Isostables in Nonlinear Dynamics

• Isostables are geometric level sets of the principal Koopman eigenfunction that capture the slowest decaying mode in nonlinear systems.
• They enable a unique global linearization via semiconjugacy, facilitating phase-amplitude reduction and transient analysis.
• The framework aids in model reduction, control design, and computational methods in high-dimensional dynamical systems.

Isostables are geometric objects in the state space of nonlinear dynamical systems, defined as the level sets of the principal Koopman eigenfunction associated with the slowest decaying mode of attraction. They provide a global foliation of the basin of attraction for both fixed points and periodic orbits, characterizing sets of initial conditions that converge synchronously to the attractor at identical asymptotic rates. The concept of isostables arises in the context of spectral operator theory and has become fundamental in phase-amplitude reduction, model reduction, numerical computation, and the analysis of transient phenomena in high-dimensional and nonlinear systems (Kvalheim et al., 2019).

1. Definition and Spectral Characterization

Formally, for a $C^1$ flow $\Phi^t$ on the basin $Q$ of a hyperbolic attracting fixed point $x_0$ or periodic orbit $\Gamma$, a Koopman eigenfunction $\psi: Q \to \mathbb{C}$ with exponent $\mu \in \mathbb{C}$ satisfies

$$\psi \circ \Phi^t(x) = e^{\mu t}\ \psi(x),\ \forall t.$$

The principal eigenfunction is defined by $\psi(x_0) = 0$ and $D\psi(x_0) \neq 0$ in the fixed point case, or $\Phi^t$0 and $\Phi^t$1 for $\Phi^t$2 in the periodic orbit case. The isostables are the level sets of $\Phi^t$3, or equivalently, of $\Phi^t$4 or $\Phi^t$5 for the real or complex-conjugate principal eigenfunctions. The corresponding Koopman exponent $\Phi^t$6 must yield the slowest decaying Floquet multiplier $\Phi^t$7, which dominates the asymptotic convergence (Kvalheim et al., 2019).

Uniqueness and global existence of principal Koopman eigenfunctions (and thus isostable coordinates) are ensured under $\Phi^t$8-nonresonance conditions on the dynamics, as established in (Kvalheim et al., 2019). When $\Phi^t$9 is simple and nonresonant, any two eigenfunctions differ only by a constant scalar multiple.

2. Existence, Uniqueness, and Explicit Construction

The core result is that, given a hyperbolic sink $Q$0 with $Q$1 of spectral radius $Q$2, for any complex $Q$3 with $Q$4 and appropriate left eigenvector $Q$5, there exists a unique $Q$6 with $Q$7 and $Q$8. Explicitly, for any “approximate” eigenfunction $Q$9 satisfying $x_0$0 and an appropriate approximate eigenrelation, the limit

$x_0$1

converges to the principal eigenfunction in $x_0$2 on compacts (Kvalheim et al., 2019).

For systems with an attracting periodic orbit, the construction is via the Poincaré map on a strong-stable leaf, extended globally by

$x_0$3

with $x_0$4 the Floquet exponent matrix. The isostables are thus globally smooth, well-defined, and uniquely determined principal Koopman eigenfunctions.

3. Hierarchy of Isostables and Spectral Decomposition

Beyond the slowest mode, "faster" isostable coordinates can be defined recursively. For each stable direction with Koopman exponent $x_0$5, the Laplace-type limit

$x_0$6

for an approximate $x_0$7 vanishing on more strongly stable directions, converges to the unique principal eigenfunction associated to $x_0$8 whenever $x_0$9 and nonresonance conditions hold (Kvalheim et al., 2019). The hierarchy of isostables thus mirrors the stable spectral decomposition of the linearized system.

This spectral hierarchy enables construction of a vector-valued factor map

$\Gamma$0

where the components of $\Gamma$1 are commuting Koopman eigenfunctions—in local coordinates, these are precisely the isostable coordinates.

4. Global Linearization, Pullback Algebra, and Coordinate Representations

Under the existence and uniqueness theorems, there exists a unique $\Gamma$2 semiconjugacy $\Gamma$3 relating the nonlinear flow to its linearization $\Gamma$4 on the basin of attraction, with the algebra $\Gamma$5 generated by the components $\Gamma$6 being independent of the particular choice of conjugacy (Kvalheim et al., 2019). The pullback algebra, constructed from the linearized principal algebra and the nonlinear coordinate transformation, is thus robust and well-defined.

Global linearization via isostable coordinates provides a mechanism for diagonalizing the (nonlinear) dynamics and for constructing global (nonlinear) action-angle or phase-amplitude coordinates, which are essential in the reduction and analysis of transient behavior even far from the attractor.

5. Illustrative Examples: Linear and Nonlinear Systems

For the linear system $\Gamma$7, $\Gamma$8, with $\Gamma$9, the slow and fast exponents are $\psi: Q \to \mathbb{C}$0, $\psi: Q \to \mathbb{C}$1. The principal (slowest) eigenfunction is $\psi: Q \to \mathbb{C}$2, and the fast isostable coordinate is $\psi: Q \to \mathbb{C}$3, both recovering classical stable subspaces (isostables $\psi: Q \to \mathbb{C}$4 and $\psi: Q \to \mathbb{C}$5, respectively). For a nonlinear perturbation $\psi: Q \to \mathbb{C}$6, the isostable $\psi: Q \to \mathbb{C}$7 is recovered via the semiconjugacy to the linear system, illustrating the persistence of the isostable framework under nonlinear conjugacy (Kvalheim et al., 2019).

The general vector-valued isostable coordinates smoothly interpolate between linear subspaces and nonlinear normal forms, providing global coordinates for the dynamics on the entire basin.

6. Relation to Other Notions: Isochrons, Koopmanism, and Pullback Structures

Isostables are complementary to isochrons, which are level sets of the phase function for periodic orbits (associated with the zero Floquet exponent). While isochrons foliate the basin by asymptotic phase, isostables organize it by amplitude of decay or approach rate to the attractor. The global action-angle formulation unifies these through the Koopman eigenfunction framework.

The pullback algebra of principal eigenfunctions is shown to be unique under nonresonance hypotheses, eliminating ambiguity in the spectral description and ensuring the robustness of isostable coordinates in applied and computational Koopmanism (Kvalheim et al., 2019).

7. Significance and Applications

Isostables articulated through the Koopman operator theory provide a natural, global framework for:

• Analysis of transient behavior and reduction of nonlinear systems to phase-amplitude coordinates;
• Computation and control strategies for high-dimensional dynamical systems, including optimal stabilization, synchronization, and suppression of transient deviations;
• Systematic organization of basins of attraction and synchronized convergence for a wide class of dissipative systems.

The existence and uniqueness theorems guarantee rigorous and robust applicability, while explicit constructions via Laplace-type limits and semiconjugacies enable practical computation in both theoretical and engineered systems. The framework generalizes and sharpens classical results such as Sternberg linearization and Floquet theory within a unifying operator-theoretic perspective (Kvalheim et al., 2019).