Simultaneous Linearization

• Simultaneous linearization is a framework that finds a single transformation or coordinate system to linearize several nonlinear objects at once, ensuring consistent dynamics.
• It spans diverse fields—from holomorphic dynamics and Lie algebra actions to delay systems and microwave photonics—using techniques like commutation, semisimplicity, and arithmetic control.
• The approach bridges local fixed-point methods and global conjugacy, enabling structure preservation in optimal control, homogenization, and variational asymptotics.

Simultaneous linearization denotes a family of procedures in which one seeks a single transformation, coordinate system, model map, or operating point that linearizes several nonlinear objects at once. In the cited literature, the phrase ranges from common Abel or Koenigs coordinates for commuting holomorphic self-maps of the disc, to analytic Darboux coordinates for whole Lie algebra actions, to one conjugacy sending a random family of hyperbolic maps to an affine action, to direct optimal-control transcriptions in which delayed variables are locally linearized, and to material limits in which linearization is performed together with homogenization and dimension reduction (Contreras et al., 4 Aug 2025, Miranda, 2015, Brown et al., 8 Jan 2026, Ritschel et al., 2024, Cherdantsev et al., 26 Nov 2025).

1. Scope and recurring structures

Across these settings, the common requirement is not merely that each object admit some linear form, but that the same linearizing mechanism work for all members of a family. What changes from field to field is the identity of the nonlinear objects and the notion of linear target.

Setting	Common linearizing object	Linear target
Holomorphic dynamics on $\mathbb D$	One holomorphic $h$	Translations $w\mapsto w+c$
Analytic symplectic or Poisson actions	One analytic coordinate system	Linear action in Darboux or Weinstein form
Nearly integrable cylinder dynamics	One conjugacy $H$	Integrable pair $(U_0,T_\alpha)$
Random hyperbolic rigidity	One conjugacy $h$	Affine or isometric action
Delay optimal control	Local Taylor linearization of delayed variables	Implicit differential equations
Microwave photonics	One DP-MZM operating point and OCB processing	Cancellation of third-order distortion families

A recurring distinction is between formal and holomorphic or smooth simultaneous linearization, between local fixed-point problems and global conjugacy problems, and between exact linearization and approximations that trade fidelity for tractability. The literature also shows that commutation, semisimplicity, Diophantine conditions, vanishing Lyapunov exponents, or structure-preserving constraints may replace one another as the decisive compatibility mechanism, depending on the domain (Raissy, 2010, DeWitt, 2020, Nowicki et al., 2023, 2207.14547).

2. Holomorphic dynamics: common Abel coordinates and centralizers

In one-variable holomorphic dynamics, a central formulation asks for a holomorphic map $h$ and constants $c_1,c_2\in\mathbb C$, not both zero, such that

$h\circ\varphi=h+c_1,\qquad h\circ\psi=h+c_2.$

For a single non-elliptic self-map of $\mathbb D$, this reduces to the Abel equation $h$0. For commuting maps, simultaneous linearization asks whether the same $h$1 linearizes both maps. The sharpest result in the provided corpus concerns parabolic self-maps of $h$2 with zero hyperbolic step: if $h$3 is such a map, then its canonical model is $h$4, and a non-identity $h$5 commutes with $h$6 if and only if $h$7 and $h$8 have the same Denjoy–Wolff point and

$h$9

for some $w\mapsto w+c$0. Equivalently, commuting maps are exactly Cowen pseudo-iterates in the Koenigs coordinate of $w\mapsto w+c$1 (Contreras et al., 4 Aug 2025).

