Dichotomy Spectrum in Nonautonomous Systems

• Dichotomy Spectrum is defined as the set of parameter values for which the shifted nonautonomous system fails to admit an exponential or nonuniform dichotomy, thus revealing key stability thresholds.
• It extends classical Sacker–Sell theory by incorporating nonuniform growth rates and various scaling functions such as exponential, polynomial, and superexponential forms.
• Its interval structure organizes invariant splittings, spectral manifolds, and reducibility properties across continuous, discrete, and stochastic dynamical systems.

The dichotomy spectrum is the spectral set determined by the failure of a dichotomy for a shifted nonautonomous linear system. In continuous time, for

$x' = A(t)x,$

it is defined by the parameters $\gamma$ for which the shifted system $x'=(A(t)-\gamma I)x$ fails to admit an exponential dichotomy, or more generally for which

$$x'=\Bigl(A(t)-\gamma \frac{\mu'(t)}{\mu(t)}I\Bigr)x$$

fails to admit a nonuniform $\mu$-dichotomy; in discrete time one analogously studies

$$x_{k+1}=e^{-y}A_kx_k \quad\text{or}\quad x(k+1)=A(k)\Bigl(\frac{\mu(k+1)}{\mu(k)}\Bigr)^{-\gamma}x(k).$$

Across deterministic differential equations, difference equations, random dynamical systems, and mean-square stochastic systems, the spectrum organizes invariant splittings, growth rates, reducibility, and stability in a way that extends classical Sacker–Sell theory to nonuniform and nonexponential regimes (Zhang, 2014, Cuong et al., 2019, Silva, 2022, Gallegos et al., 2024).

1. Classical definition and shifted systems

In the classical setting, dichotomy theory concerns invariant splittings for nonautonomous linear systems. For continuous time, an exponential dichotomy for $x'=A(t)x$ is given by an invariant projector $P(t)$ and constants $K\ge 1$, $\alpha>0$ such that

$\gamma$0

where $\gamma$1 is the evolution operator. The associated dichotomy spectrum is

$\gamma$2

For discrete time, with

$\gamma$3

Cuong–Doan use the shifted family

$\gamma$4

and define

$\gamma$5

These are the continuous and discrete versions of the Sacker–Sell spectrum (Zhang, 2014, Cuong et al., 2019, Castañeda et al., 2018).

Nonuniform versions replace the uniform estimates by bounds with explicit base-time dependence. In the continuous nonuniform exponential case one has

$\gamma$6

with $\gamma$7, $\gamma$8, and $\gamma$9. This leads to the nonuniform dichotomy spectrum

$x'=(A(t)-\gamma I)x$0

which generalizes the uniform spectrum by allowing the factors $x'=(A(t)-\gamma I)x$1 and $x'=(A(t)-\gamma I)x$2 (Zhang, 2014, Chu et al., 2014).

A broader framework replaces the exponential scale by a growth rate $x'=(A(t)-\gamma I)x$3. In that setting, a differentiable $x'=(A(t)-\gamma I)x$4 is strictly increasing, satisfies $x'=(A(t)-\gamma I)x$5, tends to $x'=(A(t)-\gamma I)x$6 as $x'=(A(t)-\gamma I)x$7, and tends to $x'=(A(t)-\gamma I)x$8 as $x'=(A(t)-\gamma I)x$9. The $$x'=\Bigl(A(t)-\gamma \frac{\mu'(t)}{\mu(t)}I\Bigr)x$$0-shifted evolution is then

$$x'=\Bigl(A(t)-\gamma \frac{\mu'(t)}{\mu(t)}I\Bigr)x$$1

and the spectrum is defined by failure of a nonuniform $$x'=\Bigl(A(t)-\gamma \frac{\mu'(t)}{\mu(t)}I\Bigr)x$$2-dichotomy for the shifted system (Silva, 2022, Gallegos et al., 2024).

