Author: Robert Galida
Date: May 2026

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Abstract

The attractor framework says that persistence under disturbance is the basic mark of reality.
To turn this idea into a formal science, we introduce the persistence functional $P(x)$P(x).

$P(x)$P(x) is a single number that measures:

• How deep a state is inside an attractor basin.
• How quickly it returns after a knock.

We define $P(x)$P(x) for three different kinds of systems:

1. Deterministic dissipative systems – here $P$P is linked to Lyapunov exponents and basin stability.
2. Stochastic systems – here $P$P is linked to escape time and quasipotential.
3. Information‑theoretic systems – here $P$P is linked to negative free energy or mutual information.

The recovery rate $-\dot{P}\mathrm{/}P$−P˙/P is a universal sign of critical slowing down – a warning that a system is about to tip.

We also discuss limitations: resilience may depend on direction (“anisotropic”), and multiple timescales may need vector or tensor persistence. We list open mathematical problems.
This paper is a roadmap, not a finished theory.

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1. Introduction

In the attractor framework, persistence under disturbance is central. But we have not had a single number to say how persistent a state is.
The persistence functional $P(x)$P(x) aims to fill that gap.

What $P(x)$P(x) should do:

• $P(x)>0$P(x)>0 for states inside an attractor basin.
• For a conservative attractor (like a free electron), $P$P is maximal (normalised to 1).
• For a dissipative attractor, $P$P drops after a disturbance and then recovers.
  The recovery rate $-\dot{P}\mathrm{/}P$−P˙/P equals:
  • the negative of the largest Lyapunov exponent (for deterministic systems)
  • the inverse return time (for stochastic systems)
  • the rate of information loss (for informational systems)
• $P$P falls as the system approaches a bifurcation, giving early warning.

We do not give one universal formula. Instead, we give a family of definitions, each suited to a different type of system, all united by the same purpose – measuring resilience.

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2. Deterministic Dissipative Systems

Consider a smooth system $\dot{x}=f(x)$x˙=f(x) with a stable attractor $A$A and its basin $B(A)$B(A).

A natural candidate for $P(x)$P(x) uses a Lyapunov function $V(x)$V(x) – a kind of energy that always decreases inside the basin ($\dot{V}<0$V˙<0).

We define:$$P(x)=1-\frac{V(x)-V_A}{V_{\text{max}}-V_A}$$P(x)=1−Vmax​−VA​V(x)−VA​​

This gives $P=1$P=1 on the attractor and $P\to 0$P→0 at the basin boundary.
Near the attractor, the recovery rate is related to the largest Lyapunov exponent ${\lambda }_1$λ1​:$$-\dot{P}\mathrm{/}P\approx -{\lambda }_1$$−P˙/P≈−λ1​

When the system approaches a tipping point, ${\lambda }_1\to 0^{-}$λ1​→0−, so the recovery rate slows down – this is critical slowing down.

Conclusion: For deterministic systems, $P$P can be built from a Lyapunov function. The recovery rate equals the negative of the largest Lyapunov exponent.

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3. Stochastic Systems

When noise is present, persistence is about how long it takes to escape from the basin.
The mean first passage time $\tau (x)$τ(x) – the average time to leave – is a natural measure.

We define:$$P(x)=\frac{\tau (x)}{{\tau }_{\text{max}}}$$P(x)=τmax​τ(x)​

where ${\tau }_{\text{max}}$τmax​ is the value at the attractor.

For weak noise, $\tau (x)$τ(x) grows exponentially with the quasipotential $U(x)$U(x) (Freidlin–Wentzell theory):$$\tau (x)\sim e^{U(x)\mathrm{/}ϵ}$$τ(x)∼eU(x)/ϵ

So:$$P(x)\propto e^{-(U_{\text{max}}-U(x))\mathrm{/}ϵ}$$P(x)∝e−(Umax​−U(x))/ϵ

The recovery rate is the inverse of the return time. As a tipping point is approached, the return time diverges, and the recovery rate goes to zero. This again gives critical slowing down – rising variance and autocorrelation.

Conclusion: For stochastic systems, $P$P is proportional to the mean exit time (or the exponential of the quasipotential). This connects persistence to large deviation theory.

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4. Information‑Theoretic Systems

For systems where information matters (neural, cognitive, social), we can define persistence using mutual information between past and future.

