Thai Son Doan

On stable manifolds for planar fractional differential equations

In this paper, we establish a local stable manifold theorem near a hyperbolic equilibrium point for planar fractional differential equations. The construction of this stable manifold is based on the associated Lyapunov-Perron operator. An example is provided to illustrate the result.

Stability radii for positive linear time-invariant systems on time scales

We deal with dynamic equations on time scales, where we characterize the positivity of a system. Uniform exponential stability of a system is determined by the spectrum of its matrix. We investigate the corresponding stability radii with respect to structured perturbations and show that, for positive systems, the complex and the real stability radius coincide.

The dichotomy spectrum for random dynamical systems and pitchfork bifurcations with additive noise

The second author was supported by a Marie Curie IEF Fellowship, the third author acknowledges the support by Nizhny Novgorod University through the grant RNF 14-41-00044, and the fourth author was supported by an EPSRC Career Acceleration Fellowship EP/I004165/1. This research has been supported by EU Marie-Curie IRSES Brazilian–European Partnership in Dynamical Systems (FP7-PEOPLE-2012-IRSES 318999 BREUDS) and EU Marie-Sklodowska-Curie ITN Critical Transitions in Complex Systems (H2020-MSCA-2014-ITN 643073 CRITICS).

Difference equations with random delay

We present a first step towards a general theory of difference and differential equations incorporating unbounded random delays. The main technical tool relies on recent work of Lian and Lu, which generalizes the multiplicative ergodic theorem by Oseledets to Banach spaces.

A Constructive Approach to Linear Lyapunov Functions for Positive Switched Systems Using Collatz-Wielandt Sets

We establish the link between linear Lyapunov functions for positive switched systems and corresponding Collatz-Wielandt sets. This leads to an algorithm to compute a linear Lyapunov function whenever a Lyapunov function exists.

Asymptotic Stability of Linear Fractional Systems with Constant Coefficients and Small Time-Dependent Perturbations

Our aim in this paper is to investigate the asymptotic behavior of solutions of the perturbed linear fractional differential system. We show that if the original linear autonomous system is asymptotically stable and then under the action of small (either linear or nonlinear) nonautonomous perturbations, the trivial solution of the perturbed system is also asymptotically stable.

Stability of positive linear switched systems on ordered Banach spaces

Abstract We provide sufficient criteria for the stability of positive linear switched systems on ordered Banach spaces. The switched systems can be generated by finitely many bounded operators in infinite-dimensional spaces with a general class of order-inducing cones. In the discrete-time case, we assume an appropriate interior point of the cone, whereas in the continuous-time case an appropriate interior point of the dual cone is sufficient for stability. This is an extension of the concept of linear Lyapunov functions for positive systems to the setting of infinite-dimensional partially ordered spaces. We illustrate our results with examples.

Hopf bifurcation with additive noise

We consider the dynamics of a two-dimensional ordinary differential equation exhibiting a Hopf bifurcation subject to additive white noise and identify three dynamical phases: (I) a random attractor with uniform synchronisation of trajectories, (II) a random attractor with non-uniform synchronisation of trajectories and (III) a random attractor without synchronisation of trajectories. The random attractors in phases (I) and (II) are random equilibrium points with negative Lyapunov exponents while in phase (III) there is a so-called random strange attractor with positive Lyapunov exponent. We analyse the occurrence of the different dynamical phases as a function of the linear stability of the origin (deterministic Hopf bifurcation parameter) and shear (ampitude-phase coupling parameter). We show that small shear implies synchronisation and obtain that synchronisation cannot be uniform in the absence of linear stability at the origin or in the presence of sufficiently strong shear. We provide numerical results in support of a conjecture that irrespective of the linear stability of the origin, there is a critical strength of the shear at which the system dynamics loses synchronisation and enters phase (III).

Finite-Time Attractivity for Diagonally Dominant Systems with Off-Diagonal Delays

We introduce a notion of attractivity for delay equations which are defined on bounded time intervals. Our main result shows that linear delay equations are finite-time attractive, provided that the delay is only in the coupling terms between different components, and the system is diagonally dominant. We apply this result to a nonlinear Lotka-Volterra system and show that the delay is harmless and does not destroy finite-time attractivity.

The Bohl Spectrum for Linear Nonautonomous Differential Equations

We develop the Bohl spectrum for nonautonomous linear differential equations on a half line, which is a spectral concept that lies between the Lyapunov and the Sacker–Sell spectra. We prove that the Bohl spectrum is given by the union of finitely many intervals, and we show by means of an explicit example that the Bohl spectrum does not coincide with the Sacker–Sell spectrum in general even for bounded systems. We demonstrate for this example that any higher-order nonlinear perturbation is exponentially stable (which is not evident from the Sacker–Sell spectrum), but we show that in general this is not true. We also analyze in detail situations in which the Bohl spectrum is identical to the Sacker–Sell spectrum.