Koopman Linearizability

• Koopman Linearizability is the transformation of a nonlinear dynamical system into an exact finite-dimensional linear flow via an appropriate state-space embedding.
• On compact invariant sets, exact linearization is characterized by torus symmetry, ensuring quasiperiodic or integrable behavior under a smooth or continuous torus action.
• For attractors, basins, and control-affine systems, additional conditions such as continuous asymptotic phase and Lie bracket commutativity are essential for achieving global linearization.

Searching arXiv for the cited papers to ground the article in current arXiv metadata. Koopman linearizability denotes the existence of a finite-dimensional linear representation of a nonlinear dynamical system through an embedding of its state space into Euclidean space, so that the embedded dynamics evolve exactly under a linear flow. For continuous-time systems with flow $\Phi$, the defining relation is $F\circ \Phi^t=e^{Bt}\circ F$, where $F$ is an embedding and $B$ is a matrix; equivalently, the nonlinear flow is realized as the restriction of a finite-dimensional linear flow on $\mathbb R^n$ (Kvalheim et al., 2023). In the finite-dimensional Koopman viewpoint, the embedding coordinates are observables that evolve linearly under the Koopman semigroup, while in computational settings the same idea appears as a lifted linear system for observables, often only approximately (Shi et al., 2023). Recent work has clarified that exact finite-dimensional Koopman linearization is highly nongeneric: on compact invariant sets it is characterized by torus symmetry, on attractor basins it additionally requires asymptotic phase and transverse linear structure, and for control-affine systems it is constrained by Lie-bracket and distribution-rank conditions (Kvalheim et al., 2023, Deka, 11 Jun 2026).

1. Exact linearizing embeddings and the Koopman formulation

The basic object is a continuous-time dynamical system

$\dot x=f(x),$

with flow $\Phi:\mathbb R\times M\to M$, where $\Phi^t(x)$ denotes the trajectory at time $t$. A map $F:X\to \mathbb R^n$ defined on a $F\circ \Phi^t=e^{Bt}\circ F$0-invariant set $F\circ \Phi^t=e^{Bt}\circ F$1 is a linearizing embedding if it is an embedding in the usual topological or smooth sense and satisfies

$F\circ \Phi^t=e^{Bt}\circ F$2

for some matrix $F\circ \Phi^t=e^{Bt}\circ F$3 (Kvalheim et al., 2023). This intertwining relation is the exact finite-dimensional Koopman representation: the nonlinear dynamics become linear in the embedded coordinates.

In Koopman terms, the state evolution is nonlinear, but observables evolve under a linear operator family. For nonlinear autonomous systems, the Koopman operator family is defined by

$F\circ \Phi^t=e^{Bt}\circ F$4

and its generator is

$F\circ \Phi^t=e^{Bt}\circ F$5

(Shi et al., 2023). Koopman eigenpairs $F\circ \Phi^t=e^{Bt}\circ F$6 satisfy

$F\circ \Phi^t=e^{Bt}\circ F$7

which implies

$F\circ \Phi^t=e^{Bt}\circ F$8

(Shi et al., 2023). In this sense, exact Koopman linearizability corresponds to the existence of a finite set of observables whose span is invariant and closes under the dynamics.

The same principle extends to nonlinear infinite-dimensional systems. For an abstract evolution equation

$F\circ \Phi^t=e^{Bt}\circ F$9

with semiflow $F$0, the Koopman semigroup on bounded continuous functionals is

$F$1

and the Lie generator is

$F$2

(Mauroy, 2021). This is again a linearization principle on observables rather than on state variables.

2. Compact invariant sets: torus actions as the exact criterion

For compact invariant sets, exact linearizability is characterized sharply by torus symmetry. If $F$3 is a compact smooth manifold and $F$4 is a smooth flow, then $F$5 is linearizable by a smooth embedding if and only if $F$6 is a $F$7-parameter subgroup of a smooth torus action on $F$8 (Kvalheim et al., 2023). Concretely, there must exist a smooth torus action

$F$9

and a vector $B$0 in the Lie algebra of $B$1 such that

$B$2

In the compact smooth case, this means that the flow must be generated by a torus action; the dynamics are quasiperiodic or integrable in a strong geometric sense (Kvalheim et al., 2023).

An analogous topological characterization holds for compact finite-dimensional metrizable spaces. If $B$3 is compact finite-dimensional metrizable and $B$4 is continuous, then $B$5 is continuously linearizable if and only if $B$6 is a $B$7-parameter subgroup of a continuous torus action on $B$8 with finitely many orbit types (Kvalheim et al., 2023). The finiteness of orbit types excludes pathological actions in the purely topological setting.

