Basin-Fraction Order Parameter

• Basin-fraction order parameter is a normalized measure of basin occupancy that quantifies the fraction of initial conditions reaching a specific attractor.
• It is applied across diverse systems, such as nonlinear skin effect, delayed dynamics, and network oscillators, to assess attractor dominance and multistability.
• Mathematically, it utilizes probability measures over tailored initial-condition ensembles to differentiate between localized and extended states.

Searching arXiv for papers on basin fraction, basin entropy, and related order-parameter formulations. A basin-fraction order parameter is a normalized measure of basin occupancy that assigns a probability, volume fraction, or aggregated basin weight to the set of initial conditions that asymptotically reach a specified attractor. The expression appears explicitly in the study of the nonlinear non-Hermitian skin effect, where it denotes the probability that trajectories launched from $\psi(0)=0$ with random initial slope $s=v(0)$ are attracted to the skin state (Lin et al., 19 Feb 2026). Closely related constructions occur in multistable delayed dynamics, basin-entropy theory, network-coupled oscillators, and mechanically stable packings, where the operative quantity is respectively called basin fraction, basin entropy, basin stability, balancing ratio, or basin volume (Tarigo et al., 2024, Daza et al., 2022, Kaiser et al., 2018, Ashwin et al., 2011). This suggests that the term is best understood as a normalized basin measure used as a compact descriptor of attractor dominance, accessibility, or unpredictability.

1. Terminology and conceptual scope

The exact phrase “basin-fraction order parameter” is not standard across all of the relevant literature. One paper introduces it directly as a probabilistic quantity over boundary slopes in a bistable nonlinear wave problem (Lin et al., 19 Feb 2026). A second treats basin fraction as the natural companion to basin entropy in a delayed Mackey–Glass system, and states that it functions as an order-parameter-like summary statistic for multistability even though it is not formally defined as an order parameter in the statistical-physics sense (Tarigo et al., 2024). A review of basin entropy does not use the phrase, and instead develops entropy-based measures of final-state unpredictability (Daza et al., 2022).

Quantity	Meaning in the cited literature	Typical role
Basin-fraction order parameter	Probability that initial slopes reach the skin attractor	Attractor selection in a coexistence window
Basin fraction	Relative volume occupied by each attractor in initial-condition space	Dominance of coexisting attractors
Basin entropy	Average Shannon entropy over coarse-grained uncertainty cells	Local intermingling and unpredictability
Basin stability / balancing ratio	Fraction of phase space flowing to stable balanced states	Graph-level classification
Basin volume	Protocol-dependent volume of initial configurations mapping to a packing	Packing probability

The common nucleus is the same: a basin is treated as a measurable subset of an initial-condition ensemble, and the quantity of interest records the normalized weight of that subset. What differs from problem to problem is the ensemble over which the normalization is taken. In the nonlinear skin-effect problem the ensemble consists of boundary slopes; in delayed systems it consists of initial functions or their discretizations; in network dynamics it is the full phase torus; in granular packings it is the configuration space at zero packing fraction (Lin et al., 19 Feb 2026, Tarigo et al., 2024, Kaiser et al., 2018, Ashwin et al., 2011).

2. Mathematical formulation as a normalized basin measure

The most explicit formulation is given for the nonlinear non-Hermitian skin effect. There, one draws initial slopes $s$ from a probability measure $\mu$ with density $\rho(s)$, and the order parameter is the probability that the trajectory is attracted to the origin, i.e. to the skin attractor (Lin et al., 19 Feb 2026). In the coexistence regime this becomes

$$p_{\mathrm{skin}}(\gamma;\mu)=\int_{-s_*(\gamma)}^{s_*(\gamma)} \rho(s)\,ds,$$

where $s_*(\gamma)$ is the separatrix threshold in slope space. For the representative Cauchy choice

$$\rho_{\rm c}(s)=\frac{1}{\pi}\frac{s_0}{s^2+s_0^2},$$

the closed form is

$$p_{\mathrm{skin}}(\gamma;\mu)=\frac{2}{\pi}\arctan\!\left(\frac{s_*(\gamma)}{s_0}\right).$$

This is a direct probability measure on a one-dimensional launch ensemble (Lin et al., 19 Feb 2026).

A formally similar normalization appears in basin stability for network dynamics. For a fixed point $\bm{\vartheta}_0$ with basin $s=v(0)$0, basin stability is defined as

$s=v(0)$1

with $s=v(0)$2. It is therefore the probability that a random initial phase configuration converges to that fixed point (Kaiser et al., 2018).