2. Interval structure, resolvent gaps, and spectral manifolds

A basic theorem across the subject is that the resolvent is open and the spectrum is interval-valued. In the continuous nonuniform exponential setting, the resolvent $$x'=\Bigl(A(t)-\gamma \frac{\mu'(t)}{\mu(t)}I\Bigr)x$$3 is open, and the spectrum is a union of at most $$x'=\Bigl(A(t)-\gamma \frac{\mu'(t)}{\mu(t)}I\Bigr)x$$4 closed intervals for an $$x'=\Bigl(A(t)-\gamma \frac{\mu'(t)}{\mu(t)}I\Bigr)x$$5-dimensional system. Under nonuniformly bounded growth,

$$x'=\Bigl(A(t)-\gamma \frac{\mu'(t)}{\mu(t)}I\Bigr)x$$6

the spectrum is nonempty and bounded, hence a finite union of compact intervals (Zhang, 2014). An analogous theorem holds for nonuniform difference equations, where $$x'=\Bigl(A(t)-\gamma \frac{\mu'(t)}{\mu(t)}I\Bigr)x$$7 is a disjoint union of at most $$x'=\Bigl(A(t)-\gamma \frac{\mu'(t)}{\mu(t)}I\Bigr)x$$8 closed intervals, and under nonuniform exponential boundedness it is contained in a compact interval of the form $$x'=\Bigl(A(t)-\gamma \frac{\mu'(t)}{\mu(t)}I\Bigr)x$$9 (Chu et al., 2014).

The discrete Sacker–Sell spectrum has the same interval structure. For $\mu$0, the dichotomy spectrum $\mu$1 is a compact set given by the union of at most $\mu$2 closed intervals, and the one-sided spectra have analogous forms (Cuong et al., 2019). In the nonuniform $\mu$3-setting, if the system has $\mu$4-growth, then

$\mu$5

is a finite union of compact intervals (Gallegos et al., 2024).

These interval decompositions are not merely set-theoretic. They induce invariant bundles and Whitney-sum decompositions. In the continuous nonuniform spectrum, choosing parameters in successive spectral gaps yields spectral manifolds

$\mu$6

with

$\mu$7

and the $\mu$8 are independent of the particular gap representatives (Zhang, 2014). The discrete nonuniform theory produces the same kind of spectral bundles for difference equations (Chu et al., 2014). On each connected component of the resolvent, the invariant projector is constant, while crossing a spectral interval changes the dimensions of stable and unstable bundles monotonically (Zhang, 2014, Cuong et al., 2019).

3. Nonuniform and growth-rate-dependent spectra

The $\mu$9-dichotomy framework replaces the exponential scale by a general growth rate. Typical examples are $$x_{k+1}=e^{-y}A_kx_k \quad\text{or}\quad x(k+1)=A(k)\Bigl(\frac{\mu(k+1)}{\mu(k)}\Bigr)^{-\gamma}x(k).$$0, polynomial scales generated by $$x_{k+1}=e^{-y}A_kx_k \quad\text{or}\quad x(k+1)=A(k)\Bigl(\frac{\mu(k+1)}{\mu(k)}\Bigr)^{-\gamma}x(k).$$1, and superexponential scales such as quadratic or cubic exponentials (Gallegos et al., 2024, Jara et al., 29 Jul 2025). In the continuous nonuniform $$x_{k+1}=e^{-y}A_kx_k \quad\text{or}\quad x(k+1)=A(k)\Bigl(\frac{\mu(k+1)}{\mu(k)}\Bigr)^{-\gamma}x(k).$$2-setting, one asks for an invariant projector $$x_{k+1}=e^{-y}A_kx_k \quad\text{or}\quad x(k+1)=A(k)\Bigl(\frac{\mu(k+1)}{\mu(k)}\Bigr)^{-\gamma}x(k).$$3 and constants $$x_{k+1}=e^{-y}A_kx_k \quad\text{or}\quad x(k+1)=A(k)\Bigl(\frac{\mu(k+1)}{\mu(k)}\Bigr)^{-\gamma}x(k).$$4, $$x_{k+1}=e^{-y}A_kx_k \quad\text{or}\quad x(k+1)=A(k)\Bigl(\frac{\mu(k+1)}{\mu(k)}\Bigr)^{-\gamma}x(k).$$5, $$x_{k+1}=e^{-y}A_kx_k \quad\text{or}\quad x(k+1)=A(k)\Bigl(\frac{\mu(k+1)}{\mu(k)}\Bigr)^{-\gamma}x(k).$$6, $$x_{k+1}=e^{-y}A_kx_k \quad\text{or}\quad x(k+1)=A(k)\Bigl(\frac{\mu(k+1)}{\mu(k)}\Bigr)^{-\gamma}x(k).$$7, with $$x_{k+1}=e^{-y}A_kx_k \quad\text{or}\quad x(k+1)=A(k)\Bigl(\frac{\mu(k+1)}{\mu(k)}\Bigr)^{-\gamma}x(k).$$8 and $$x_{k+1}=e^{-y}A_kx_k \quad\text{or}\quad x(k+1)=A(k)\Bigl(\frac{\mu(k+1)}{\mu(k)}\Bigr)^{-\gamma}x(k).$$9, such that

$x'=A(t)x$0

$x'=A(t)x$1

This includes the exponential case $x'=A(t)x$2 and the polynomial case as specializations (Gallegos et al., 2024, Silva, 2022).