This zero-step parabolic case is rigid because the model domain is the whole plane $w\mapsto w+c$2, not a strip or half-plane. The centralizer

$w\mapsto w+c$3

becomes an abelian semigroup modeled by addition of complex constants. If

$w\mapsto w+c$4

then $w\mapsto w+c$5, so commutativity is immediate in model coordinates. The same work shows that if $w\mapsto w+c$6 is univalent then every element of $w\mapsto w+c$7 is univalent, while if $w\mapsto w+c$8 is not univalent, the identity is an isolated point of the centralizer. It also gives the coefficient formula

$w\mapsto w+c$9

and identifies the centralizer with a closed additive subsemigroup of $H$0 via $H$1. The paper states explicitly that these conclusions fail in positive-step parabolic dynamics, so the zero-step theorem is not a generic parabolic statement. It also answers a question of Gentili and Vlacci and refines Cowen’s pseudo-iteration framework (Contreras et al., 4 Aug 2025).

A related but distinct global phenomenon appears for uniformly quasiregular Schröder equations. If $H$2 with $H$3 strongly automorphic and $H$4, and if the relevant infinitesimal spaces of $H$5 are simple and $H$6-homogeneous, then for every repelling periodic point $H$7 there exists a translation $H$8 such that $H$9 is a linearizer of an iterate of $(U_0,T_\alpha)$0 at $(U_0,T_\alpha)$1. In that sense, the same global map $(U_0,T_\alpha)$2 simultaneously linearizes many periodic points, exactly as $(U_0,T_\alpha)$3 does for $(U_0,T_\alpha)$4 in the classical planar prototype (Fletcher et al., 2018).

3. Local analytic and KAM formulations for commuting actions

For germs of biholomorphisms of $(U_0,T_\alpha)$5 fixing the origin, simultaneous linearization is the existence of one formal or holomorphic change of coordinates $(U_0,T_\alpha)$6 such that

$(U_0,T_\alpha)$7

Raissy proves that if $(U_0,T_\alpha)$8 commute and their linear parts are almost simultaneously Jordanizable, then they are simultaneously formally linearizable. In the diagonalizable case, he introduces a simultaneous Brjuno-type condition and proves that commutation plus that joint arithmetic condition imply holomorphic simultaneous linearization. The same work also identifies a canonical non-resonant simultaneous formal linearization by setting the coefficients on simultaneous resonances to zero, and presents this as a multidimensional answer to a problem raised by Moser (Raissy, 2010).

In analytic symplectic geometry, simultaneous linearization acquires an equivariant geometric form. Miranda proves that if a semisimple Lie algebra acts analytically by symplectic vector fields and fixes a point $(U_0,T_\alpha)$9, then there exist local analytic coordinates

$h$0

such that the whole representation is linear and

$h$1

This yields simultaneous analytic linearization for Hamiltonian vector fields near a common zero. The proof combines analytic linearization of semisimple Lie algebra actions, a Moser path, and averaging over a compact real form of the complexified action. The same note also emphasizes a limitation: the smooth analogue fails in general for noncompact semisimple algebras, and survives only for semisimple Lie algebras of compact type (Miranda, 2015).

A KAM-type version appears for commuting nearly integrable diffeomorphisms of the cylinder. If $h$2 is close to a twist map $h$3, $h$4 is close to $h$5, $h$6 is Diophantine, $h$7 has the intersection property, and $h$8 is Lipschitz semi-conjugate to $h$9, then there exists one $h$0 diffeomorphism $h$1 such that

$h$2

The paper also provides counterexamples showing that the intersection property and semi-conjugacy condition are not decorative hypotheses. This makes simultaneous linearization a local rigidity statement for a $h$3-action generated by commuting twist-type maps (Chen et al., 2021).

4. Global rigidity: quasiregular, random, and isotropic-manifold settings

A global rigidity interpretation of simultaneous linearization is especially explicit in random dynamics. For small smooth perturbations of a generating family of isometries on a closed isotropic manifold, DeWitt shows that the perturbations are simultaneously conjugate to isometries if and only if the associated uniform Bernoulli random walk has vanishing Lyapunov exponents in the sense encoded by the condition $h$4 for a sequence of ergodic stationary measures. The result extends a theorem of Dolgopyat and Krikorian from $h$5 to real, complex, and quaternionic projective spaces and the Cayley plane, and the paper identifies and repairs a gap in the earlier proof by introducing the strain tensor as the correct object for measuring nearness to an isometry (DeWitt, 2020).