The compact-case proof uses equivariant embedding theorems of Mostow–Palais. If a torus action exists, one can embed equivariantly into $B$9, producing a linear flow. Conversely, if the flow is already linearly embedded in $\mathbb R^n$0, compactness forces the relevant linear dynamics to lie in the purely imaginary spectral part, and the closure of the one-parameter subgroup is a torus (Kvalheim et al., 2023). After a change of basis, the compact linear dynamics reduce to

$\mathbb R^n$1

on $\mathbb R^n$2, with $\mathbb R^n$3; the closure of $\mathbb R^n$4 is then a torus $\mathbb R^n$5 (Kvalheim et al., 2023).

A useful recognition criterion is also available. If there exists a smooth map

$\mathbb R^n$6

and a frequency vector $\mathbb R^n$7 with rationally independent components such that

$\mathbb R^n$8

then $\mathbb R^n$9 is smoothly conjugate to a constant vector field on $\dot x=f(x),$0, hence smoothly linearizable (Kvalheim et al., 2023). The same result is equivalent to having $\dot x=f(x),$1 smooth nowhere-zero Koopman eigenfunctions with rationally independent imaginary parts. This suggests that a finite set of Koopman eigenfunctions can certify exact linearizability, but only in a very restrictive torus-based setting.

3. Attractors and basins: asymptotic phase and transverse linear structure

For noncompact invariant sets that arise as basins of attraction, the conditions are stronger. Let $\dot x=f(x),$2 be the basin of attraction of a globally asymptotically stable compact invariant set $\dot x=f(x),$3, with $\dot x=f(x),$4 finite-dimensional, locally compact, and metrizable. Then $\dot x=f(x),$5 is linearizable by a topological embedding if and only if two conditions hold: $\dot x=f(x),$6 has continuous asymptotic phase, and $\dot x=f(x),$7 is a $\dot x=f(x),$8-parameter subgroup of a continuous torus action on $\dot x=f(x),$9 with finitely many orbit types (Kvalheim et al., 2023). Continuous asymptotic phase means there exists a continuous retraction

$\Phi:\mathbb R\times M\to M$0

Any such topological linearizing embedding is automatically proper (Kvalheim et al., 2023).

The smooth theorem is stricter. If $\Phi:\mathbb R\times M\to M$1 is a smooth manifold and $\Phi:\mathbb R\times M\to M$2 is a globally asymptotically stable compact invariant set, then $\Phi:\mathbb R\times M\to M$3 is linearizable by a smooth embedding if and only if three conditions hold: $\Phi:\mathbb R\times M\to M$4 is a smoothly embedded submanifold with a smooth asymptotic phase map $\Phi:\mathbb R\times M\to M$5; $\Phi:\mathbb R\times M\to M$6 is a $\Phi:\mathbb R\times M\to M$7-parameter subgroup of a smooth torus action on $\Phi:\mathbb R\times M\to M$8; and there exist $\Phi:\mathbb R\times M\to M$9, a matrix $\Phi^t(x)$0 with all eigenvalues having negative real part, an open neighborhood $\Phi^t(x)$1, and a smooth map

$\Phi^t(x)$2

such that

$\Phi^t(x)$3

whenever $\Phi^t(x)$4 (Kvalheim et al., 2023). In other words, smooth basin linearization requires smooth phase synchronization to the attractor, torus dynamics on the attractor, and a smooth linearization of the transverse stable directions.

The global embedding for basin problems is assembled from an embedding on the attractor, a transverse coordinate, and a time-to-impact function $\Phi^t(x)$5. One explicit form is

$\Phi^t(x)$6

and in the smooth case the intertwining relation becomes

$\Phi^t(x)$7

(Kvalheim et al., 2023). The requirement that the transverse matrix $\Phi^t(x)$8 have eigenvalues with negative real part reflects stability off the attractor.

These basin theorems extend the spirit of Hartman–Grobman and Floquet theory beyond their classical local settings. Instead of a local conjugacy near a hyperbolic equilibrium or periodic orbit, one obtains a global embedding of the entire basin into a linear flow, provided the attractor has the required torus structure, asymptotic phase, and transverse linearization (Kvalheim et al., 2023).

4. Topological obstructions, equilibria, and geometric normal forms

Exact smooth linearizability on compact manifolds imposes strong topological restrictions. If $\Phi^t(x)$9 is smoothly linearizable on a connected compact manifold and has an isolated equilibrium, then $t$0 must be even-dimensional and the Hopf index at any isolated equilibrium must be $t$1 (Kvalheim et al., 2023). This is obtained by reducing locally to a skew-symmetric linear vector field coming from a torus action.

