis of systems built from elements that are not commonly available, such as memristors, via classical dual elements.

Higher-Order Hamiltonian for Circuits with (α,β) Elements

“…However, the corresponding application circuit must contain, in addition to this memristor, only the other appropriate HOEs from the Σ = −2 diagonal-for example, (−2, 0), (0, −2) and other elements. If some of these elements are linear, then they can be moved in Chua's table along the corresponding ∆-diagonals [31]. Then, the set of these admissible linear elements will grow, as is illustrated in Section 5.…”

“…where g 2 () is, according to (31), the inverse of the constitutive relation f 2 () of the memristor. The modeling diagram is the same as in Figure 5, but the functional block in the feedback will now correspond to the function g 2 ().…”

Lagrangian for Circuits with Higher-Order Elements

“…Due to its time-domain differentiation or integration, the state function S add loses the ability to be a scalar potential function, which can generate the monogenic quantity f add [1]. The generalized voltages f add (d) and f add (-d) , by which the terms below and above the Σ-diagonal contribute by (32) to KV (α) L, are polygenic quantities [1] that cannot be included in the Lagrangian. That is why Hamilton's principle holds only for d = 0, i.e., when all the elements of the circuit are located on the same Σ-diagonal.…”

“…The necessary and sufficient condition for the validity of Hamilton's principle, namely that all the circuit elements must occupy one common Σ-diagonal, strictly holds only for nonlinear elements, i.e., elements with nonlinear constitutive relations. It is well known that the character of the linear element is not changed during the element movement along the ∆-diagonal [8,32]. Such a movement changes the α and β indices by the same value k, thus both the original (F(v (α) ,i (β) ) = 0) and the new (F(v (α+k) ,i (β+k) ) = 0) relations hold simultaneously.…”

Hamilton’s Principle for Circuits with Dissipative Elements

“…e inner sum in (13) gives the total contribution of resistors to KV (0) L law along the i-th loop. e variation of the velocity takes place in (13) instead of the variation of coordinates.…”