Polyattractors in Dynamical Systems

Updated 9 July 2026

Polyattractors are configurations with multiple attractors and corresponding basins, defined variably across dynamical systems, self-affine geometry, and cosmological models.
They emerge in classical dynamics via coexisting periodic sinks and in logarithmic iterations using distinct Lambert W branches, with implications for stability and bifurcation analysis.
In geometric settings, polyattractors appear as self-affine attractors generated by multiple digits, while in hyperchaos and cosmology they reveal composite, multifractal, or polynomial asymptotic behavior.

Polyattractors are attractor configurations or attractor-like structures that occur in several distinct research programs, but the term is not used uniformly across them. In one line of work it denotes multiple coexisting attractors together with their basins of attraction; in iterative logarithmic dynamics it refers to the coexistence of several attracting fixed points for the same logarithmic base; in self-affine geometry it can denote an attractor generated by multiple digits; and in some later literatures it is extended to dual-multifractal hyperchaotic sets or to polynomial subclasses of cosmological $\alpha$ -attractors (Wallisch, 2010, Negaard-Paper, 2019, Protasov et al., 2020). This suggests that “polyattractor” is best understood as a family of related usages centered on multiplicity, branching, or composite attractor geometry, rather than as a single invariant definition.

1. Terminology and conceptual scope

In the most standard dynamical-systems sense represented here, a polyattractor configuration is one in which a system admits multiple coexisting attractors together with their basins of attraction. That formulation is explicit in the multiflow literature, where the point is not merely the existence of several invariant sets, but the simultaneous organization of forward asymptotics by several attracting neighborhoods and their associated $\omega$ -limit sets (Negaard-Paper, 2019). By contrast, the self-affine literature uses “polyattractor” in a combinatorial-geometric sense: a self-affine attractor with $|D|=m\ge 2$ digits, with the extensively studied $2$-attractors corresponding to the case $m=2$ (Protasov et al., 2020).

A further extension appears in hyperchaos, where “polyattractor” is used for a single invariant set whose invariant measure behaves as a superposition of two intermingled multifractals, rather than for several separate attractors with disjoint basins (Harikrishnan et al., 2016). In inflationary cosmology, “polyattractors” or “polynomial $\alpha$ -attractors” denote models whose potentials approach a plateau polynomially and can interpolate between polynomial and exponential attractor regimes as a parameter varies (Kallosh et al., 8 Jul 2026, Kallosh et al., 2 Dec 2025).

Literature	Meaning of “polyattractor”	Representative source
Multiflows and relations	Multiple coexisting attractors with basins	(Negaard-Paper, 2019)
Self-affine geometry	Attractor with $m\ge 2$ digits	(Protasov et al., 2020)
Hyperchaos	One invariant set with dual multifractal structure	(Harikrishnan et al., 2016)
Cosmology	Polynomial $\alpha$ -attractor regime	(Kallosh et al., 8 Jul 2026)

The terminological spread matters because claims about existence, stability, basin structure, or robustness depend strongly on which usage is intended. In some contexts the multiplicity is spatial and basin-theoretic; in others it is branch-theoretic, combinatorial, multifractal, or asymptotic.

2. Coexisting attractors in classical dynamics

A direct realization of polyattractors in the coexistence sense is provided by the almost-conservative Hénon map. For the parameter choice $a=1.0176$ and $b=1-10^{-5}$ , the system exhibits $\omega$ 0 coexisting attracting periodic orbits with a total of $\omega$ 1 periodic points. Two of these are low-period attractors—an attracting fixed point $\omega$ 2 and an attracting period- $\omega$ 3 orbit $\omega$ 4—while the remaining $\omega$ 5 sinks are organized into three families: $\omega$ 6 with $\omega$ 7 attracting orbits of periods $\omega$ 8, $\omega$ 9 with $|D|=m\ge 2$ 0 attracting orbits of periods $|D|=m\ge 2$ 1, and $|D|=m\ge 2$ 2 with $|D|=m\ge 2$ 3 attracting orbits of periods $|D|=m\ge 2$ 4 (Falcolini et al., 8 Jun 2026). Here the polyattractor picture is literal: many separate periodic sinks coexist in a bounded region of phase space.

