All Pinched Hysteresis Loops Generated by (α, β) Elements: in What Coordinates They May be Observable

Abstract

Two existing theorems for studying pinched hysteresis loops generated by nonlinear higherorder elements from Chua's table are reformulated, namely the generalized homothety theorem and the associated Loop Location Rule, specifying the coordinates where the hysteresis may occur, and the ω 2 criterion theorem for computing the corresponding loop areas. It is demonstrated in this work that the pinched hysteresis loops are also generated in other coordinates than those predicted by the Loop Location Rule, and all these possible coordinates are found. The ω 2 criterion is generalized to computing the areas of all hysteresis loops that may be observable. INDEX TERMSConstitutive relation, content, co-content, higher-order elements, homothety theorem, pinched hysteresis loop.

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“…The fingerprints of memristors have been introduced in the early paper on memristive systems [ 5 ] in order to define constitutive relations of the circuit element. One of the main characteristics of a memristor is its pinched (zero-crossing) loop hysteresis curve in the i-v plane under sinusoidal excitation and the behavior of the hysteresis curve upon frequency and input amplitude changes [ 6 ].…”

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