We investigate noise-induced transitions in non-gradient systems when complex invariant sets emerge. Our example is the Lorenz system in three representative Rayleigh number regimes. It is found that before the homoclinic explosion bifurcation, the only transition state is the saddle point, and the transition is similar to that in gradient systems. However, when the chaotic invariant set emerges, an unstable limit cycle continues from the homoclinic trajectory. This orbit, which is embedded in a local tube-like manifold around the initial stable stationary point as a relative attractor, plays the role of the most probable exit set in the transition process. This example demonstrates how limit cycles, the next simplest invariant set beyond fixed points, can be involved in the transition process in smooth dynamical systems.

Recently the author developed a numerical method for the multidimensional momentconstrained maximum entropy problem, which is practically capable of solving maximum entropy problems in the two-dimensional domain with moment constraints of order up to 8, in the threedimensional domain with moment constraints of order up to 6, and in the four-dimensional domain with moment constraints of order up to 4, corresponding to the total number of moment constraints of 44, 83 and 69, respectively. In this work, the author brings together key algorithms and observations from his previous works as well as other literature in an attempt to present a comprehensive exposition of the current methods and results for the multidimensional maximum entropy moment problem.

We consider application of the stochastic mode-reduction strategy to a particular class of coupled models where a part of self-interactions of the slow variables is given by a rotationally invariant gradient system. The stochastic mode-reduction strategy is utilized to derive stochastic reduced models which yield a simple description of the phenomena resulting from breaking the original rotational symmetry. It is demonstrated that the direction of the symmetry breaking can be predicted a priori without any knowledge of the statistical behavior of the fast modes.

We study the impact of stochastic mechanisms on a coupled hybrid system consisting of a general advection-diffusion-reaction partial differential equation and a spatially distributed stochastic lattice noise model. The stochastic dynamics include both spin-flip and spin-exchange type interparticle interactions. Furthermore, we consider a new, asymmetric, single exclusion pro- cess, studied elsewhere in the context of traffic flow modeling, with an one-sided interaction potential which imposes advective trends on the stochastic dynamics. This look-ahead stochastic mechanism is responsible for rich nonlinear behavior in solutions. Our approach relies heavily on first deriving approximate differential mesoscopic equations. These approximations become exact either in the long range, Kac interaction partial differential equation case, or, given sufficient time separation con- ditions, between the partial differential equation and the stochastic model giving rise to a stochastic averaging partial differential equation. Although these approximations can in some cases be crude, they can still give a first indication, via linearized stability analysis, of the interesting regimes for the stochastic model. Motivated by this linearized stability analysis we choose particular regimes where interacting nonlinear stochastic waves are responsible for phenomena such as random switching, convective instability, and metastability, all driven by stochasticity. Numerical kinetic Monte Carlo simulations of the coarse grained hybrid system are implemented to assist in producing solutions and understanding their behavior.

We consider stochastically perturbed gradient flows in the limit when the amplitude of random fluctuations is small relative to the typical energy scale in the system and the minima of the energy are not isolated but form submanifolds of the phase space. In this case the limiting dynamics may be described in terms of a diffusion process on these manifolds. We derive explicit equations for this limiting dynamics and illustrate them on a few finite-dimensional examples. Finally, we formally extrapolate the reduction technique to several infinite-dimensional examples and derive equations of the stochastic kink motion in Allen-Cahn-type systems.

We study the diffusion limit of the Vlasov-Poisson-Fokker-Planck System. Here, we generalize the local in time results and the two dimensional results of Poupaud-Soler and of Goudon to the case of several space dimensions. Renormalization techniques, the method of moments and a velocity averaging lemma are used to prove the convergence of free energy solutions (renormalized solutions) to the Vlasov-Poisson-Fokker- Planck system towards a global weak solution of the Drift-Diffusion-Poisson model.