The discrete theory distinguishes three levels: uniform $x'=A(t)x$3-dichotomy, nonuniform $x'=A(t)x$4-dichotomy, and slow nonuniform $x'=A(t)x$5-dichotomy. They satisfy

$x'=A(t)x$6

For uniform and nonuniform $x'=A(t)x$7-dichotomy, the system has the Unbounded Solutions Property and the Unique Projector Property. By contrast, the slow nonuniform theory admits unconventional situations in which there are nontrivial bounded solutions and more than one invariant projector with which the system has a dichotomy (Castañeda et al., 8 Jan 2025).

That phenomenon motivates a further refinement: the unique-projector slow nonuniform $x'=A(t)x$8-dichotomy spectrum

$x'=A(t)x$9

defined through slow nonuniform $P(t)$0-dichotomy together with the Unique Projector Property. Assuming the USPP conjecture, this spectrum also admits a spectral theorem: it is a union of nonoverlapping intervals, and under $P(t)$1-growth it is nonempty and bounded (Castañeda et al., 8 Jan 2025).

These growth-rate-dependent spectra can reveal behavior invisible to the exponential theory. For example, the system $P(t)$2 has exponential dichotomy spectrum $P(t)$3 in Rasmussen’s convention, while for the quadratic-exponential growth rate $P(t)$4 the $P(t)$5-dichotomy spectrum is $P(t)$6 (Jara et al., 29 Jul 2025).

4. Bohl, Lyapunov, subspace-uniform spectra, and reducibility

The dichotomy spectrum is closely tied to asymptotic growth rates. Under nonuniformly bounded growth,

$P(t)$7

so the Lyapunov spectrum is contained in the nonuniform dichotomy spectrum, which is itself contained in the Sacker–Sell spectrum (Zhu, 2019). For discrete systems on $P(t)$8, if

$P(t)$9

then every trajectory starting in the $K\ge 1$0-th invariant bundle has its lower and upper Bohl exponents in $K\ge 1$1 (Pinto et al., 2018). On $K\ge 1$2, the Bohl dichotomy spectrum satisfies

$K\ge 1$3

and all three spectra are unions of at most $K\ge 1$4 intervals (Czornik et al., 2023). In the nonuniform $K\ge 1$5-setting, bounded spectral intervals also localize $K\ge 1$6-Lyapunov exponents: $K\ge 1$7 for $K\ge 1$8 in the corresponding spectral manifold (Silva, 2022).

A recent refinement defines dichotomies that are uniform not on the whole stable and unstable bundles, but on prescribed families of subspaces. For admissible uniformity dimensions $K\ge 1$9, the associated spectrum $\alpha>0$0 is again a nonempty union of at most $\alpha>0$1 compact intervals. If

$\alpha>0$2

and

$\alpha>0$3

is the spectral flag, then the endpoints satisfy

$\alpha>0$4

with $\alpha>0$5. This formulation specializes both to the Bohl dichotomy spectrum and to the exponential dichotomy spectrum, yielding new endpoint formulas in both cases (Czornik et al., 2024).

Spectral decompositions also drive reducibility and normal forms. For continuous systems with nonuniform exponential dichotomy, the spectral manifolds yield block diagonalization aligned with spectral intervals, and for nonlinear systems

$\alpha>0$6

they lead to finite jet normal forms governed by interval-based resonance conditions (Zhang, 2014). For nonautonomous difference equations with nonuniform dichotomy, weak kinematical similarity produces block diagonal systems whose blocks carry the spectral intervals of the original system (Chu et al., 2014). In the nonuniform $\alpha>0$7-framework, if $\alpha>0$8 is differentiable, the evolution operator has nonuniformly bounded growth, and the similarity loss $\alpha>0$9 satisfies

$\gamma$00

then the system is $\gamma$01-nonuniformly kinematically similar to a block-diagonal system whose blocks have spectra equal to the spectral intervals (Silva, 2022).

5. Invariance, growth-rate comparison, and computation

In the uniform theory, the dichotomy spectrum is invariant under uniformly bounded kinematic similarity. In particular, if two systems are uniformly kinematically similar, then their nonuniform $\gamma$02-resolvents coincide, hence so do their spectra (Gallegos et al., 2024). The nonuniform theory is different. A continuous scalar counterexample takes $\gamma$03,

$\gamma$04

Then the systems are nonuniformly $\gamma$05-kinematically similar with $\gamma$06, but

$\gamma$07

The mechanism is quantitative: under similarity, the parameters transform as

$\gamma$08

and near the spectral boundary the optimal ratios $\gamma$09 and $\gamma$10 tend to $\gamma$11, so the transferred system can lose dichotomy (Gallegos et al., 2024).

