599 research outputs found
A variational method for analyzing stochastic limit cycle oscillators
We introduce a variational method for analyzing limit cycle oscillators in
driven by Gaussian noise. This allows us to derive exact
stochastic differential equations (SDEs) for the amplitude and phase of the
solution, which are accurate over times over order
, where is the amplitude of the noise
and the magnitude of decay of transverse fluctuations. Within the
variational framework, different choices of the amplitude-phase decomposition
correspond to different choices of the inner product space . For
concreteness, we take a weighted Euclidean norm, so that the minimization
scheme determines the phase by projecting the full solution on to the limit
cycle using Floquet vectors. Since there is coupling between the amplitude and
phase equations, even in the weak noise limit, there is a small but non-zero
probability of a rare event in which the stochastic trajectory makes a large
excursion away from a neighborhood of the limit cycle. We use the amplitude and
phase equations to bound the probability of it doing this: finding that the
typical time the system takes to leave a neighborhood of the oscillator scales
as
Synchronization of stochastic hybrid oscillators driven by a common switching environment
Many systems in biology, physics and chemistry can be modeled through
ordinary differential equations, which are piecewise smooth, but switch between
different states according to a Markov jump process. In the fast switching
limit, the dynamics converges to a deterministic ODE. In this paper we suppose
that this limit ODE supports a stable limit cycle. We demonstrate that a set of
such oscillators can synchronize when they are uncoupled, but they share the
same switching Markov jump process. The latter is taken to represent the effect
of a common randomly switching environment. We determine the leading order of
the Lyapunov coefficient governing the rate of decay of the phase difference in
the fast switching limit. The analysis bears some similarities to the classical
analysis of synchronization of stochastic oscillators subject to common white
noise. However the discrete nature of the Markov jump process raises some
difficulties: in fact we find that the Lyapunov coefficient from the
quasi-steady-state approximation differs from the Lyapunov coefficient one
obtains from a second order perturbation expansion in the waiting time between
jumps. Finally, we demonstrate synchronization numerically in the radial
isochron clock model and show that the latter Lyapinov exponent is more
accurate
Direct and Indirect Couplings in Coherent Feedback Control of Linear Quantum Systems
The purpose of this paper is to study and design direct and indirect
couplings for use in coherent feedback control of a class of linear quantum
stochastic systems. A general physical model for a nominal linear quantum
system coupled directly and indirectly to external systems is presented.
Fundamental properties of stability, dissipation, passivity, and gain for this
class of linear quantum models are presented and characterized using complex
Lyapunov equations and linear matrix inequalities (LMIs). Coherent
and LQG synthesis methods are extended to accommodate direct couplings using
multistep optimization. Examples are given to illustrate the results.Comment: 33 pages, 7 figures; accepted for publication in IEEE Transactions on
Automatic Control, October 201
A study of synchronization of nonlinear oscillators: Application to epileptic seizures
This dissertation focuses on several problems in neuroscience from the perspective of nonlinear dynamics and stochastic processes. The first part concerns a method to visualize the idea of the power spectrum of spike trains, which has an educational value to introductory students in biophysics. The next part consists of experimental and computational work on drug-induced epileptic seizures in the rat neocortex. In the experimental part, spatiotemporal patterns of electrical activities in the rat neocortex are measured using voltage-sensitive dye imaging. Epileptic regions show well-synchronized, in-phase activity during epileptic seizures. In the computational part, a network of a Hodgkin-Huxley type neocortical neural model is constructed. Phase reduction, which is a dimension reduction technique for a stable limit cycle, is applied to the system. The results propose a possible mechanism for the initiation of the drug-induced seizure as a result of a bifurcation. In the last part, a theoretical framework is developed to obtain the statistics for the period of oscillations of a stable limit cycle under stochastic perturbation. A stochastic version of phase reduction and first passage time analysis are utilized for this purpose. The method presented here shows a good agreement with numerical results for the weak noise regime --Abstract, page iii
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