Let $I_{\text{past,future}}$Ipast,future​ be the predictive information. Then:$$P(t)=I(\text{past};\text{future at time }t)\text{or}P=e^{-\text{surprisal}}$$P(t)=I(past;future at time t)orP=e−surprisal

The decay of $P(t)$P(t) over time measures memory loss.
Landauer’s principle connects information loss to entropy production:$$\dot{P}\mathrm{/}P\le -\frac{\dot{S}}{k_B\ln 2}$$P˙/P≤−kB​ln2S˙​

Alternatively, in the free energy principle (Friston), the negative free energy $-F$−F acts like a Lyapunov function. We can set:$$P=e^{-F\mathrm{/}kT}\text{or}P=-F$$P=e−F/kTorP=−F

Then $-\dot{P}\mathrm{/}P$−P˙/P is the rate of free energy minimisation, which slows near bifurcations.

Conclusion: For information‑theoretic systems, $P$P can be defined via mutual information decay or negative free energy, linking persistence to entropy production and predictive coding.

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5. Unifying Recovery Rate and Critical Slowing Down

Across all types of systems, the recovery rate ${\lambda }_{\text{rec}}=-\dot{P}\mathrm{/}P$λrec​=−P˙/P (just after a small disturbance) is a universal indicator:

• Deterministic dissipative: ${\lambda }_{\text{rec}}=\mathrm{\mid }{\lambda }_1\mathrm{\mid }$λrec​=∣λ1​∣ (absolute value of the largest Lyapunov exponent)
• Stochastic: ${\lambda }_{\text{rec}}$λrec​ = inverse of the return time, related to the quasipotential’s curvature
• Information‑theoretic: ${\lambda }_{\text{rec}}$λrec​ = rate of free energy minimisation or information loss

As the system approaches a bifurcation, ${\lambda }_{\text{rec}}\to 0$λrec​→0. This is critical slowing down.
It shows up as rising lag‑1 autocorrelation and variance (Scheffer et al., 2009).
So $P$P and its recovery rate give early warnings.

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6. Normalisation for Conservative Attractors

For a perfect conservative attractor (e.g., an electron in its ground state, no decay), the persistence functional should be constant and maximal:$$P_{\text{cons}}=1\text{for all times}$$Pcons​=1for all times

No recovery rate is defined (or it is zero). This anchors the scale.

For emergent approximate conservative systems (like atomic clocks), $P$P is very close to 1 and decays extremely slowly.

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7. Limitations – Scalar Collapse and Anisotropic Resilience

A single scalar $P(x)$P(x) may not be enough for systems where resilience is anisotropic – that is, recovery speed depends on the direction of the perturbation.
High‑dimensional systems can have multiple timescales (fast and slow modes). A scalar average can miss important structure.

Future work may need:

• Vector persistence – a list of recovery rates along different directions.
• Tensor persistence – a metric that captures the full shape of the basin.
• Persistence manifold – the geometry of the basin in state space.

We accept this limitation. The scalar $P$P is a useful first approximation for systems with isotropic resilience or for early‑warning applications where a single number is enough. For complex systems, a multidimensional generalisation is an open research problem.

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8. Open Mathematical Problems

1. Derive P(x)P(x) from first principles for a given class of systems (e.g., from a variational principle).
2. Prove that −P˙/P=∣λ1∣−P˙/P=∣λ1​∣ for a wide class of dissipative systems.
3. Extend the definition to systems with multiple attractors and chaotic basins (where basin stability is fractal).
4. Establish a rigorous relationship between PP and the mutual information decay rate for non‑equilibrium processes.
5. Formulate a universal persistence functional that works across all regimes – or prove it’s impossible.
6. Test the predictive power of PP in controlled experiments (e.g., ecological microcosms, neural cultures, social media sentiment).
7. Develop vector/tensor persistence for anisotropic resilience.

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9. Conclusion

The persistence functional $P(x)$P(x) gives a mathematical language for attractor resilience.

We have given operational definitions for three regimes:

• Deterministic dissipative → Lyapunov / basin stability
• Stochastic → escape time / quasipotential
• Information‑theoretic → mutual information / free energy

The recovery rate $-\dot{P}\mathrm{/}P$−P˙/P unifies critical slowing down across all these domains.

We have explicitly noted limitations (scalar collapse, anisotropy) as open problems.

This paper is a roadmap, not a final theory. The framework now has a quantitative step.

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Suggested citation: Galida, R. S. (2026). The Persistence Functional: Towards a Mathematical Measure of Attractor Resilience (Reader‑Friendly Version). Fantasy Attractor.