A more recent random hyperbolic rigidity theorem proves an analogous statement for expanding maps on $h$6 and Anosov diffeomorphisms on $h$7. If the Lyapunov spectrum of the $h$8-stationary SRB measure coincides with that of the algebraic action, then one can simultaneously linearize $h$9-almost every system to an affine action. On $c_1,c_2\in\mathbb C$0, equality in

$c_1,c_2\in\mathbb C$1

is equivalent to the existence of $c_1,c_2\in\mathbb C$2 such that every generator in $c_1,c_2\in\mathbb C$3 is conjugated to $c_1,c_2\in\mathbb C$4. On $c_1,c_2\in\mathbb C$5, the conclusion takes the form

$c_1,c_2\in\mathbb C$6

for $c_1,c_2\in\mathbb C$7-almost every $c_1,c_2\in\mathbb C$8 near the algebraic model. This is not local fixed-point linearization; it is global affine conjugacy of a random family by one common $c_1,c_2\in\mathbb C$9 (Brown et al., 8 Jan 2026).

These results suggest a useful distinction. In local complex dynamics, simultaneous linearization typically concerns a common coordinate near a distinguished point or orbit. In random rigidity, the same phrase refers to a single manifold-wide conjugacy recovering an affine or isometric action from Lyapunov data. The common structural feature is still the same: linearization is a property of a family rather than of an individual map (Fletcher et al., 2018, DeWitt, 2020, Brown et al., 8 Jan 2026).

5. Estimation, delay systems, and multi-signal engineering

In numerical optimal control of delay differential equations, simultaneous linearization does not mean conjugacy. The relevant operation is the local Taylor expansion

$h\circ\varphi=h+c_1,\qquad h\circ\psi=h+c_2.$0

which converts a DDE with possibly input-dependent delays into an implicit differential system. A direct simultaneous transcription then treats state values and controls as decision variables in one nonlinear program, with implicit Euler residuals as equality constraints. A notable structural property is that the approximate system has exactly the same steady states as the original DDE, because the delayed-state approximation is exact when $h\circ\varphi=h+c_1,\qquad h\circ\psi=h+c_2.$1, but the local stability criterion is altered by replacing $h\circ\varphi=h+c_1,\qquad h\circ\psi=h+c_2.$2 with $h\circ\varphi=h+c_1,\qquad h\circ\psi=h+c_2.$3 in the characteristic equation (Ritschel et al., 2024).

The distributed-delay counterpart applies the same idea to convolutions. If

$h\circ\varphi=h+c_1,\qquad h\circ\psi=h+c_2.$4

the delayed variable is linearized as

$h\circ\varphi=h+c_1,\qquad h\circ\psi=h+c_2.$5

which yields

$h\circ\varphi=h+c_1,\qquad h\circ\psi=h+c_2.$6

The resulting implicit system is discretized by implicit Euler and transcribed into a simultaneous NLP solved with Matlab’s fmincon. The paper illustrates the approach on a molten salt nuclear fission reactor and reports that accuracy degrades as the effective delay becomes larger, because the first-order local approximation becomes less faithful (Ritschel, 2024).

In Bayesian SLAM for 5G mmWave sensing, the relevant object is a joint posterior linearization of the combined UE–landmark state for each data-association hypothesis. The nonlinear measurement function $h\circ\varphi=h+c_1,\qquad h\circ\psi=h+c_2.$7 is approximated by

$h\circ\varphi=h+c_1,\qquad h\circ\psi=h+c_2.$8

but $h\circ\varphi=h+c_1,\qquad h\circ\psi=h+c_2.$9 are recomputed iteratively with respect to an approximation of the posterior, not the prior. This “iterated posterior linearization” is then embedded into a Poisson multi-Bernoulli SLAM filter. The paper reports mapping and positioning improvements over an EK-PMB baseline, with the trade-off of increased update cost (Ge et al., 2021). A contrasting SLAM formulation replaces nonlinear raw measurements by exact linear time-varying “virtual synthetic measurements,” thereby avoiding EKF-style linearization altogether; this contrast is useful because it shows that not every nonlinear estimation framework uses simultaneous linearization in the same sense (Tan et al., 2015).

