American Institute of Mathematical Sciences

Advanced
January  2010, 26(1): 63-74. doi: 10.3934/dcds.2010.26.63

Quadratic Lyapunov sequences and arbitrary growth rates

Luis Barreira 1, and  Claudia Valls 2,

1. 

Departamento de Matemática, Instituto Superior Técnico, Universidade de Lisboa, 1049-001 Lisboa
2. 

Departamento de Matemática, Instituto Superior Técnico, 1049-001 Lisboa

Received  August 2008 Revised  August 2009 Published  October 2009

We characterize completely in terms of strict quadratic Lyapunov sequences when a linear nonautonomous dynamics admits a nonuniform exponential dichotomy. We allow asymptotic rates of the form $e^{c\rho(m)}$ determined by an arbitrary increasing sequence $\rho(m)$, which thus may correspond to infinite Lyapunov exponents, and not only the usual exponential behavior with $\rho(m)=m$. In particular, we obtain inverse theorems in this general setting, by constructing explicitly a strict quadratic Lyapunov sequence for each nonuniform exponential dichotomy. Furthermore, for a large class of perturbations and using only quadratic Lyapunov sequences, we show in a simple manner that if the linear dynamics admits a nonuniform exponential dichotomy, then the perturbed dynamics remains unstable.
Keywords: Lyapunov sequences, nonuniform exponential dichotomies., arbitrary growth rates.
Mathematics Subject Classification: Primary: 34D09, 34D1.
Citation: Luis Barreira, Claudia Valls. Quadratic Lyapunov sequences and arbitrary growth rates. Discrete & Continuous Dynamical Systems - A, 2010, 26 (1) : 63-74. doi: 10.3934/dcds.2010.26.63
[1]

Luis Barreira, Claudia Valls. Center manifolds for nonuniform trichotomies and arbitrary growth rates. Communications on Pure & Applied Analysis, 2010, 9 (3) : 643-654. doi: 10.3934/cpaa.2010.9.643
[2]

Luis Barreira, Claudia Valls. Nonuniform exponential dichotomies and admissibility. Discrete & Continuous Dynamical Systems - A, 2011, 30 (1) : 39-53. doi: 10.3934/dcds.2011.30.39
[3]

Luis Barreira, Claudia Valls. Characterization of stable manifolds for nonuniform exponential dichotomies. Discrete & Continuous Dynamical Systems - A, 2008, 21 (4) : 1025-1046. doi: 10.3934/dcds.2008.21.1025
[4]

Luis Barreira, Claudia Valls. Growth rates and nonuniform hyperbolicity. Discrete & Continuous Dynamical Systems - A, 2008, 22 (3) : 509-528. doi: 10.3934/dcds.2008.22.509
[5]

Dung Le. Exponential attractors for a chemotaxis growth system on domains of arbitrary dimension. Conference Publications, 2003, 2003 (Special) : 536-543. doi: 10.3934/proc.2003.2003.536
[6]

Luis Barreira, Claudia Valls. Noninvertible cocycles: Robustness of exponential dichotomies. Discrete & Continuous Dynamical Systems - A, 2012, 32 (12) : 4111-4131. doi: 10.3934/dcds.2012.32.4111
[7]

Christian Pötzsche. Smooth roughness of exponential dichotomies, revisited. Discrete & Continuous Dynamical Systems - B, 2015, 20 (3) : 853-859. doi: 10.3934/dcdsb.2015.20.853
[8]

Luis Barreira, Claudia Valls. Delay equations and nonuniform exponential stability. Discrete & Continuous Dynamical Systems - S, 2008, 1 (2) : 219-223. doi: 10.3934/dcdss.2008.1.219
[9]

Sebastian J. Schreiber. Expansion rates and Lyapunov exponents. Discrete & Continuous Dynamical Systems - A, 1997, 3 (3) : 433-438. doi: 10.3934/dcds.1997.3.433
[10]

Luis Barreira, Claudia Valls. Admissibility versus nonuniform exponential behavior for noninvertible cocycles. Discrete & Continuous Dynamical Systems - A, 2013, 33 (4) : 1297-1311. doi: 10.3934/dcds.2013.33.1297
[11]

Peter Giesl, Sigurdur Hafstein. Existence of piecewise linear Lyapunov functions in arbitrary dimensions. Discrete & Continuous Dynamical Systems - A, 2012, 32 (10) : 3539-3565. doi: 10.3934/dcds.2012.32.3539
[12]

Oǧuz Yayla. Nearly perfect sequences with arbitrary out-of-phase autocorrelation. Advances in Mathematics of Communications, 2016, 10 (2) : 401-411. doi: 10.3934/amc.2016014
[13]

Wilhelm Schlag. Regularity and convergence rates for the Lyapunov exponents of linear cocycles. Journal of Modern Dynamics, 2013, 7 (4) : 619-637. doi: 10.3934/jmd.2013.7.619
[14]

Barbara Kaltenbacher, Irena Lasiecka. Global existence and exponential decay rates for the Westervelt equation. Discrete & Continuous Dynamical Systems - S, 2009, 2 (3) : 503-523. doi: 10.3934/dcdss.2009.2.503
[15]

Zhaoli Liu, Jiabao Su. Solutions of some nonlinear elliptic problems with perturbation terms of arbitrary growth. Discrete & Continuous Dynamical Systems - A, 2004, 10 (3) : 617-634. doi: 10.3934/dcds.2004.10.617
[16]

Sigurdur Freyr Hafstein. A constructive converse Lyapunov theorem on exponential stability. Discrete & Continuous Dynamical Systems - A, 2004, 10 (3) : 657-678. doi: 10.3934/dcds.2004.10.657
[17]

A. Giambruno and M. Zaicev. Minimal varieties of algebras of exponential growth. Electronic Research Announcements, 2000, 6: 40-44.

[18]

Frank Blume. Realizing subexponential entropy growth rates by cutting and stacking. Discrete & Continuous Dynamical Systems - B, 2015, 20 (10) : 3435-3459. doi: 10.3934/dcdsb.2015.20.3435
[19]

John A. D. Appleby, Denis D. Patterson. Subexponential growth rates in functional differential equations. Conference Publications, 2015, 2015 (special) : 56-65. doi: 10.3934/proc.2015.0056
[20]

Denis Dmitriev, Jonathan Jedwab. Bounds on the growth rate of the peak sidelobe level of binary sequences. Advances in Mathematics of Communications, 2007, 1 (4) : 461-475. doi: 10.3934/amc.2007.1.461

2018 Impact Factor: 1.143

Tools

Metrics

  • PDF downloads (9)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences