Nonuniform Exponential Dichotomy

• Nonuniform exponential dichotomy is a property of linear nonautonomous systems that allows subexponential weights to characterize nonuniform growth and decay rates.
• The concept extends spectral analysis by defining spectral intervals and invariant projections, enabling precise decomposition and robustness under perturbations.
• NUED underpins applications such as invariant manifold theorems, smooth linearization, and attractor theory in both finite and infinite-dimensional dynamical systems.

A nonuniform exponential dichotomy (NUED) is a structural property of linear nonautonomous dynamical systems that generalizes the classical exponential dichotomy by permitting certain nonuniform growth in the dichotomy estimates. This concept plays a central role in the modern theory of nonuniform hyperbolicity, invariant manifold theory, spectral analysis of nonautonomous differential and difference equations, and smooth linearization theorems. The feature distinguishing NUED from its uniform counterpart is the introduction of tempered (subexponential) weights or drift factors in the dichotomy inequalities, enabling analysis of systems exhibiting time-dependent, nonautonomous, or even infinite-dimensional dynamical behavior beyond classical hyperbolic theory.

1. Definition and Fundamental Properties

Let $\dot{x} = A(t)x$ be a linear nonautonomous system on $[0, \infty)$ (or $\mathbb{R}$) with evolution operator $\Phi(t,s)$. The system is said to admit a nonuniform exponential dichotomy if there exists a (strongly) continuous family of invariant projections $P(t)$, together with constants $K \geq 1$, $\alpha > 0$, and $\mu \geq 0$ (with $\mu < \alpha$), such that for all $t, s \geq 0$,

$$\begin{aligned} P(t)\,\Phi(t,s)&= \Phi(t,s)P(s), \ \|\Phi(t,s) P(s)\| &\leq K e^{-\alpha (t-s) + \mu s}, \qquad t \geq s, \ \|\Phi(t,s) (I - P(s))\| &\leq K e^{\alpha (t-s) + \mu s}, \qquad t \leq s. \end{aligned}$$

Here, $P(t)$ identifies a dynamically invariant splitting into stable (decaying) and unstable (growing) subspaces, but decay/growth rates are modulated by a subexponential weight $e^{\mu s}$. The case $\mu=0$ recovers the classical exponential dichotomy with uniform, time-independent rates (Castañeda et al., 28 Sep 2024).

In Banach spaces, a general version employs a family $K(s)=D e^{\nu|s|}$ and extends the invariance and bounds accordingly, requiring the restriction of evolution to the unstable bundle to be an isomorphism (Caraballo et al., 2020).

Nonuniformity quantitatively measures the allowable nonautonomous drift or nonuniform temporal behavior in the dichotomy; large values of $\mu$ or $\nu$ correspond to more pronounced time variability.

2. Spectral Theory and the Nonuniform Dichotomy Spectrum

Associated to NUED is the nonuniform exponential dichotomy spectrum, a generalization of the Sacker–Sell spectrum. For each real parameter $\lambda$, consider the shifted system $\dot{x} = [A(t) - \lambda I]x$. The spectrum $\Sigma_{NUED}(A)$ is defined as

$\Sigma_{NUED}(A) = \{ \lambda \in \mathbb{R} : \dot{x} = (A(t)-\lambda I)x \text{ does \emph{not} admit a NUED}\}$

and its resolvent set is open. Under nonuniformly bounded growth, $\Sigma_{NUED}(A)$ is a finite union of closed intervals: $$\Sigma_{NUED}(A) = \bigcup_{i=1}^k [a_i, b_i], \qquad 0 \leq k \leq n,$$ with $$a_1 \leq b_1 < a_2 \leq b_2 < \cdots < a_k \leq b_k$$ (Zhu, 2019, Zhang, 2014, Silva, 2022). These spectral intervals encode spectral gaps within which NUED projections and invariant subbundles (spectral manifolds) can be constructed. Spectral theory for higher-order or generalized nonuniform dichotomies (e.g., with arbitrary growth rates $\mu$) is developed in (Castañeda et al., 8 Jan 2025), including discrete and slow nonuniform variants.

Moreover, there is an inclusion: $$\Sigma_L(A) \subseteq \Sigma_{NUED}(A) \subseteq \Sigma_{ED}(A)$$ where $\Sigma_L$ is the Lyapunov spectrum and $\Sigma_{ED}$ is the uniform dichotomy spectrum (Zhu, 2019).

3. Robustness and Parameter Dependence

NUED admits various roughness properties, i.e., stability under perturbations. If a linear system with a nonuniform exponential dichotomy is perturbed by another (possibly nonlinear or stochastic) term, the dichotomy persists provided the perturbation is sufficiently small, typically measured in a weighted norm or integrability condition:

$$q := 2K \sup_{t \in \mathbb{R}} \int_{-\infty}^\infty e^{-\alpha |t-s|} \|B(s)\| e^{\varepsilon |s|} ds < 1,$$

implies that the perturbed evolution family admits a NUED with possibly altered constants (Dragicevic, 14 Dec 2025). Explicit quantitative robustness results are available for both linear (Caraballo et al., 2020, Dragicevic, 14 Dec 2025) and stochastic (Zhu, 2019) cases, as well as for nonuniform polynomial or arbitrary growth-dichotomies (Bento et al., 2011, Bento et al., 2014).

In addition, the projections associated with the splitting are unique and depend continuously (even Lipschitz continuously) on the evolution family and system parameters (Caraballo et al., 2020, Zhang et al., 2015), a crucial property for applications involving parameter-dependent families of differential equations.

