In a hypercycle of dimension n ≥ 5 under constant organization, the concentrations evolve according to the elementary hypercycle equations. The parameter is the catalytic rate constant for production of species by its predecessor x_i−1 in the cyclic order → → … → → .

(k_i x_i−1 − Φ),   _j k_j x_j x_j−1,   , indices modulo n.

Rotational symmetry and its breaking

When all catalytic rate constants are equal, = … = , the hypercycle possesses a symmetric rotating-wave limit cycle. Every species shows an identical pulse, and the pulses are equally spaced in phase. The whole pattern rotates around the catalytic cycle without changing shape.

Unequal rate constants break this rotational symmetry. The cyclic order of pulses remains → → … → → , but pulse height, width, and phase lag become species-specific. A species with a larger catalytic advantage reaches a higher peak concentration. Peak order still follows the catalytic cycle.

Phase lags encode rate-limiting steps

Phase differences need not be uniform. A fast step shortens the delay between the predecessor pulse and the product pulse; a slow step lengthens it. Thus pulse timing encodes which catalytic step is relatively rate-limiting. For example, if is large, the pulse follows with a shorter lag than in the symmetric case.

Time-averaged concentrations

Time-averaged concentrations obey ⟨⟩ = ⟨Φ⟩ / k_i+1. A species' mean level therefore depends on the downstream rate constant, not simply on its own peak height. A tall pulse can coexist with a modest time average if the next catalytic step is fast.

Example: five-membered hypercycle

Consider with . The pulse tends to peak higher and follow with a shorter lag. Its average is not simply proportional to its peak. The relation ⟨⟩ = ⟨Φ⟩ / shows that the mean level of is coupled to the downstream rate driving .

The pulses remain coupled through Φ and constant organization. Unequal constants modulate hypercyclic organization without destroying its cyclic topology.