## Lyapunov functions for trichotomies with growth rates.(English)Zbl 1197.34094

Let $$T(t, \tau)$$ be the linear evolution operator associated to the linear equation $x' = A(t)x$
in the finite-dimensional space $${\mathbb R}^n$$. We say that $$x' = A(t)x$$ admits a $$\rho$$-nonuniform exponential contraction if there exist constants $$d$$, $$D > 0$$ and $$\varepsilon \geq 0$$ such that $$\| T(t, \tau) \| \leq D e^{-d(\rho(t) - \rho(\tau)) + \varepsilon|\rho(\tau)|}$$ for $$t \geq \tau > 0$$. When the equation admits a $$\rho$$-nonuniform exponential contraction with $$\varepsilon = 0$$, we say that it admits a $$\rho$$-uniform exponential contraction. In this paper, in strong contrast with the usual stable, unstable, and central behavior, the authors consider asymptotic rates of the form $$e^{c \rho(t)}$$ determined by an arbitrary function $$\rho(t)$$.
The usual notion of nonuniform exponential contraction corresponds to the particular case when $$\rho(t) = t$$, that is, $$\| T(t, \tau) \| \leq D e^{-d(t - \tau) + \varepsilon \tau}$$ for $$t \geq \tau > 0$$ [see L. Barreira and C. Valls, J. Differ. Equations 217, No. 1, 204–248 (2005; Zbl 1088.34053)]. The notion of nonuniform polynomial contraction is recovered when $$\rho(t) = \log (1 + t)$$, that is, $\| T(t, \tau) \| \leq D \left( \frac{1 + t}{1 + \tau} \right)^{-d} (1 + \tau)^\varepsilon$ for $$t \geq \tau > 0$$ [see L. Barreira and C. Valls, Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 71, No. 11, 5208–5219 (2009; Zbl 1181.34046)].
Let $$V : {\mathbb R}_0^+ \times {\mathbb R}^n \to {\mathbb R}_0^-$$ be a function, and assume that there exist $$C > 0$$ and $$\delta \geq 0$$ such that $$|V(t, x)| \leq C e^{\delta |\rho(t)|} \| x \|$$ for every $$t \geq 0$$ and $$x \in {\mathbb R}^n$$. Given $$\alpha > 0$$ and $$\gamma \geq 0$$, we say that $$V$$ is a strict Lyapunov function for the equation $$x' = A(t)x$$ if
$V(t, T(t, \tau) x) \leq \alpha^{\rho(t) - \rho(\tau)} V(\tau, x)$
for $$t \geq \tau$$, and
$V(\tau, x) \geq \frac1C e^{-\gamma \rho(\tau)} \| x \|$
for every $$\tau \geq 0$$ and $$x \in {\mathbb R}^n$$.
The main objective is to give a complete characterization in terms of strict Lyapunov functions of the linear equations admitting a $$\rho$$-nonuniform exponential trichotomy. This includes criteria for the existence of a $$\rho$$-nonuniform exponential trichotomy, as well as inverse theorems providing explicit strict Lyapunov functions for each given exponential trichotomy. In the particular case of quadratic Lyapunov functions, it is shown that the existence of strict Lyapunov sequences can be deduced from more algebraic relations between the quadratic forms defining the Lyapunov functions. As an application of the characterization of nonuniform exponential trichotomy in terms of strict Lyapunov functions, the authors establish the robustness of $$\rho$$-nonuniform exponential trichotomy under sufficiently small linear perturbations. It is emphasized that, in comparison with former works, the proof of the robustness is much simpler even when $$\rho(t) = t$$.

### MSC:

 34D09 Dichotomy, trichotomy of solutions to ordinary differential equations 34D20 Stability of solutions to ordinary differential equations 34D08 Characteristic and Lyapunov exponents of ordinary differential equations

### Keywords:

Lyapunov functions; nonuniform exponential trichotomy

### Citations:

Zbl 1088.34053; Zbl 1181.34046
Full Text:

