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 higher-order 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 ω2 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 ω2 criterion is generalized to computing the areas of all hysteresis loops that may be observable.

Keywords

• Constitutive relation
• content
• co-content
• higher-order elements
• homothety theorem
• pinched hysteresis loop