In microwave photonics, the term acquires yet another meaning. For a dual-parallel Mach–Zehnder modulator, simultaneous linearization means using one DP-MZM bias configuration together with optical carrier band processing so that all relevant third-order distortion terms generated by simultaneous multi-tone excitation are cancelled. The derived operating point is

$\mathbb D$0

which cancels crossed IMD$\mathbb D$1 terms by bias tuning and direct IMD$\mathbb D$2 terms by carrier-band attenuation and phase inversion. Simulations give a fundamental-to-IMD$\mathbb D$3 ratio above $\mathbb D$4 dB in the complete linearization setting, while experiments report about $\mathbb D$5 dB suppression of direct third-order distortion and about $\mathbb D$6 dB SFDR improvement (2207.14547).

6. Simultaneous linearization with structure preservation, homogenization, and reduction

For mechanical control systems, simultaneous input-output feedback linearization and decoupling becomes structure-sensitive. Classical nonlinear control theory would ask only for full rank of the decoupling matrix on the full state space. In the mechanical setting studied by Mureddu, one insists that both the coordinate change and the feedback preserve the second-order mechanical form. This introduces additional conditions on the configuration manifold, notably a vector relative half-degree and the covariant linearity requirement

$\mathbb D$7

When these conditions hold, one can find a mechanical coordinate change $\mathbb D$8 and a mechanical feedback law that put the observable part into decoupled even-dimensional chains while leaving the internal dynamics mechanical. The paper also gives examples where classical input-output decoupling is possible but mechanical simultaneous linearization is not, because the required coordinates would mix positions and velocities in a non-mechanical way (Nowicki et al., 2023).

In continuum mechanics and variational asymptotics, simultaneous linearization refers to commutation of linearization with other singular limits. In magnetoelasticity, a mixed Eulerian–Lagrangian energy with deformation $\mathbb D$9 and magnetization $h$00 is studied in a regime $h$01. The nonlinear magnetostrictive density is scaled by $h$02 and reduced to a quadratic form $h$03 around the identity, while exchange and magnetostatic terms are carried through the limit. The resulting $h$04-limit is

$h$05

and the paper shows that the same limit is obtained by first linearizing and then homogenizing (Cherdantsev et al., 26 Nov 2025).

A dynamic discrete-to-continuum analogue appears in atomistic elastodynamics. Here the simultaneous limit is $h$06 for lattice spacing and $h$07 for strain amplitude, with rescaling $h$08. The rescaled atomistic energy

$h$09

converges to linearized elasticity, and solutions of the atomistic equation of motion converge to solutions of the continuum momentum equation

$h$10

The proof uses an energy-dissipation-inertia principle, which yields pointwise-in-time convergence of the energies and, under stronger assumptions, strong convergence of the discrete gradients (Friedrich et al., 2023).

For thin periodic plates, the simultaneous limit combines homogenization, dimension reduction, and linearization. The total elastic energy is assumed of order $h$11 with $h$12; the main analysis focuses on the critical case $h$13. The paper proves that the normalized minima converge to a linearized homogenized plate functional

$h$14

and that the limit is unchanged whether one first performs linearization and then homogenization plus dimension reduction, or instead lets $h$15 simultaneously in the nonlinear model (Chakrabortty et al., 23 Oct 2025).

Taken together, these works show that simultaneous linearization is not a single theorem but a family of structurally similar ideas. In one direction it is a rigidity principle for commuting or random dynamical systems; in another it is a structure-preserving normal-form problem in geometry and control; in a third it is a multi-limit variational principle in elasticity and homogenization; and in engineering it can denote a single design or transcription that suppresses or approximates several nonlinear effects at once. The most useful question is therefore not whether simultaneous linearization exists in the abstract, but which compatibility mechanism—commutation, semisimplicity, arithmetic control, Lyapunov rigidity, variational scaling, or structure preservation—governs it in the specific setting under study.