A related but structurally different coexistence mechanism appears in the $|D|=m\ge 2$ 5D polymatrix replicator family on $|D|=m\ge 2$ 6. In that model, the route to chaos proceeds through a Belyakov transition, a supercritical Hopf bifurcation, a Shilnikov-type homoclinic mechanism, and suspended horseshoes. Over parameter ranges such as $|D|=m\ge 2$ 7 and $|D|=m\ge 2$ 8, a boundary sink coexists with an interior chaotic set, and the basins are separated by the two-dimensional invariant manifold $|D|=m\ge 2$ 9 (Peixe et al., 2021). The same parameter value can therefore support qualitatively distinct long-time regimes: convergence to a boundary equilibrium in one basin and strange-attractor dynamics in another.

For polynomial maps of $2$0, coexistence can be organized not only pointwise in parameter space but as large-scale geometry in parameter space. There are codimension-$2$1 laminations of maps with at least $2$2 period-doubling Cantor attractors, and codimension-$2$3 loci with two period-doubling Cantor attractors; the leaves are real-analytic, have uniform diameter, and accumulate on the codimension-$2$4 tangency locus of a saddle point (Palmisano, 2019). This places polyattractors within homoclinic-tangency theory and renormalization, rather than only within isolated examples.

The multichaos framework pushes this logic to an explicitly replicated form. In that setting, multichaos means that there exist two or more disjoint positive invariant sets and the solution in every disjoint set is chaotic. The multiple logistic map realizes multistability, multiperiodicity, and multichaos by tiling $2$5 into invariant unit intervals; the sawtooth-modified Lorenz system similarly constructs infinitely many disjoint invariant sets, each carrying Lorenz-type chaos (Liu et al., 2014). The defining feature is that initial conditions select among different invariant cells, and each cell reproduces the same dynamical type.

3. Branch-induced polyattractors in logarithmic iteration

The paper “The attractor structure of logarithmic iterations in the complex plane” gives perhaps the most explicit use of the word “polyattractor” in a concrete complex map. The iteration is

$2$6

with $2$7 and $2$8, numerically taken on the principal branch with $2$9 (Wallisch, 2010). In this setting, a polyattractor means that multiple distinct attracting fixed points coexist for the same base $m=2$ 0.

The fixed points satisfy

$m=2$ 1

which can be rewritten via Lambert $m=2$ 2 as

$m=2$ 3

This representation makes the branch structure explicit: each branch $m=2$ 4 gives a candidate fixed point, and the local stability condition is

$m=2$ 5

Accordingly, polyattractor behavior is controlled by which Lambert $m=2$ 6 branches yield $m=2$ 7.

For real $m=2$ 8, the main bifurcation occurs at $m=2$ 9, i.e.

$\alpha$ 0

When $\alpha$ 1, there are two real fixed points but only one is attracting, namely the $\alpha$ 2 branch; as $\alpha$ 3, that attracting point becomes very large, with examples $\alpha$ 4 for $\alpha$ 5, $\alpha$ 6 for $\alpha$ 7, $\alpha$ 8 for $\alpha$ 9, and $m\ge 2$ 0 for $m\ge 2$ 1. When $m\ge 2$ 2, the two relevant branches become complex conjugates, producing two attracting fixed points. For $m\ge 2$ 3, the attractors are

$m\ge 2$ 4

and the local multiplier has modulus about $m\ge 2$ 5 and argument about $m\ge 2$ 6, explaining the empirically observed logarithmic spirals into the fixed points.

Under the principal-branch rule, the basins are cleanly organized. For $m\ge 2$ 7, the upper half-plane converges to the attractor with positive imaginary part and the lower half-plane to its conjugate. The negative real axis acts as a branch cut; crossing it changes $m\ge 2$ 8 by $m\ge 2$ 9 and can redirect the orbit to the conjugate basin. In that sense, the polyattractor structure is not generated by critical-point dynamics, as in polynomial iteration, but by multivaluedness of $\alpha$ 0 and the associated Lambert $\alpha$ 1 branches.