We examine stochastic coarse-graining strategies for two biomolecular systems. First, we compute the large-scale transport properties of the basic flashing ratchet mathematical model for (Brownian) molecular motors and consider in this light whether the underlying continuous-space, continuous-time Markovian model can be coarse-grained as a discrete-state, continuous-time Markovian random walk model. Through careful computation of associated statistical signatures of Markovianity, we find that such a discrete coarse-graining is an excellent approximation over much but not all of the parameter regime. In particular, for the parameter values associated with the fastest transport by the flashing ratchet, the discretized model displays non-Markovian features such as waiting times between jumps which are not exponentially distributed. We provide a theoretical framework for understanding the conditions under which Markovianity is to be expected in the discretized model and two mechanisms by which the flashing ratchet model coarse-grains to a non-Markovian discretized model. Next we turn to a basic question of how the dynamics of water molecules near the surface of a solute can be represented by a simple drift-diffusion stochastic model. This question is of most interest for the purpose of accelerating molecular dynamics simulations of proteins, but for simplicity, here we examine the simple case where the solute is a C60 buckyball, which has a homogenous, roughly isotropic form. We compare the mathematical drift-diffusion framework with a statistical quantification of water dynamics near a solute discussed in the biophysical literature. A key concern is the choice of time interval on which to sample the molecular dynamics data to generate estimators for the drift and diffusivity. We use a simple mathematical toy model to establish insight and a strategy, but find for the actual molecular dynamics data that the sampling times which produce the most faithful drift coefficient and the sampling times which produce the most faithful diffusion coefficient do not overlap, so that sacrifice of quality in one or the other parameter appears necessary.

We derive a multi-scale model of moist tropical dynamics which is valid on horizontal synoptic scales, zonal planetary scales, and synoptic and intraseasonal time scales. The Intraseasonal Multi-Scale Moist Dynamics (IMMD) framework builds on the IPESD framework of A.J. Majda and R. Klein, J. Atmos. Sci., 60, 393–408, 2003. It generalizes the latter by allowing for strong zonal winds (the Trade Winds) and the pressure and stratification variations that they generate. The framework consists of three pieces. The first, called TH, are planetary scale climatology modulation equations which govern the Trade Winds and Hadley Circulation. Self-consistency of the asymptotic theory requires that the meridional component of the Hadley Circulation is an order of magnitude weaker than the zonal component. The second piece, S, is a linear system of equations which govern synoptic scale velocity, temperature, and pressure fluctuations forced by synoptic scale heating fluctuations. Unlike the IPESD theory, these fluctuations are advected by part of the planetary scale climatology from TH. Since the meridional component of TH is an order of magnitude weaker than the zonal component, the synoptic scale fluctuations are only advected by the latter. The third, P, govern the planetary scale anomalies which, like IPESD, are driven both by planetary scale mean heating and by upscale fluxes from the synoptic scales. These planetary scale anomalies are advected both by the zonal component of the Trade Winds and by the meridional component of the Hadley Circulation and, furthermore, respond to an in-scale flux from the mean climatology. We also present an asymptotic analysis of the equations of bulk cloud thermodynamics in order to lay out a self-contained path for incorporating synoptic scale cloud models into the IMMD framework. This framework has potentially important implications for the development of models describing the Madden-Julian Oscillation (MJO) since the MJO manifests itself as planetary scale anomalies from a mean climatology which it modulates on intraseasonal time scales.

Synchronous and asynchronous dynamics in all-to-all coupled networks of identical, excitatory, current-based, integrate-and-fire (I&F) neurons with delta-impulse coupling currents and Poisson spike-train external drive are studied. Repeating synchronous total firing events, during which all the neurons fire simultaneously, are observed using numerical simulations and found to be the attracting state of the network for a large range of parameters. Mechanisms leading to such events are then described in two regimes of external drive: superthreshold and subthreshold. In the former, a probabilistic argument similar to the proof of the Central Limit Theorem yields the oscillation period, while in the latter, this period is analyzed via an exit time calculation utilizing a diffusion approximation of the Kolmogorov forward equation. Asynchronous dynamics are observed computationally in networks with random transmission delays. Neuronal voltage probability density functions (PDFs) and gain curves—graphs depicting the dependence of the network firing rate on the external drive strength—are analyzed using the steady solutions of the self-consistency problem for a Kolmogorov forward equation. All the voltage PDFs are obtained analytically, and asymptotic solutions for the gain curves are obtained in several physiologically relevant limits. The absence of chaotic dynamics is proved for the type of network under investigation by demonstrating convergence in time of its trajectories.

In this note we revisit the homogenization theory of Hamilton-Jacobi and “viscous”- Hamilton-Jacobi partial differential equations with convex nonlinearities in stationary ergodic envi- ronments. We present a new simple proof for the homogenization in probability. The argument uses some a priori bounds (uniform modulus of continuity) on the solution and the convexity and coer- civity (growth) of the nonlinearity. It does not rely, however, on the control interpretation formula of the solution as was the case with all previously known proofs. We also introduce a new formula for the effective Hamiltonian for Hamilton-Jacobi and “viscous” Hamilton-Jacobi equations.