Jinqiao Duan

Smooth Stable and Unstable Manifolds for Stochastic Evolutionary Equations

Invariant manifolds are fundamental tools for describing and understanding nonlinear dynamics. In this paper, we present a theory of stable and unstable manifolds for infinite dimensional random dynamical systems generated by a class of stochastic partial differential equations. We first show the existence of Lipschitz continuous stable and unstable manifolds by the Lyapunov–Perron’s method. Then, we prove the smoothness of these invariant manifolds.

Fractional Fokker-Planck equation for nonlinear stochastic differential equations driven by non-Gaussian Levy stable noises

The Fokker–Planck equation has been very useful for studying dynamic behavior of stochastic differential equations driven by Gaussian noises. However, there are both theoretical and empirical reasons to consider similar equations driven by strongly non-Gaussian noises. In particular, they yield strongly non-Gaussian anomalous diffusion which seems to be relevant in different domains of Physics. In this paper, we therefore derive a fractional Fokker–Planck equation for the probability distribution of particles whose motion is governed by a nonlinear Langevin-type equation, which is driven by a Levy stable noise rather than a Gaussian. We obtain in fact a general result for a Markovian forcing. We also discuss the existence and uniqueness of the solution of the fractional Fokker–Planck equation.

Three-Dimensional Turbulent Bottom Density Currents from a High-Order Nonhydrostatic Spectral Element Model

Overflows are bottom gravity currents that supply dense water masses generated in high-latitude and marginal seas into the general circulation. Oceanic observations have revealed that mixing of overflows with ambient water masses takes place over small spatial and time scales. Studies with ocean general circulation models indicate that the strength of the thermohaline circulation is strongly sensitive to representation of overflows in these models. In light of these results, overflow-induced mixing emerges as one of the prominent oceanic processes. In this study, as a continuation of an effort to develop appropriate process models for overflows, nonhydrostatic 3D simulations of bottom gravity are carried out that would complement analysis of dedicated observations and large-scale ocean modeling. A parallel high-order spectral-element Navier‐Stokes solver is used as the basis of the simulations. Numerical experiments are conducted in an idealized setting focusing on the startup phase of a dense water mass released at the top of a sloping wedge. Results from 3D experiments are compared with results from 2D experiments and laboratory experiments, based on propagation speed of the density front, growth rate of the characteristic head at the leading edge, turbulent overturning length scales, and entrainment parameters. Results from 3D experiments are found to be in general agreement with those from laboratory tank experiments. In 2D simulations, the propagation speed is approximately 20% slower than that of the 3D experiments and the head growth rate is 3 times as large, Thorpe scales are 1.3‐1.5 times as large, and the entrainment parameter is up to 2 times as large as those in the 3D experiments. The differences between 2D and 3D simulations are entirely due to internal factors associated with the truncation of the Navier‐Stokes equations for 2D approximation.

Restoration of rhythmicity in diffusively coupled dynamical networks

Oscillatory behaviour is essential for proper functioning of various physical and biological processes. However, diffusive coupling is capable of suppressing intrinsic oscillations due to the manifestation of the phenomena of amplitude and oscillation deaths. Here we present a scheme to revoke these quenching states in diffusively coupled dynamical networks, and demonstrate the approach in experiments with an oscillatory chemical reaction. By introducing a simple feedback factor in the diffusive coupling, we show that the stable (in)homogeneous steady states can be effectively destabilized to restore dynamic behaviours of coupled systems. Even a feeble deviation from the normal diffusive coupling drastically shrinks the death regions in the parameter space. The generality of our method is corroborated in diverse non-linear systems of diffusively coupled paradigmatic models with various death scenarios. Our study provides a general framework to strengthen the robustness of dynamic activity in diffusively coupled dynamical networks.

Fluid Exchange across a Meandering Jet Quasiperiodic Variability

In this paper, the motion of fluid parcels in the two-dimensional kinematic model of a meandering jet developed by Bower and Samelson is studied. The earlier work is extended by considering quasiperiodic spatiotemporal variability in a reference frame moving with the phase speed of the meander. This necessitates the introduction of recently developed techniques in dynamical systems theory for analyzing transport in velocity fields with quasiperiodic variability. A detailed comparison between exchange for variability with one and two independent frequencies is given, and it is shown that the exchange rates may be very different for periodic and quasiperiodic variability.

Entrainment in bottom gravity currents over complex topography from three-dimensional nonhydrostatic simulations

[1] By recognizing that oceanic overflows follow the seafloor morphology, which shows a self-similar structure at spatial scales ranging from 100 km to 1 m, the impact of topographic bumps on entrainment in gravity currents is investigated using a 3D nonhydrostatic spectral element model. It is found that a bumpy surface can lead to a significant enhancement of entrainment compared to a smooth surface. The change in entrainment is parameterized as a function of statistical estimates of the amplitude and wavenumber parameters of bumps with respect to the background slope.

Lévy noise-induced stochastic resonance in a bistable system

AbstractThe stochastic resonance phenomenon induced by Lévy noise in a second-order and under-damped bistable system is investigated. The signal-to-noise ratio for different parameters is computed by an efficient numerical scheme. The influences of the intensity and stability index of Lévy noise, as well as the amplitude of external signal on the occurrence of stochastic resonance phenomenon are characterized. The results imply that higher signal amplitude not only enhances the output power spectrum of system but also promotes stochastic resonance, and a proper adjustment of noise intensity in a certain range enlarges the peak value of output power spectrum which is significant for stochastic resonance. Moreover, with an appropriate damping parameter, lowering the stability index leads to larger fluctuations of Lévy noise, and further weakens the occurrence of the stochastic resonance.

Probability and partial differential equations in modern applied mathematics

Nonnegative Markov Chains with Applications.- Phase Changes with Time and Multi-Scale Homogenizations of a Class of Anomalous Diffusions.- Semi-Markov Cascade Representations of Local Solutions to 3-D Incompressible Navier-Stokes.- Amplitude Equations for Spdes: Approximate Centre Manifolds and Invariant Measures.- Enstrophy and Ergodicity Of Gravity Currents.- Stochastic Heat and Burgers Equations and Their Singularities.- A Gentle Introduction to Cluster Expansions.- Continuity of the Ito-Map for Holder Rough Paths with Applications to the Support Theorem in Holder Norm.- Data-Driven Stochastic Processes in Fully Developed Turbulence.- Stochastic Flows on the Circle.- Path Integration: Connecting Pure Jump and Wiener Processes.- Random Dynamical Systems in Economics.- A Geometric Cascade for the Spectral Approximation of the Navier-Stokes Equations.- Inertial Manifolds for Random Differential Equations.- Existence and Uniqueness of Classical, Nonnegative, Smooth Solutions of a Class of Semi-Linear Spdes.- Nonlinear Pdes Driven by Levy Diffusions and Related Statistical Issues.

Mean exit time and escape probability for dynamical systems driven by Lévy Noises

The mean first exit time and escape probability are utilized to quantify dynamical behaviors of stochastic differential equations with non-Gaussian

Uniform Attractors for Nonautonomous Wave Equations with Nonlinear Damping

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