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Linear And Nonlinear Arabesques: A Study Of Closed Chains Of Negative 2-Element Circuits

In this paper we consider a family of dynamical systems that we call "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 and one negative, depending on the parity (even or odd) of the dimension $n$. In view of the absence of diagonal terms in their Jacobian matrices, all these dynamical systems are conservative and consequently, they can not possess any attractor. First, we analyze a linear variant of them which we call "arabesque 0" or for short "A0". For increasing dimensions, the trajectories are increasingly complex open tori. Next, we inserted a single cubic nonlinearity that does not affect the signs of its circuits (that we call "arabesque 1" or for short "A1"). These systems have three steady states, whatever the dimension is, in agreement with the order of the nonlinearity. All three are unstable, as there can not be any attractor in their state-space. The 3D variant (that we call for short "A1\_3D") has been analyzed in some detail and found to display a complex mixed set of quasi-periodic and chaotic trajectories. Inserting $n$ cubic nonlinearities (one per equation) in the same way as above, we generate systems "A2\_$n$D". A2\_3D behaves essentially as A1\_3D, in agreement with the fact that the signs of the circuits remain identical. A2\_4D, as well as other arabesque systems with even dimension, has two positive $n$-circuits and nine steady states. Finally, we investigate and compare the complex dynamics of this family of systems in terms of their symmetries.