Manifolds of equilibria and bifurcations without parameters in memristive circuits (Q2910868)

The author deals with the dynamics of a new electrical device which keeps track of the device history, the so-called memory resistor. The special form of the circuit leads to a semi-explicit differential-algebraic equation (DAE) with a manifold of equilibria. Under certain topological restrictions it is shown that the manifold of equilibria is normally hyperbolic and its dimension is equal to the number of memory resistors.NEWLINENEWLINE Moreover, the dimension of the state-space is defined by the number of memory resistors plus the number of reactive elements and the normal hyperbolicity of these manifolds of equilibria is characterized in graph-theoretic terms. When the assumptions of the normal hyperbolicity fails, the model structure of the DAE leads to bifurcations of equilibria without parameter. Some examples based on the design of quantum bits are provided to illustrate the theoretical results.