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Register a new userStochastic limit-cycle oscillations of a nonlinear system under random perturbations
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Dynamical systems with $epsilon$ small random perturbations appear in both continuous mechanical motions and discrete stochastic chemical kinetics. The present work provides a detailed analysis of the central limit theorem (CLT), with a time-inhomogeneous Gaussian process, near a deterministic limit cycle in $mathbb{R}^n$. Based on the theory of random perturbations of dynamical systems and the WKB approximation respectively, results are developed in parallel from both standpoints of stochastic trajectories and transition probability density and their relations are elucidated. We show rigorously the correspondence between the local Gaussian fluctuations and the curvature of the large deviation rate function near its infimum, connecting the CLT and the large deviation principle of diffusion processes. We study uniform asymptotic behavior of stochastic limit cycles through the interchange of limits of time $ttoinfty$ and $epsilonto 0$. Three further characterizations of stochastic limit cycle oscillators are obtained: (i) An approximation of the probability flux near the cycle; (ii) Two special features of the vector field for the cyclic motion; (iii) A local entropy balance equation along the cycle with clear physical meanings. Lastly and different from the standard treatment, the origin of the $epsilon$ in the theory is justified by a novel scaling hypothesis via constructing a sequence of stochastic differential equations.
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We consider the dynamics of a rigid filament in a motor protein assay under external loading. The motor proteins are modeled as active harmonic linkers with tail ends immobilized on a substrate. Their heads attach to the filament stochastically to extend along it, resulting in a force on the filament, before detaching. The rate of extension and detachment are load dependent. Here we formulate and characterize the governing dynamics in the mean field approximation using linear stability analysis, and direct numerical simulations of the motor proteins and filament. Under constant loading, the system shows transition from a stable configuration to instability towards detachment of the filament from motor proteins. Under elastic loading, we find emergence of stable limit cycle oscillations via a supercritical Hopf bifurcation with change in activity and the number of motor proteins. Numerical simulations of the system for large number of motor proteins show good agreement with the mean field predictions.
We analyze the behavior of quantum dynamical entropies production from sequences of quantum approximants approaching their (chaotic) classical limit. The model of the quantized hyperbolic automorphisms of the 2-torus is examined in detail and a semi-classical analysis is performed on it using coherent states, fulfilling an appropriate dynamical localization property. Correspondence between quantum dynamical entropies and the Kolmogorov-Sinai invariant is found only over time scales that are logarithmic in the quantization parameter.
This paper compares the results of applying a recently developed method of stochastic uncertainty quantification designed for fluid dynamics to the Born-Infeld model of nonlinear electromagnetism. The similarities in the results are striking. Namely, the introduction of Stratonovich cylindrical noise into each of their Hamiltonian formulations introduces stochastic Lie transport into their dynamics in the same form for both theories. Moreover, the resulting stochastic partial differential equations (SPDE) retain their unperturbed form, except for an additional term representing induced Lie transport by the set of divergence-free vector fields associated with the spatial correlations of the cylindrical noise. The explanation for this remarkable similarity lies in the method of construction of the Hamiltonian for the Stratonovich stochastic contribution to the motion in both cases; which is done via pairing spatial correlation eigenvectors for cylindrical noise with the momentum map for the deterministic motion. This momentum map is responsible for the well-known analogy between hydrodynamics and electromagnetism. The momentum map for the Maxwell and Born-Infeld theories of electromagnetism treated here is the 1-form density known as the Poynting vector. Two Appendices treat the Hamiltonian structures underlying these results.
We investigate a dynamical system consisting of $N$ particles moving on a $d$-dimensional torus under the action of an electric field $E$ with a Gaussian thermostat to keep the total energy constant. The particles are also subject to stochastic collisions which randomize direction but do not change the speed. We prove that in the van Hove scaling limit, $Eto 0$ and $tto t/E^2$, the trajectory of the speeds $v_i$ is described by a stochastic differential equation corresponding to diffusion on a constant energy sphere. This verifies previously conjectured behavior. Our results are based on splitting the systems evolution into a slow process and an independent noise. We show that the noise, suitably rescaled, converges a Brownian motion, enhanced in the sense of rough paths. Then we employ the It^o-Lyons continuity theorem to identify the limit of the slow process.
In the pattern matching approach to imaging science, the process of emph{metamorphosis} in template matching with dynamical templates was introduced in cite{ty05b}. In cite{HoTrYo2009} the metamorphosis equations of cite{ty05b} were recast into the Euler-Poincare variational framework of cite{HoMaRa1998} and shown to contain the equations for a perfect complex fluid cite{Holm2002}. This result related the data structure underlying the process of metamorphosis in image matching to the physical concept of order parameter in the theory of complex fluids cite{GBHR2013}. In particular, it cast the concept of Lagrangian paths in imaging science as deterministically evolving curves in the space of diffeomorphisms acting on image data structure, expressed in Eulerian space. (In contrast, the landmarks in the standard LDDMM approach are Lagrangian.) For the sake of introducing an Eulerian uncertainty quantification approach in imaging science, we extend the method of metamorphosis to apply to image matching along emph{stochastically} evolving time dependent curves on the space of diffeomorphisms. The approach will be guided by recent progress in developing stochastic Lie transport models for uncertainty quantification in fluid dynamics in cite{holm2015variational,CrFlHo2017}.
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