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 ω <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> 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 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> criterion is generalized to computing the areas of all hysteresis loops that may be observable.
Highlights
Nonlinear (α,β) two-terminal elements, referred to as the higher-order elements, organized in Chua’s table of fundamental electrical elements [1], are an important tool for the so-called predictive modeling of complex nonlinear dynamical systems and processes including the phenomena in molecular and nanoscale devices [2]
The synthesis of the model of a concrete system is based on the selection of the (α,β) elements with the relevant nonlinear constitutive relations and on their proper interconnection
This approach facilitates the physical insight into the mechanism of complex phenomena and significantly decreases the simulation times
Summary
Nonlinear (α,β) two-terminal elements, referred to as the higher-order elements, organized in Chua’s table of fundamental electrical elements [1], are an important tool for the so-called predictive modeling of complex nonlinear dynamical systems and processes including the phenomena in molecular and nanoscale devices [2]. If the PHLs are observed in voltage-current, v(0)–i(0) coordinates, the source of such a behavior can be the (– 1,–1) element, which is the memristor defined by the constitutive relation between the v(–1)–i(–1) variables (time integrals of the voltage and current, denoted as the flux and charge). For the PHLs studied in the stress-strain characteristics of cyclically stressed polymers, the (–1,–2) mechanical element, whose constitutive variables are the integral of force vs the second integral of velocity, can be responsible for the hysteretic behavior: Considering that the effort is the force and the flow is the velocity, the stress-strain coordinates can be interpreted as a force-displacement pair (the first integral of the velocity).
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