The same study also reports that principal-branch iteration for $\alpha$ 2 typically yields a single real attractor approaching $\alpha$ 3 along a nearly linear locus as $\alpha$ 4, while negative and purely imaginary bases produce attractors lying on smooth curves. This restricts the polyattractor phenomenon: it is prominent for many real bases $\alpha$ 5, but not universal across all bases under the principal-branch convention.

4. Topological, set-valued, and complex-analytic frameworks

The multiflow framework provides a rigorous topological formalism for polyattractors in systems without forward uniqueness. A multiflow is a closed subset $\alpha$ 6, equivalently a family of closed relations $\alpha$ 7 satisfying $\alpha$ 8 and $\alpha$ 9. An attracting neighborhood is a compact set $a=1.0176$ 0 for which there exists $a=1.0176$ 1 with $a=1.0176$ 2 for all $a=1.0176$ 3, and the associated attractor is

$a=1.0176$ 4

Its basin is

$a=1.0176$ 5

Because $a=1.0176$ 6 aggregates all admissible futures, basins in multiflows need not partition the phase space: if some forward selections from $a=1.0176$ 7 approach $a=1.0176$ 8 and others approach $a=1.0176$ 9, then $b=1-10^{-5}$ 0 belongs to neither basin under the “all futures” definition (Negaard-Paper, 2019). This is one of the sharpest formal differences between classical multistability and polyattractors in set-valued dynamics.

A related hyperspace formalism arises for continuous multivalued maps $b=1-10^{-5}$ 1. The induced operator

$b=1-10^{-5}$ 2

acts on the hyperspace $b=1-10^{-5}$ 3 with Hausdorff metric. If $b=1-10^{-5}$ 4 is an attractor of $b=1-10^{-5}$ 5, then it is asymptotically stable; moreover, in locally compact complete metric spaces there exists a metric equivalent to the Hausdorff metric on which $b=1-10^{-5}$ 6 is a contraction on the basin (Rypka, 2017). This gives a Banach-type converse theorem for multivalued attractors and shows that “compound” attracting sets in hyperspace inherit a strong stability theory.

In several complex variables, codimension-one attracting sets in $b=1-10^{-5}$ 7 furnish another rigorous polyattractor setting. For a holomorphic endomorphism $b=1-10^{-5}$ 8 of degree $b=1-10^{-5}$ 9, an attracting set is

$\omega$ 00

for some trapping region $\omega$ 01 with $\omega$ 02. When $\omega$ 03 has small topological degree on $\omega$ 04, there is an attracting current $\omega$ 05 and the invariant measure

$\omega$ 06

is mixing with entropy $\omega$ 07; the attracting current has a continuous potential with logarithmic modulus

$\omega$ 08

and equidistribution toward $\omega$ 09 holds with exponential speed (Daurat et al., 2015). A complementary result shows that, in codimension one, small topological degree implies bounded quasi-potential for the attracting current and therefore non-pluripolarity of the attracting set; abundant examples are constructed in $\omega$ 10 (Daurat, 2013).

A further composite-attractor viewpoint is given by heteroclinic networks in generalized Lotka–Volterra models. There the “heteroclinic channel” formed by the strongest unstable connections among a finite set of hyperbolic saddles constitutes part of a Milnor attractor and is predominantly asymptotically stable under explicit eigenvalue inequalities such as $\omega$ 11 and $\omega$ 12 (Rodrigues, 2017). In this usage, a polyattractor is not a collection of disjoint sinks but a single attracting object assembled from multiple invariant pieces and connecting trajectories.

5. Geometric and self-affine polyattractors

In self-affine geometry, the basic object is the compact set

$\omega$ 13

equivalently the solution of

$\omega$ 14

Here a polyattractor is a self-affine attractor with $\omega$ 15 digits, and the extensively analyzed case of $\omega$ 16-attractors corresponds to $\omega$ 17 (Protasov et al., 2020). This is a genuinely different meaning from coexistence of several basins: multiplicity is encoded in the digit set and the associated refinement equation

$\omega$ 18

The $\omega$ 19-attractor literature obtains a particularly sharp classification. If $\omega$ 20 is isotropic and $\omega$ 21 is odd, all isotropic $\omega$ 22-attractors in $\omega$ 23 are parallelepipeds. If $\omega$ 24 is even, then up to affine similarity there are exactly three isotropic $\omega$ 25-attractors: a parallelepiped, the direct product of $\omega$ 26 planar dragons, and the direct product of $\omega$ 27 planar bears. More generally, a $\omega$ 28-attractor is uniquely defined up to affine similarity by the spectrum of the dilation matrix, and the number $\omega$ 29 of distinct $\omega$ 30-attractors satisfies

$\omega$ 31

This is a classification theory for digit-generated attractors, not for multistable dynamics.