Several corollaries follow. If $t$2 is odd-dimensional and has at least one isolated equilibrium, then it cannot be smoothly linearizable. If a compact smooth manifold supports a smooth linearizable flow with finitely many equilibria, then

$t$3

If $t$4 is a compact connected surface and there are only isolated equilibria, then smooth linearizability is possible only for

$t$5

(Kvalheim et al., 2023). These are checkable obstructions derived from parity, Hopf index, and Euler characteristic.

The topological compact case also admits a geometric normal form. A compact continuous flow is topologically linearizable if and only if it is topologically conjugate to a quasiperiodic pinched torus family (Kvalheim et al., 2023). This description organizes topologically linearizable compact flows as quotients of torus-like families in which some circle factors are collapsed over specified closed subsets.

For basins, asymptotic phase yields another obstruction. A necessary condition for topological linearization is the existence of a continuous asymptotic phase map. The paper gives a planar spiral-to-circle flow where the attractor circle is linearizable, but the basin is not, because asymptotic phase fails (Kvalheim et al., 2023). A plausible implication is that linearizability of the attractor alone is insufficient; the geometry of the stable foliation must also be compatible with the embedding.

5. Computational linearization: spectral approximations and comparison with Carleman methods

Computational work often uses the language of Koopman linearization more broadly, referring to finite-dimensional approximations rather than exact embeddings. In one such formulation, the nonlinear autonomous system

$t$6

is lifted to a linear system for observables through a spectral discretization of the Koopman generator using Chebyshev or Gauss–Lobatto collocation and the differentiation matrix (Shi et al., 2023). For a scalar observable $t$7, the lifted system is

$t$8

with solution

$t$9

(Shi et al., 2023).

In one dimension, the finite matrix is

$F:X\to \mathbb R^n$0

In two dimensions,

$F:X\to \mathbb R^n$1

and for general dimension $F:X\to \mathbb R^n$2,

$F:X\to \mathbb R^n$3

(Shi et al., 2023). This is explicitly a linear system for observables rather than a direct state-space linearization.

The method is compared with Carleman linearization. The relationship is precise: the Carleman matrix is a transposed finite section approximation of the Koopman operator generator in the polynomial basis (Shi et al., 2023). Carleman linearization uses polynomial or tensor-product monomials $F:X\to \mathbb R^n$4, truncating the monomial hierarchy at order $F:X\to \mathbb R^n$5, whereas Koopman spectral linearization uses spectral collocation and a Chebyshev–Lagrange basis on Gauss–Lobatto points (Shi et al., 2023). The reported matrix sizes are $F:X\to \mathbb R^n$6 for the Koopman spectral method and $F:X\to \mathbb R^n$7, approximated as $F:X\to \mathbb R^n$8, for Carleman lifting (Shi et al., 2023).

The numerical experiments cover five systems: $F:X\to \mathbb R^n$9, $F\circ \Phi^t=e^{Bt}\circ F$00, the simple pendulum, the Lotka–Volterra model, and the Kraichnan–Orszag model (Shi et al., 2023). Reported observations are that error generally decays exponentially with truncation order $F\circ \Phi^t=e^{Bt}\circ F$01; error versus radius $F\circ \Phi^t=e^{Bt}\circ F$02 has a bowl-shaped or V-shaped curve; Koopman spectral linearization is typically faster than Carleman, especially in $F\circ \Phi^t=e^{Bt}\circ F$03; and for non-polynomial systems Carleman performs much worse because of the polynomial or Taylor truncation issue (Shi et al., 2023). At the same time, the method has important limitations: it is built for nonlinear autonomous dynamical systems, tailored to smooth scalar observables, relies critically on choosing a suitable radius $F\circ \Phi^t=e^{Bt}\circ F$04, and remains expensive in high dimension because of tensor-product growth $F\circ \Phi^t=e^{Bt}\circ F$05 (Shi et al., 2023).

This computational literature therefore concerns finite-dimensional approximation of Koopman linearization, not the exact geometric notion characterized by torus actions and asymptotic phase. A common misconception is to conflate successful lifted linear prediction with exact finite-dimensional Koopman linearizability. The comparison suggests that the two notions should be kept distinct.

6. Infinite-dimensional systems and control-affine extensions

For nonlinear infinite-dimensional systems such as PDEs and integro-differential equations, the Koopman framework operates on bounded continuous functionals rather than on finite-dimensional state coordinates. The finite-dimensional approximation is obtained by compressing the Koopman semigroup or Lie generator onto a subspace

$F\circ \Phi^t=e^{Bt}\circ F$06

yielding

$F\circ \Phi^t=e^{Bt}\circ F$07

with matrix representations $F\circ \Phi^t=e^{Bt}\circ F$08 and $F\circ \Phi^t=e^{Bt}\circ F$09 (Mauroy, 2021). Given data pairs $F\circ \Phi^t=e^{Bt}\circ F$10, the least-squares approximation is

$F\circ \Phi^t=e^{Bt}\circ F$11

which generalizes EDMD to infinite-dimensional systems (Mauroy, 2021). Eigenvalues of $F\circ \Phi^t=e^{Bt}\circ F$12 approximate eigenvalues of $F\circ \Phi^t=e^{Bt}\circ F$13, generator eigenvalues are estimated by

$F\circ \Phi^t=e^{Bt}\circ F$14

and Koopman eigenfunctionals are approximated by right eigenvectors of $F\circ \Phi^t=e^{Bt}\circ F$15 (Mauroy, 2021).

