Toggling energy landscape to enable commutative and non-commutative responses in multistable structures

• History-dependent bistability enabled by kinematic constraints from neighboring bistable building blocks • Breaking structural symmetry elicits loading sequence–dependent responses • Toggling energy landscape allows switching between commutative and non-commutative multistability The deformation of a structure may be either insensitive or sensitive to the sequence of the applied forces: the former corresponds to a commutative response, whereas the latter indicates a non-commutative response. Among diverse physical systems, an elastic multistable structure tessellated from bistable units or bits can display commutative or non-commutative responses, each dictated by the manner of interaction between the units. In this work, we elucidate the underlying physics of commutative and non-commutative responses through the lens of the energy landscape of a multistable structure, and demonstrate that initially non-commutative units can be toggled to become commutative by altering the number and distribution of local energy minima in their state space, thus enabling reprogrammable sensitivity to the loading sequence. The concept is realized and demonstrated as a multistable switch in an electrical circuit capable of offering in situ adjustable levels of information protection through reprogrammable sequence sensitivity.