In differential systems, the logical structure responsible for essential aspects of the dynamics can be characterized by the feedback circuits present in the jacobian matrix. It has been found that a positive circuit is a necessary condition for multistationarity, a negative circuit (with at least two elements), a necessary condition for stable periodicity, and that both positive and negative circuits are required for deterministic chaos. However, we show here that a single, three-element circuit can generate chaotic dynamics if the nature of nonlinearities is such that this unique circuit is positive or negative depending on the location in phase space.
Concretely, systems such as x' = - b x + sin y
y' = - b y + sin z
z' = - b z + sin x
generate aesthetically admirable chaotic attractors (see figure) whose size and complexity gradually increase as parameter b decreases. More specifically, for b = 0.15 or so, phase space is cut into 27 (3) domains in which the three-element circuit is alternatively positive and negative. Each of these domains contains an unstable steady state which belongs to either of the two complementary types of saddle foci, depending on whether the three-element circuit is positive or negative in the domain considered.
As one decreases the value of parameter b, the number of steady states increases, as well as the size and complexity of the attractor, which tends to occupy the totality of phase space as b tends to zero. In the limit case b = 0, phases space has become an infinite three-dimensional lattice with an infinite number of steady states, all unstable, between wich trajectories endlessly percolate in a perfectly deterministic, altough long term unpredictable way.
The general interest of this system resides in the fact that, starting with a system with a single steady state and elementary dynamical properties, the mere decrease of the value of a parameter results in the progressive structuration of phase space, which is partitionned into an increasing number of domains, each containing an unstable steady state, and in the emergence of a chaotic attractor of increasing size and complexity.
A trajectory of the system described above, with b = 0.15. Initial state (0, 0.1, 0.1), integraton step : 0.1. Numeric integration (grind) was run for 400 time units but recorded only from t = 200 in ordrer to discard transitories. The two images are tilted by 15¡ around the vertical axis in order to permit a stereoscopical view.