Attractors and Limit Cycles in Higher Dimensions

Cyclic Attractors and Limit Cycles in Higher Dimensions 

In order for Eq. (32) to display cyclic behavior it is necessary that there be a cycle in the state transition diagram. The cycle we have discussed for N = 3 (Fig. 4) has a special property. All transitions between a state on the cycle and a neighboring state not on the cycle are oriented toward the states on the cycle. If this is true, we call the cycle a cyclic attractor For N=3, the number of distinct cyclic attractors is 1. Since this case has been of chemical interest, it is of potential interest to investigate the cyclic attractors in higher dimensions. 

From the definition, each state on a qrclic attractor specifies the orientations of N — 1 edges on the state transition diagram. Therefore, the maximum length Lmax for a cyclic attractor in N dimensions must satisfy the relation  

This upper bound has been sharpened for higher dimensions, = 8 for N = 4 and 2 - 2 for iV  

3d and 4, since it represents the cyclic attractor found in the state transition diagram of Eq. (27), iV = 4. For any number of dimensions there will always be a cyclic attractor through 2N volumes corresponding to the cyclic attractor found for Eq. (27). Numerical integration of Eq. (27) for n = 8, N = 5,6, 7 has indicated stable limit cycle attractors for each case where both the period of the oscillation and the amplitude increase as N increases. 

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