Linear and nonlinear Arabesques: a Study of Closed Chains of Negative 2-Element Circuits

In this paper we consider a family of dynamical systems (“arabesques”) defined as closed chains of 2-element negative circuits. An n-dimensional arabesque system has n 2-element circuits, but in addition, it displays by construction, two n-element circuits which are both positive vs one positive, one negative, depending on the parity (even vs odd) of the dimensionality n. In view of the absence of diagonal terms in their Jacobian matrices, all these dynamical systems are conservative and consequently, they cannot display any attractor. First, we analyze a linear variant of them which we call “arabesque 0” or for short “A0”. For increasing dimensionalities, the trajectories are increasingly complex open tori. Next, we inserted a single, cubic, nonlinearity that does not affect the signs of its circuits (“arabesque 1”, for short “A1”). These systems have three steady states, whatever the dimensionality, in agreement with the order of the nonlinearity. All three are unstable, as there cannot be any attractor in their state-space. The 3D variant (“A1 3D”) has been analyzed in some detail and found to display a complex mixed set of quasiperiodic and chaotic trajectories. Inserting n cubic nonlinearities (one per equation) in the same way as above, generates “A2 nD” systems. System A2 3D behaves essentially as A1 3D,