In delayed systems, the same idea is implemented empirically rather than through a closed-form definition. The initial condition is a function over a finite interval, not a point, and in the discretized Mackey–Glass system this becomes an $s=v(0)$3-dimensional initial state. Basin fraction is then the relative volume occupied by each attractor in that initial-condition space, estimated by the fraction of sampled initial conditions that end in the attractor (Tarigo et al., 2024). The paper uses both cross-sections,

$s=v(0)$4

and random sampling of initial conditions from a uniform distribution on $s=v(0)$5 (Tarigo et al., 2024).

A plausible unifying interpretation is that the basin-fraction order parameter is a normalized basin measure whose specific domain is problem-dependent, but whose operational meaning is always the same: the fraction of admissible initial data flowing to a selected asymptotic state.

3. Relation to basin entropy and final-state unpredictability

Basin fraction and basin entropy quantify different aspects of basin organization. In the delayed Mackey–Glass analysis, the distinction is stated explicitly: basin entropy measures the degree of uncertainty or intermingling of attractors inside a region of initial-condition space, whereas basin fraction measures how much of that space is occupied by each attractor (Tarigo et al., 2024). The entropy is the Shannon average

$s=v(0)$6

computed over sampled boxes, with $s=v(0)$7 in each box (Tarigo et al., 2024).

This nonredundancy is central. A region can contain many attractors yet have low entropy if one attractor dominates the local volume. The Mackey–Glass paper gives exactly such a case: in the bottom-right corner of the relevant parameter plot, the number of coexisting attractors is highest, but the basin entropy is nearly zero because most attractors in that region have very low basin fraction compared to the strange attractor (Tarigo et al., 2024). Basin fraction therefore explains why attractor count alone is not a reliable proxy for unpredictability.

The basin-entropy review develops the same point at a more abstract level. It defines basin entropy on uncertainty cells of radius $s=v(0)$8, and under simplifying assumptions gives the approximation

$s=v(0)$9

where $s$0 is associated with lacunarity, $s$1 captures the uncertain fraction of cells as a function of scale, and $s$2 reflects the number of final states (Daza et al., 2022). The paper states that this factor is the closest thing in its framework to a basin-fraction quantity, because it acts as an effective occupancy fraction of boundary-hit cells. It follows that entropy combines three ingredients—occupancy, boundary complexity, and multiplicity of outcomes—whereas basin fraction isolates attractor occupancy itself (Daza et al., 2022).

In this sense, basin-fraction order parameters are especially informative when multistability is present but practical predictability depends on whether one basin overwhelmingly dominates the admissible initial-condition ensemble.

4. Basin-fraction order parameter in the nonlinear non-Hermitian skin effect

The clearest named realization of the concept occurs in the continuum nonlinear Hatano–Nelson model with saturating nonlinear nonreciprocity (Lin et al., 19 Feb 2026). For real stationary fields, the stationary problem is cast as the planar flow

$s$3

with

$s$4

This phase-space reduction makes the basin geometry explicit (Lin et al., 19 Feb 2026).

The relevant invariant sets are the origin $s$5, corresponding to a skin-localized state when attracting, a stable outer limit cycle corresponding to an extended state, and an unstable inner limit cycle that acts as the basin separatrix in the coexistence window $s$6 (Lin et al., 19 Feb 2026). For the semi-infinite boundary condition $s$7, trajectories are launched by choosing the slope $s$8. The unstable inner cycle projects onto the slope axis and defines a threshold $s$9: for $\mu$0, trajectories flow to the origin; for $\mu$1, they flow to the outer cycle (Lin et al., 19 Feb 2026).

The bifurcation structure consists of a subcritical Hopf bifurcation at $\mu$2 and a saddle-node of limit cycles (SNLC) at $\mu$3. The averaged amplitude equation is

$\mu$4

with nontrivial branches

$\mu$5

and predicted threshold

$\mu$6

The order parameter exhibits a first-order-like jump at SNLC because the appearance of the stable outer cycle and unstable inner separatrix immediately assigns a finite measure of slopes to the extended attractor (Lin et al., 19 Feb 2026).

Outside the coexistence window the quantity saturates: $\mu$7 for $\mu$8 and $\mu$9 for $\rho(s)$0. Within $\rho(s)$1, it decreases continuously as the separatrix shrinks, and as $\rho(s)$2 the unstable inner cycle collapses into the origin so that $\rho(s)$3 (Lin et al., 19 Feb 2026). The authors further state that the jump is robust for any admissible density $\rho(s)$4 with full support. In physical terms, the order parameter distinguishes skin-localized and extended stationary states by basin membership in phase space rather than by a spectral mobility-edge mechanism. The same basin geometry also underlies separatrix-induced long-lived spatial transients and hysteresis under adiabatic sweeps of $\rho(s)$5 (Lin et al., 19 Feb 2026).

