Lyapunov Stability and Broken Hypercycles

A hypercycle's central fixed point can be certified as asymptotically stable without solving the full nonlinear system, by exhibiting a Lyapunov function V(x) that vanishes at the equilibrium, stays strictly positive everywhere else, and decreases steadily along every trajectory the system can follow.

The Lyapunov Criterion

A function V(x) qualifies as a Lyapunov function for the central fixed point x̂ when V(x̂) = 0 and V(x) > 0 for all ̂. If, in addition, V decreases along trajectories (dV/dt < 0), then every nearby orbit is drawn toward x̂ and the fixed point is asymptotically stable. The test is qualitative: only the sign of the change matters, not an explicit closed-form solution of the dynamics.

Stability for , 3 and 4

For small hypercycles of dimension , 3 and 4 a suitable Lyapunov function can be constructed explicitly, and it proves that the central fixed point x̄₀ = (c₀/n)(1, …, 1)ᵀ is asymptotically stable. This complements the normal-mode picture, in which the equilibrium changes character with dimension — a focus at , a spiral sink at , a center at — and upgrades a local description into a global stability statement.

Broken Hypercycles and Invariant Subsystems

If a single catalytic link is broken, the network no longer closes into one self-reinforcing cycle. Instead it partitions into globally invariant subsystems: sets of species whose dynamics never leak into the rest of the system. Because orbits are trapped inside them, each subsystem can be analysed on its own.

2A edge — a flowing edge that transmits catalytic influence.
2B edge — a fixed-point edge that settles to rest.
3A face — a two-dimensional subsystem spanning the remaining members.

Studying these lower-dimensional objects recovers the stability and selection behaviour of the broken cycle piece by piece, and links the Lyapunov analysis of intact hypercycles to the dynamical decomposition of broken ones.

Key contrast

An intact small hypercycle is one cooperating unit with a certified central fixed point; a broken hypercycle is a family of independent invariant subsystems. Same species, two qualitatively different fates.

Broken Hypercycle Subsystems

A hypercycle with one catalytic link removed no longer forms a single closed loop; its remaining architecture appears as globally invariant subsystems — the 2A edge, the 2B edge and the 3A face — each analysable for stability on its own. On the left, the intact three-membered cycle carries a central fixed point x̂ that a Lyapunov function certifies as asymptotically stable for

, 3 and 4.

QUESTION 1

Which pair of conditions makes a function V(x) a valid Lyapunov function for the central fixed point x̂?

V takes the same constant value everywhere in the simplex

V(x̂) = 0 and V(x) > 0 for every

̂, with V decreasing along trajectories

V reaches its maximum exactly at x̂

V(x̂) > 0 and V increasing along every trajectory

QUESTION 2

For which hypercycle dimensions can a suitable Lyapunov function prove asymptotic stability of the central fixed point?

Every dimension n without exception

, 3 and 4

only

and

only

QUESTION 3

What happens to a hypercycle when a single catalytic link is broken?

It becomes a single, stronger closed cycle

It partitions into globally invariant subsystems that can be analysed separately

It instantly collapses onto one corner fixed point

It loses every fixed point and every invariant set

QUESTION 4

Why are the globally invariant subsystems of a broken hypercycle analytically useful?

They remove any need for fixed-point analysis

Orbits cannot leave them, so stability and selection can be studied in lower dimension

They prove that the broken network is chaotic

They force every concentration to stay constant over time

QUESTION 5

In the three-membered example with one link removed, which components describe the remaining catalytic architecture?

A single closed hypercycle

A stable limit cycle of order five

Only the corner fixed points

The 2A edge, the 2B edge and the 3A face

Case Study: A Broken Three-Membered Hypercycle

Read the scenario and answer the questions.

A self-replicating system is modelled as a three-membered hypercycle whose species 1, 2 and 3 form a single catalytic cycle 1→2→3→1, with the central fixed point located at x̄₀ = (c₀/3)(1, 1, 1)ᵀ. An experimental perturbation permanently removes the catalytic link from species 3 to species 1.

1. Before the perturbation, how would you justify that the intact three-membered hypercycle has an asymptotically stable central fixed point?

Answer:
Construct a Lyapunov function V(x) with V(x̂) = 0 and V(x) > 0 for all

̂, and show that V decreases along trajectories (dV/dt < 0). For

such a function exists, so the central fixed point x̄₀ = (c₀/3)(1, 1, 1)ᵀ is asymptotically stable: nearby states are drawn toward equal concentrations of all three species.

2. After removing the 3→1 link, why can the broken hypercycle be analysed through invariant subsystems, and what does this reveal?

Answer:
Removing the link destroys the single closed cycle, so the network decomposes into globally invariant subsystems — the 2A edge, the 2B edge and the 3A face. Because orbits cannot leave these subsystems, each one can be examined independently at lower dimension, which recovers the stability and selection behaviour of the broken cycle piece by piece instead of as one cooperating whole.