The same framework yields a linear identification method for nonlinear PDEs. If

$F\circ \Phi^t=e^{Bt}\circ F$16

then with basis functionals $F\circ \Phi^t=e^{Bt}\circ F$17 and $F\circ \Phi^t=e^{Bt}\circ F$18, one estimates

$F\circ \Phi^t=e^{Bt}\circ F$19

(Mauroy, 2021). Under the stated linear-independence assumptions, the paper proves that

$F\circ \Phi^t=e^{Bt}\circ F$20

(Mauroy, 2021). This identifies the Lie generator as the linear object through which nonlinear infinite-dimensional dynamics can be analyzed and learned.

For control-affine nonlinear systems,

$F\circ \Phi^t=e^{Bt}\circ F$21

the question becomes whether there exists a diffeomorphism $F\circ \Phi^t=e^{Bt}\circ F$22 into a lifted space such that

$F\circ \Phi^t=e^{Bt}\circ F$23

with the same input $F\circ \Phi^t=e^{Bt}\circ F$24 and no input feedback transformation (Deka, 11 Jun 2026). Recent differential-geometric results show that exact finite-dimensional Koopman linearizability is highly constrained. The central distribution is

$F\circ \Phi^t=e^{Bt}\circ F$25

where

$F\circ \Phi^t=e^{Bt}\circ F$26

Necessary conditions for Koopman linearization are: constant rank of the span of $F\circ \Phi^t=e^{Bt}\circ F$27 on a neighborhood, and commutativity

$F\circ \Phi^t=e^{Bt}\circ F$28

for all $F\circ \Phi^t=e^{Bt}\circ F$29 on that neighborhood (Deka, 11 Jun 2026). The same conditions are sufficient for a weaker notion of Koopman linearizability on a control-invariant manifold, and with the added condition that the span has dimension $F\circ \Phi^t=e^{Bt}\circ F$30, they become necessary and sufficient for Koopman linearizability to a controllable linear system (Deka, 11 Jun 2026).

These controlled results show that linearizability of the autonomous drift alone is not enough. One example has the same drift as a Koopman-linearizable case but a different input direction, causing the rank condition to fail; the system is then not Koopman linearizable even though the autonomous part can be Koopman-lifted (Deka, 11 Jun 2026). Another example is feedback linearizable but not Koopman linearizable because the bracket condition fails (Deka, 11 Jun 2026). This establishes a sharp distinction between Koopman linearization and classical feedback linearization.

7. Conceptual scope and limits

Across compact flows, attractor basins, infinite-dimensional systems, and control-affine dynamics, a common theme is that exact finite-dimensional Koopman linearizability is a structural property rather than a generic approximation phenomenon. On compact invariant sets, it is equivalent to hidden torus symmetry; on basins, it requires torus dynamics on the attractor together with asymptotic phase and transverse stable linearization; on control-affine systems, it requires constant-rank and commutativity conditions on iterated Lie brackets of the control vector fields with the drift (Kvalheim et al., 2023, Deka, 11 Jun 2026).

The role of topology is central. Compactness forces the relevant spectral part to be purely imaginary in compact linearizable cases, while finite-dimensionality and metrizability are required for the topological characterization (Kvalheim et al., 2023). The role of invariant manifold theory is equally central: for smooth basins, the attractor must be a smoothly embedded invariant manifold with smooth asymptotic phase, and the paper remarks more generally that it must be a normally hyperbolic invariant manifold and even an eventually relatively $F\circ \Phi^t=e^{Bt}\circ F$31-NHIM (Kvalheim et al., 2023). In the control setting, invariant manifolds reappear as the natural domains on which weaker Koopman linearizations can exist (Deka, 11 Jun 2026).

A plausible implication is that the phrase “Koopman linearizability” has two distinct usages in the literature. One usage refers to exact finite-dimensional embeddings satisfying an intertwining relation, with stringent geometric and topological prerequisites. The other refers to finite-dimensional lifted approximations of the Koopman generator or semigroup, often used for prediction, spectral estimation, or identification (Shi et al., 2023, Mauroy, 2021). The former is an exact equivalence problem; the latter is an approximation methodology. Maintaining this distinction is essential for interpreting both positive computational results and impossibility theorems.