4. Spectral, Reducibility, and Splitting Theory

The existence of a NUED enables a spectral decomposition of the underlying space via invariant projections, leading to block-diagonal (normal form) or almost-reducible representations of nonautonomous systems (Zhang, 2014, Castañeda et al., 2017, Silva, 2022).

If the system admits a full spectral decomposition, one can construct a nonuniform Lyapunov transformation that block-diagonalizes the system so that each block supports only the dynamics associated with a single spectral interval: $$x = S(t) y, \quad y' = \operatorname{diag}(B_1(t), \ldots, B_m(t))y,$$ with the dichotomy spectrum of $B_j$ coinciding with $[a_j,b_j]$ (Zhang, 2014, Silva, 2022, Castañeda et al., 2017). Techniques for almost-reducibility (diagonalization up to a NUED small remainder) are also available (Castañeda et al., 2017).

Weak integral separation of invariant directions constitutes a necessary and sufficient condition for the spectral splitting and for the Lyapunov exponents' stability under exponentially decaying perturbations (Zhu et al., 2019).

5. Applications: Linearization, Invariant Manifolds, and Attractors

NUED is foundational in proofs of invariant manifold theorems, Hartman–Grobman linearization, and smooth conjugacy for nonautonomous ordinary and partial differential equations.

• Linearization: Sufficient spectral gap and NUED imply global or local $C^1$ (sometimes Hölder) conjugacy between a nonlinear perturbation and its linearization, even in the presence of nonuniform hyperbolicity. Quantitative criteria for smoothness (via combinations of dichotomy, bounded-growth, and perturbation-decay parameters) determine the size of the interval on which smooth invertibility of conjugacies is available (Castañeda et al., 28 Sep 2024, Zhu, 2019, Zhu et al., 2019, Zhu, 2019, Dragicevic et al., 2019, Dragicevic et al., 2017).
• Stable Manifolds: The existence of a stable manifold for a focus or saddle in nonautonomous or even infinite-dimensional settings relies on NUED for the linearization. Results extend to semilinear PDEs and admit nonuniform $p$-polynomial or generalized dichotomy rates (Bento et al., 2010, Bento et al., 2011, Zhang et al., 2015).
• Attractor Theory: Nonuniform exponential dichotomies underpinings the construction of pullback and forward attractors in nonautonomous systems, with the admissibility pairs characterized via the dichotomy constants and growth rates (Langa et al., 2021).
• Mean-Square Dichotomies in SDEs: For stochastic systems, the mean-square (L^2)-version of NUED controls invariant splitting and asymptotic behavior of solutions under noise, with robustness under perturbations (Zhu, 2019).

The triangular and block-diagonal criteria, including for upper triangular and block systems, enable explicit spectral analysis and sharp criteria for global asymptotic stability in nonautonomous and time-varying settings (Castañeda et al., 2022).

6. Generalizations, Nonuniform Growth Rates, and Recent Developments

Generalizations of NUED encompass $\mu$-dichotomies and $(h,k,\mu,\nu)$-dichotomies, where the power-type or arbitrary growth sequences/polynomials replace the exponential (Weijie et al., 3 Jun 2025, Silva, 2022, Castañeda et al., 8 Jan 2025, Zhang et al., 2015, Bento et al., 2014). These frameworks describe systems with polynomial or more exotic time dependence and cover nonuniform hyperbolicity outside the classical exponential setting.

Additionally, recent investigations highlight subtle differences between uniform, nonuniform, and slow nonuniform dichotomies, especially in discrete time, including failures of invariance of the dichotomy spectrum under weak kinematic similarity—showing the loss of spectral invariance unless the intertwining transformation is tempered to match the nonuniformity weights (Castañeda et al., 8 Jan 2025).

Recent Lyapunov function and integral characterizations of NUED, generalizing Datko-type theorems, connect the existence of a dichotomy to integral bounds on the Green kernel and to structurally adapted Lyapunov functions, providing robust tools for both analysis and computation (Bento et al., 2014).

7. Illustrative Examples and Normal Forms

Standard classes of systems with NUED include:

• Planar systems with nonautonomous oscillatory coefficients (e.g., $\dot{x} = (10 + a t \sin t)x$) (Castañeda et al., 2017),
• Block-diagonal and triangular systems with time-dependent coefficients (Castañeda et al., 2022, Silva, 2022),
• Systems with time-dependent drift satisfying only nonuniform polynomial or slow exponential growth (Bento et al., 2010, Bento et al., 2011).

The normal form (finite jet) theory for nonautonomous and nonlinear systems leverages the nonuniform dichotomy spectrum for the elimination of nonresonant terms and provides a complete structure up to a desired order of nonlinearity (Zhang, 2014).

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In summary, the theory of nonuniform exponential dichotomy is central to contemporary study of nonautonomous and infinite-dimensional hyperbolic dynamics. It delivers the structural framework necessary for spectral theory, invariant manifold constructions, linearization, and robust stability under parameter variation or perturbation, with applications throughout nonlinear analysis, ergodic theory, and mathematical physics (Castañeda et al., 28 Sep 2024, Caraballo et al., 2020, Castañeda et al., 2017, Zhu et al., 2019, Langa et al., 2021, Dragicevic, 14 Dec 2025, Weijie et al., 3 Jun 2025, Castañeda et al., 2022, Zhang et al., 2015, Silva, 2022, Bento et al., 2010, Dragicevic et al., 2019, Zhang, 2014, Bento et al., 2011, Zhu, 2019, Bento et al., 2014, Zhu, 2019, Dragicevic et al., 2017, Castañeda et al., 8 Jan 2025).