### References:

 [1] Barreira, L.; Pesin, Ya., Nonuniform hyperbolicity, Encyclopedia math. appl., vol. 115, (2007), Cambridge Univ. Press · Zbl 1144.37002 [2] Barreira, L.; Valls, C., Stability of nonautonomous differential equations in Hilbert spaces, J. differential equations, 217, 204-248, (2005) · Zbl 1088.34053 [3] Barreira, L.; Valls, C., Growth rates and nonuniform hyperbolicity, Discrete contin. dyn. syst., 22, 509-528, (2008) · Zbl 1202.34096 [4] Barreira, L.; Valls, C., Stability of nonautonomous differential equations, Lecture notes in math., vol. 1926, (2008), Springer · Zbl 1152.34003 [5] L. Barreira, C. Valls, Polynomial growth rates, Nonlinear Anal., in press · Zbl 1181.34046 [6] Brin, M.; Pesin, Ja., Partially hyperbolic dynamical systems, Math. USSR izv., 8, 177-218, (1974) · Zbl 0309.58017 [7] Chicone, C.; Latushkin, Yu., Center manifolds for infinite dimensional nonautonomous differential equations, J. differential equations, 141, 356-399, (1997) · Zbl 0992.34033 [8] Chicone, C.; Latushkin, Yu., Evolution semigroups in dynamical systems and differential equations, Math. surveys monogr., vol. 70, (1999), Amer. Math. Soc. · Zbl 0970.47027 [9] Chow, S.-N.; Leiva, H., Existence and roughness of the exponential dichotomy for skew-product semiflow in Banach spaces, J. differential equations, 120, 429-477, (1995) · Zbl 0831.34067 [10] Chow, S.-N.; Liu, W.; Yi, Y., Center manifolds for invariant sets, J. differential equations, 168, 355-385, (2000) · Zbl 0972.34033 [11] Chow, S.-N.; Liu, W.; Yi, Y., Center manifolds for smooth invariant manifolds, Trans. amer. math. soc., 352, 5179-5211, (2000) · Zbl 0953.34038 [12] Conley, C.; Miller, R., Asymptotic stability without uniform stability: almost periodic coefficients, J. differential equations, 1, 333-336, (1965) · Zbl 0145.11401 [13] Coppel, W., Dichotomies and reducibility, J. differential equations, 3, 500-521, (1967) · Zbl 0162.39104 [14] Coppel, W., Dichotomies in stability theory, Lecture notes in math., vol. 629, (1978), Springer · Zbl 0376.34001 [15] Dalec’kiĭ, Ju.; Kreĭn, M., Stability of solutions of differential equations in Banach space, Transl. math. monogr., vol. 43, (1974), Amer. Math. Soc. [16] Elaydi, S.; Hájek, O., Exponential trichotomy of differential systems, J. math. anal. appl., 129, 362-374, (1988) · Zbl 0651.34052 [17] Eliasson, L., Almost reducibility of linear quasi-periodic systems, (), 679-705 · Zbl 1015.34028 [18] Hahn, W., The present state of Lyapunov’s direct method, (), 195-205 · Zbl 0111.28403 [19] Hale, J., Asymptotic behavior of dissipative systems, Math. surveys monogr., vol. 25, (1988), Amer. Math. Soc. · Zbl 0642.58013 [20] Henry, D., Geometric theory of semilinear parabolic equations, Lecture notes in math., vol. 840, (1981), Springer · Zbl 0456.35001 [21] Kelley, A., The stable, center-stable, center, center-unstable, unstable manifolds, J. differential equations, 3, 546-570, (1967) · Zbl 0173.11001 [22] Massera, J.; Schäffer, J., Linear differential equations and functional analysis, I, Ann. of math. (2), 67, 517-573, (1958) · Zbl 0178.17701 [23] Massera, J.; Schäffer, J., Linear differential equations and function spaces, Pure appl. math., vol. 21, (1966), Academic Press · Zbl 0202.14701 [24] Mielke, A., A reduction principle for nonautonomous systems in infinite-dimensional spaces, J. differential equations, 65, 68-88, (1986) · Zbl 0601.35018 [25] Naulin, R.; Pinto, M., Admissible perturbations of exponential dichotomy roughness, Nonlinear anal., 31, 559-571, (1998) · Zbl 0902.34041 [26] Oseledets, V., A multiplicative ergodic theorem. Liapunov characteristic numbers for dynamical systems, Trans. Moscow math. soc., 19, 197-221, (1968) · Zbl 0236.93034 [27] Perron, O., Die stabilitätsfrage bei differentialgleichungen, Math. Z., 32, 703-728, (1930) · JFM 56.1040.01 [28] Pesin, Ya., Families of invariant manifolds corresponding to nonzero characteristic exponents, Math. USSR izv., 10, 1261-1305, (1976) · Zbl 0383.58012 [29] Pesin, Ya., Characteristic ljapunov exponents, and smooth ergodic theory, Russian math. surveys, 32, 55-114, (1977) · Zbl 0383.58011 [30] Pliss, V., A reduction principle in the theory of stability of motion, Izv. akad. nauk SSSR ser. mat., 28, 1297-1324, (1964) · Zbl 0131.31505 [31] Pliss, V.; Sell, G., Robustness of exponential dichotomies in infinite-dimensional dynamical systems, J. dynam. differential equations, 11, 471-513, (1999) · Zbl 0941.37052 [32] Popescu, L., Exponential dichotomy roughness on Banach spaces, J. math. anal. appl., 314, 436-454, (2006) · Zbl 1093.34022 [33] Sacker, R.; Sell, G., Existence of dichotomies and invariant splittings for linear differential systems, I, J. differential equations, 15, 429-458, (1974) · Zbl 0294.58008 [34] Sacker, R.; Sell, G., Existence of dichotomies and invariant splittings for linear differential systems, III, J. differential equations, 22, 497-522, (1976) · Zbl 0338.58016 [35] Sell, G.; You, Y., Dynamics of evolutionary equations, Appl. math. sci., vol. 143, (2002), Springer · Zbl 1254.37002 [36] Vanderbauwhede, A., Centre manifolds, normal forms and elementary bifurcations, (), 89-169 · Zbl 0677.58001 [37] Vanderbauwhede, A.; Iooss, G., Center manifold theory in infinite dimensions, (), 125-163 · Zbl 0751.58025 [38] Vanderbauwhede, A.; van Gils, S., Center manifolds and contractions on a scale of Banach spaces, J. funct. anal., 72, 209-224, (1987) · Zbl 0621.47050
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.