The same noninvariance occurs in the discrete nonuniform exponential setting under weak kinematic similarity. In the scalar example of (Castañeda et al., 8 Jan 2025), one system has

$\gamma$12

while its weakly kinematically similar partner has

$\gamma$13

This contradicts the belief that the nonuniform exponential dichotomy spectrum is preserved under weak kinematic similarity (Castañeda et al., 8 Jan 2025).

Growth-rate comparison provides a different kind of structural control. If $\gamma$14 is faster than $\gamma$15, then faster growth rates compress the spectrum, while slower ones expand it. In particular, if a system has $\gamma$16-bounded growth and $\gamma$17, then

$\gamma$18

whereas if the system has $\gamma$19-dichotomy and $\gamma$20, then the $\gamma$21-spectrum reduces to one of $\gamma$22, $\gamma$23, or $\gamma$24. The same paper introduces equivalence relations on growth rates and proves that, on any totally ordered chain of equivalence classes, there exists at most one growth rate for which the system has both bounded growth and dichotomy (Jara et al., 29 Jul 2025).

There is also a computation theory under nonuniformly bounded growth. For diagonal systems

$\gamma$25

the weak integral separation spectrum equals the nonuniform dichotomy spectrum, and Steklov averages

$\gamma$26

provide interval bounds for spectral components. The theory distinguishes the uniform regime, where one works with $\gamma$27, from the nonuniform regime, where one needs $\gamma$28 to damp the base-time bias; using the wrong regime can lead to incorrect identification of $\gamma$29 versus $\gamma$30 (Zhu, 2019).

6. Applications in control, stability, and stochastic dynamics

In control theory, the dichotomy spectrum is a closed-loop design target. For the discrete time-varying system

$\gamma$31

uniform complete controllability implies arbitrary assignability of the dichotomy spectrum of the closed-loop system: for any collection of $\gamma$32 disjoint closed intervals $\gamma$33, with $\gamma$34, there exists bounded state feedback $\gamma$35 such that

$\gamma$36

This strictly strengthens earlier arbitrary assignability results for the Lyapunov spectrum (Cuong et al., 2019).

In nonlinear stability theory, the spectrum functions as a nonautonomous replacement for the Hurwitz condition. On $\gamma$37, a nonautonomous Markus–Yamabe conjecture was formulated by requiring

$\gamma$38

for every piecewise continuous $\gamma$39. The conjecture is verified for broad triangular classes by uniformization methods and block-triangular criteria for nonuniform exponential dichotomy (Castañeda et al., 2022). In the uniform exponential setting on $\gamma$40, the same spectral negativity condition yields scalar and triangular global asymptotic stability results (Castañeda et al., 2018).

Stochastic extensions preserve the interval picture while changing the ambient topology. For linear mean-field SDEs, the mean-square dichotomy spectrum is the set of growth rates $\gamma$41 for which the linear mean-square random dynamical system has no exponential dichotomy with growth rate $\gamma$42. Under bounded coefficients, it is the disjoint union of at most $\gamma$43 compact intervals (Doan et al., 2014). In random dynamical systems, the dichotomy spectrum of a linear cocycle is again a finite union of intervals, and it detects hyperbolicity by whether $\gamma$44 lies in the spectrum (Callaway et al., 2013).

A notable example is the noisy pitchfork. For the variational equation along the unique attracting random fixed point, the dichotomy spectrum is

$\gamma$45

Thus $\gamma$46 for $\gamma$47 and $\gamma$48 for $\gamma$49. The top Lyapunov exponent remains negative for all $\gamma$50, so the transition is not seen at the level of asymptotic Lyapunov exponents; it appears instead as a transition from hyperbolic to non-hyperbolic dichotomy spectrum, accompanied by loss of uniform attractivity and loss of experimental observability of the finite-time Lyapunov exponent (Callaway et al., 2013).

A distinct operator-theoretic line studies dichotomous and strictly dichotomous operators under uniform resolvent boundedness along the imaginary axis, constructing invariant subspaces $\gamma$51 and projections $\gamma$52 by resolvent integrals. Although formulated in a different language, it addresses the same underlying issue: spectral splitting into stable and unstable parts (Winklmeier et al., 2014).