5. Basin fraction in delayed and effectively infinite-dimensional systems

In the Mackey–Glass delay system, basin fraction is used to characterize basin organization when the initial condition is a function on a finite interval and the state space is therefore effectively very high-dimensional (Tarigo et al., 2024). The paper extends basin entropy to randomly sampled high-dimensional spaces and pairs it with basin fraction to understand how the basins are structured and how they are intermixed. This joint use is motivated as a way to quantify predictability and to infer the long-term evolution of trajectories as a function of the initial conditions (Tarigo et al., 2024).

The computational strategy is stochastic. Instead of a full regular grid, the box centers and the initial conditions inside boxes are selected randomly; the paper presents this as a Monte-Carlo-like approximation that converges rapidly for the Mackey–Glass model. With $\rho(s)$6, convergence is reported before $\rho(s)$7 boxes and $\rho(s)$8 trajectories per box (Tarigo et al., 2024). The multistability map shows up to 16 coexisting solutions in some parameter regions, but the basin-fraction analysis demonstrates that a high attractor count does not automatically imply high entropy or high unpredictability (Tarigo et al., 2024).

The paper also uses basin fraction as an order-parameter-like summary over parameter cuts. The figure titled “Basin fraction (top) and basin entropy (bottom) for 9 lines in the parameter space” shows that attractors emerge and disappear as parameters change, that basin entropy does not always correlate with basin fraction, that entropy spikes often coincide with the appearance of new attractors, and that constant basin fractions across parameter changes can indicate effective immutability of basin organization even when the attractor itself mutates (Tarigo et al., 2024). In that framework, basin fraction provides the dominance axis missing from raw attractor counts, while basin entropy provides the intermingling axis.

6. Related constructions, extensions, and limitations

A graph-level analogue appears in the Kuramoto literature as the balancing ratio

$\rho(s)$9

defined as the sum of the basin stabilities of all stable balanced states on a graph (Kaiser et al., 2018). Here a balanced state is one with vanishing Kuramoto order parameter,

$$p_{\mathrm{skin}}(\gamma;\mu)=\int_{-s_*(\gamma)}^{s_*(\gamma)} \rho(s)\,ds,$$0

The balancing ratio is explicitly interpreted as the fraction of basin stability occupied by balanced states, or equivalently the probability of ending up in a balanced state from a randomly chosen initial condition (Kaiser et al., 2018). Although the target state is now a class of attractors rather than a single attractor, the construction is basin-fractional in precisely the same sense: it aggregates normalized basin measures into a graph-classification parameter.

A cautionary counterpoint comes from mechanically stable packings of frictionless grains. There the basin of attraction of a packing is the collection of initial points in configuration space at zero packing fraction that map to that packing under a protocol $$p_{\mathrm{skin}}(\gamma;\mu)=\int_{-s_*(\gamma)}^{s_*(\gamma)} \rho(s)\,ds,$$1, and the packing probability is

$$p_{\mathrm{skin}}(\gamma;\mu)=\int_{-s_*(\gamma)}^{s_*(\gamma)} \rho(s)\,ds,$$2

so the full basin volume acts directly as a probabilistic weight (Ashwin et al., 2011). However, the paper also defines a core volume

$$p_{\mathrm{skin}}(\gamma;\mu)=\int_{-s_*(\gamma)}^{s_*(\gamma)} \rho(s)\,ds,$$3

associated with a small hyperspherical region around the packing, and shows that this core volume is only very weakly correlated with the actual packing probability (Ashwin et al., 2011). For $$p_{\mathrm{skin}}(\gamma;\mu)=\int_{-s_*(\gamma)}^{s_*(\gamma)} \rho(s)\,ds,$$4, the reported fit

$$p_{\mathrm{skin}}(\gamma;\mu)=\int_{-s_*(\gamma)}^{s_*(\gamma)} \rho(s)\,ds,$$5

still exhibits enormous scatter. The conclusion is explicit: the small core basin fraction is not a reliable single order parameter for packing probability, because the dominant contributions arise from distant, complicated, thread-like regions of the basin (Ashwin et al., 2011).

The broader implication is that basin-fraction order parameters are powerful when the question is attractor dominance under a specified ensemble of initial conditions, but they are not universally sufficient. In some systems they sharply encode coexistence and bifurcation structure, as in the nonlinear skin-effect problem (Lin et al., 19 Feb 2026). In others they must be paired with entropy to resolve intermingling and practical unpredictability, as in delayed multistability (Tarigo et al., 2024, Daza et al., 2022). In still others, only the full basin geometry—not a local or core fraction—controls observable probabilities (Ashwin et al., 2011).