Luu Hoang Duc

HYPERBOLICITY AND INVARIANT MANIFOLDS FOR PLANAR NONAUTONOMOUS SYSTEMS ON FINITE TIME INTERVALS

The method of invariant manifolds was originally developed for hyperbolic rest points of autonomous equations. It was then extended from fixed points to arbitrary solutions and from autonomous equations to nonautonomous dynamical systems by either the Lyapunov–Perron approach or Hadamards graph transformation. We go one step further and study meaningful notions of hyperbolicity and stable and unstable manifolds for equations which are defined or known only for a finite time, together with matching notions of attraction and repulsion. As a consequence, hyperbolicity and invariant manifolds will describe the dynamics on the finite time interval. We prove an analog of the Theorem of Linearized Asymptotic Stability on finite time intervals, generalize the Okubo–Weiss criterion from fluid dynamics and prove a theorem on the location of periodic orbits. Several examples are treated, including a double gyre flow and symmetric vortex merger.

On stability of linear time-varying second-order differential equations

We derive sufficient conditions for stability and asymptotic stability of second order, scalar differential equations with differentiable coefficients.

TOWARDS A MORSE THEORY FOR RANDOM DYNAMICAL SYSTEMS

A generalization of the concepts of deterministic Morse theory to random dynamical systems is presented. Using the notions of attraction and repulsion in probability, the main building blocks of Morse theory such as attractor–repeller pairs, Morse sets, and the Morse decomposition are obtained for random dynamical systems.

A note on the generation of random dynamical systems from fractional stochastic delay differential equations

In this note we prove that a fractional stochastic delay differential equation which satisfies natural regularity conditions generates a continuous random dynamical system on a subspace of a Holder space which is separable.

Finite-time Lyapunov exponents and metabolic control coefficients for threshold detection of stimulus–response curves

ABSTRACT In biochemical networks transient dynamics plays a fundamental role, since the activation of signalling pathways is determined by thresholds encountered during the transition from an initial state (e.g. an initial concentration of a certain protein) to a steady-state. These thresholds can be defined in terms of the inflection points of the stimulus–response curves associated to the activation processes in the biochemical network. In the present work, we present a rigorous discussion as to the suitability of finite-time Lyapunov exponents and metabolic control coefficients for the detection of inflection points of stimulus–response curves with sigmoidal shape.

Asymptotic Behavior of Linear Almost Periodic Differential Equations

The present paper is concerned with strong stability of solutions of non-autonomous equations of the form \(\dot{u}(t) = A(t)u(t)\), where A(t) is an unbounded operator in a Banach space depending almost periodically on t. A general condition on strong stability is given in terms of Perron conditions on the solvability of the associated inhomogeneous equation.

Dynamics of SPDEs Driven by a Small Fractional Brownian Motion with Hurst Parameter Larger than 1/2

We consider mild solutions of an SPDE driven by a time dependent perturbation which is Holder continuous with a Holder exponent larger than 1/2. In particular, such a perturbation is given by a fractional Brownian motion with Hurst parameter larger than 1/2. The coefficient in front of this noise is an operator with bounded first and second derivatives. We formulate conditions such that the equation has a unique pathwise solution. Further we investigate the globally exponential stability of the trivial solution.

The stability of Try-Once-Discard for stochastic communication channels: Theory and validation

This paper concerns communication protocols for nonlinear networked control systems in a stochastic setting. Motivated by a recent implementation of the Maximum-Error-First/Try-Once-Discard (MEF/TOD) protocol for wireless networks, we analyze network control protocols in a stochastic framework. Specifically, the stochastic stability notions of almost sure attractivity and stability in probability can be guaranteed provided a bound on the maximum allowable transfer interval (MATI) is satisfied. We briefly present the implementation of TOD for wireless networks and experimental data validating the assumptions for the stochastic analysis.

ON THE ABSOLUTE REGULARITY OF LINEAR RANDOM DYNAMICAL SYSTEMS

We introduce a concept of absolute regularity of linear random dynamical systems (RDS) that is stronger than Lyapunov regularity. We prove that a linear RDS that satisfies the integrability conditions of the multiplicative ergodic theorem of Oseledets is not merely Lyapunov regular but absolutely regular.

A concept of local metric entropy for finite-time nonautonomous dynamical systems

We introduce a concept of entropy for difference and differential equations which is a local-in-space and transient-in-time version of the classical concept of metric entropy. Based on that, a finite-time (or transient) version of Pesin’s entropy theorem and also an explicit formula of finite-time entropy for 2-D systems are derived.