Non-explosion by Stratonovich noise for ODEs

We show that the addition of a suitable Stratonovich noise prevents the explosion for ODEs with drifts of super-linear growth, in dimension $d\ge 2$. We also show the existence of an invariant measure and the geometric ergodicity for the corresponding SDE.Comment: 16 page

On the asymptotic stability and numerical analysis of solutions to nonlinear stochastic differential equations with jumps

This paper is concerned with the stability and numerical analysis of solution to highly nonlinear stochastic differential equations with jumps. By the Itô formula, stochastic inequality and semi-martingale convergence theorem, we study the asymptotic stability in the pth moment and almost sure exponential stability of solutions under the local Lipschitz condition and nonlinear growth condition. On the other hand, we also show the convergence in probability of numerical schemes under nonlinear growth condition. Finally, an example is provided to illustrate the theor

Particles and fields in fluid turbulence

The understanding of fluid turbulence has considerably progressed in recent years. The application of the methods of statistical mechanics to the description of the motion of fluid particles, i.e. to the Lagrangian dynamics, has led to a new quantitative theory of intermittency in turbulent transport. The first analytical description of anomalous scaling laws in turbulence has been obtained. The underlying physical mechanism reveals the role of statistical integrals of motion in non-equilibrium systems. For turbulent transport, the statistical conservation laws are hidden in the evolution of groups of fluid particles and arise from the competition between the expansion of a group and the change of its geometry. By breaking the scale-invariance symmetry, the statistically conserved quantities lead to the observed anomalous scaling of transported fields. Lagrangian methods also shed new light on some practical issues, such as mixing and turbulent magnetic dynamo.Comment: 165 pages, review article for Rev. Mod. Phy

Razumikhin-type theorem for stochastic functional differential systems via vector Lyapunov function

This paper is concerned with input-to-state stability of SFDSs. By using stochastic analysis techniques, Razumikhin techniques and vector Lyapunov function method, vector Razumikhin-type theorem has been established on input-to-state stability for SFDSs. Novel sufficient criteria on the pth moment exponential input-to-state stability are obtained by the established vector Razumikhin-type theorem. When input is zero, an improved criterion on exponential stability is obtained. Two examples are provided to demonstrate validity of the obtained results

The kinematics, dynamics and statistics of three-wave interactions in models of geophysical flow

We study the dynamics, kinematics and statistics of resonant and quasiresonant three-wave interactions appearing in models of geophysical flow. In these dispersive wave systems, the phenomenon of nonlinear resonance broadening plays a significant role across all three different branches of wave turbulence theory: from the statistical, to the discrete, and even the mesoscopic, formed as an intermediate regime between the two. The principal aim of this thesis is to understand the processes by which resonance broadening can induce a transition between each of these three different regimes. Beginning with the discrete case, we study two variants of the isolated triad: one with a constant additive forcing term; and the other in the presence of detuning. We provide a detailed analysis of both of these systems, covering their integrability and boundedness properties, showing that for almost all initial conditions the motion remains quasi-periodic and periodic respectively. Interestingly, we show that moderate amounts of detuning can actually promote energy exchange, increase the period and in rare instances cease to be periodic at all; each of these statements are contrary to what was previously thought. This motivates a more detailed study into the kinematics of resonance broadening. By analysing how the set of quasi-resonant modes develops under increased broadening, we show that a percolation-like transition exists, independent of the dispersion relationship used. At critical levels of broadening, we see the emergence of a single quasi-resonant cluster that begins to dominate the entire system. We argue that the formation of this cluster provides a way of characterising the turbulent state of the system, distinguishing between the discrete and statistical regimes. Through direct numerical simulation of the Charney-Hasegawa-Mima equation, we then assess whether this view is truly representative of the underlying dynamics. Here we find that the generation of quasi-resonantly excited modes can be detected through the statistical measures of total correlation and mutual information. We conclude by suggesting that these techniques have an incredible potential to infer the signature of both resonant and quasi-resonant clusters in fully realised turbulent systems, and yet are also subtle enough to detect qualitative changes in the underlying dynamics between different interacting modes

Dynamics of Macrosystems; Proceedings of a Workshop, September 3-7, 1984

There is an increasing awareness of the important and persuasive role that instability and random, chaotic motion play in the dynamics of macrosystems. Further research in the field should aim at providing useful tools, and therefore the motivation should come from important questions arising in specific macrosystems. Such systems include biochemical networks, genetic mechanisms, biological communities, neutral networks, cognitive processes and economic structures. This list may seem heterogeneous, but there are similarities between evolution in the different fields. It is not surprising that mathematical methods devised in one field can also be used to describe the dynamics of another. IIASA is attempting to make progress in this direction. With this aim in view this workshop was held at Laxenburg over the period 3-7 September 1984. These Proceedings cover a broad canvas, ranging from specific biological and economic problems to general aspects of dynamical systems and evolutionary theory