DCDS-B
An instability theorem for nonlinear fractional differential systems
Nguyen Dinh Cong Doan Thai Son Stefan Siegmund Hoang The Tuan
Discrete & Continuous Dynamical Systems - B 2017, 22(8): 3079-3090 doi: 10.3934/dcdsb.2017164
In this paper, we give a criterion on instability of an equilibrium of a nonlinear Caputo fractional differential system. More precisely, we prove that if the spectrum of the linearization has at least one eigenvalue in the sector
$\left\{ \lambda \in \mathbb{C}\setminus \{0\}:|\arg (\lambda )| < \frac{\alpha \pi }{2} \right\},$
where
$α∈ (0,1)$
is the order of the fractional differential system, then the equilibrium of the nonlinear system is unstable.
keywords: Fractional differential equations qualitative theory stability theory instability condition
DCDS-S
On integral separation of bounded linear random differential equations
Nguyen Dinh Cong Doan Thai Son
Discrete & Continuous Dynamical Systems - S 2016, 9(4): 995-1007 doi: 10.3934/dcdss.2016038
Our aim in this paper is to investigate the openness and denseness for the set of integrally separated systems in the space of bounded linear random differential equations equipped with the $L^{\infty}$-metric. We show that in the general case, the set of integrally separated systems is open and dense. An exception is the case when the base space is isomorphic to the ergodic rotation flow of the unit circle, in which the set of integrally separated systems is open but not dense.
keywords: Lyapunov exponents multiplicative ergodic theorem Random differential equations integral separation genericity.
DCDS-B
On the spectrum of the one-dimensional Schrödinger operator
Nguyen Dinh Cong Roberta Fabbri
Discrete & Continuous Dynamical Systems - B 2008, 9(3&4, May): 541-554 doi: 10.3934/dcdsb.2008.9.541
The spectral theory of the one-dimensional Schrödinger operator with a quasi-periodic potential can be fruitfully studied considering the corresponding differential system. In fact the presence of an exponential dichotomy for the system is equivalent to the statement that the energy $E$ belongs to the resolvent of the operator. Starting from results already obtained for the spectrum in the continuous case, we show that in the discrete case a generic bounded measurable Schrödinger cocycle has Cantor spectrum.
keywords: Quasi-periodic Schrödinger operator Cantor spectrum exponential dichotomy.
PROC
Coincidence of Lyapunov exponents and central exponents of linear Ito stochastic differential equations with nondegenerate stochastic term
Nguyen Dinh Cong Nguyen Thi Thuy Quynh
Conference Publications 2011, 2011(Special): 332-342 doi: 10.3934/proc.2011.2011.332
In this paper we show that under a nondegeneracy condition Lyapunov exponents and central exponents of linear Ito stochastic di erential equation coincide. Furthermore, as the stochastic term is small and tends to zero the highest Lyapunov exponent tends to the highest central exponent of the ordinary di erential equation which is the deterministic part of the system.
keywords: central exponents Lyapunov exponents Lyapunov spectrum two-parameter stochastic ow nonautonomous stochastic di erential equation
DCDS-B
On Lyapunov exponents of difference equations with random delay
Nguyen Dinh Cong Thai Son Doan Stefan Siegmund
Discrete & Continuous Dynamical Systems - B 2015, 20(3): 861-874 doi: 10.3934/dcdsb.2015.20.861
The multiplicative ergodic theorem by Oseledets on Lyapunov spectrum and Oseledets subspaces is extended to linear random difference equations with random delay. In contrast to the general multiplicative ergodic theorem by Lian and Lu, we can prove that a random dynamical system generated by a difference equation with random delay cannot have infinitely many Lyapunov exponents.
keywords: Random difference equations random delay multiplicative ergodic theorem Lyapunov exponent.

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