The closely related theory of simple tiles and attractors further restricts which polyhedral self-similar attractors can occur. Every convex attractor in $\omega$ 32 is a parallelepiped, and every polygonal attractor in $\omega$ 33 is a parallelogram. In dimension one, finite unions of intervals are classified via direct sums of arithmetic progressions, and tensor products of such one-dimensional constructions produce higher-dimensional disconnected polyattractors (Zaitseva, 2020). The emphasis here is on geometric realizability, digit sets, and tiling, rather than on basin decomposition.

Piecewise translation maps provide a different “poly-” geometry. For a piecewise translation $\omega$ 34 on a compact region $\omega$ 35, one defines nested compact sets

$\omega$ 36

and the attractor

$\omega$ 37

When the number of pieces is $\omega$ 38 and the associated torus rotation is ergodic, the system is finite type, $\omega$ 39 is reached in uniformly bounded finite time, and $\omega$ 40 is a region exchange on finitely many pieces. In stochastic piecewise translations, by contrast, the random attractor for double rotations on the circle has Lebesgue measure zero almost surely (Volk, 2017). This literature uses “polyattractor” in a natural descriptive sense for multi-piece polygonal or Cantor-like attractors induced by piecewise isometries.

The hyperchaos literature explicitly warns that a polyattractor need not mean several separate invariant sets. For hyperchaotic Chen, Mackey–Glass, and Ikeda systems, the evidence is a crossover in the scaling of weighted box-counting or correlation sums, yielding two effective $\omega$ 41 values and two overlapping singularity spectra $\omega$ 42. In that geometric sense, a hyperchaotic attractor is described as a single invariant set composed of two intermingled multifractals, distinct from multistability and also distinct from multi-scroll systems (Harikrishnan et al., 2016). The multiplicity is therefore internal to the invariant measure rather than external in basin structure.

In inflationary cosmology, “polyattractors” are again a different object: polynomial $\omega$ 43-attractors or P-models. A representative family is

$\omega$ 44

embedded into $\omega$ 45-attractor geometry by

$\omega$ 46

For small $\omega$ 47, the last $\omega$ 48 e-folds occur away from the pole and one obtains the polynomial-attractor prediction

$\omega$ 49

whereas for large $\omega$ 50 the model reduces to the standard exponential $\omega$ 51-attractor regime

$\omega$ 52

The interpolation parameter $\omega$ 53 therefore scans continuously between polynomial and exponential attractor regimes (Kallosh et al., 8 Jul 2026). A related classification places P-models within a broader family of singular $\omega$ 54-attractors: T- and E-models are non-singular with exponential approach to the plateau, P-models have regular potentials but derivative singularities at the boundary and hence polynomial approach, and S-models allow the potential itself to become singular (Kallosh et al., 2 Dec 2025).

A separate terminological caution concerns “zero attractors” of polynomial sequences. In that literature, the attractor is not dynamical at all but the accumulation set of zeros of a sequence of polynomials. For the recurrence

$\omega$ 55

the zero attractor is characterized by modulus-balance curves among residue contributions in a Cauchy-integral representation (Belbachir et al., 2020). This is a distinct usage of “attractor” and is not a polyattractor notion in the dynamical-systems sense.

Taken together, these literatures show that “polyattractor” is a technically productive but semantically non-uniform term. In its strongest and most stable meaning, it denotes coexistence of several attractors and basins. In other settings it denotes branch multiplicity, digit multiplicity, composite invariant sets, dual multifractality, or polynomial asymptotics of cosmological plateau potentials. The common thread is structured multiplicity in asymptotic organization; the precise ontology of that multiplicity is context-dependent.