Nonlinear Stochastic Dynamics and Control in Hamiltonian Formulation

2006 ◽  
Vol 59 (4) ◽  
pp. 230-248 ◽  
Author(s):  
W. Q. Zhu
Keyword(s):  
Hamiltonian Systems ◽  
Stochastic Dynamics ◽  
First Passage Time ◽  
Hamiltonian Formulation ◽  
Stochastic Averaging ◽  
Passage Time ◽  
Stochastic Bifurcation ◽  
Dynamics And Control ◽  
Nonlinear Stochastic Dynamics ◽  
And Control

The significant advances in nonlinear stochastic dynamics and control in Hamiltonian formulation during the past decade are reviewed. The exact stationary solutions and equivalent nonlinear system method of Gaussian-white -noises excited and dissipated Hamiltonian systems, the stochastic averaging method for quasi Hamiltonian systems, the stochastic stability, stochastic bifurcation, first-passage time and nonlinear stochastic optimal control of quasi Hamiltonian systems are summarized. Possible extension and applications of the theory are pointed out. This review article cites 158 references.

Nonlinear Stochastic Optimal Control of Quasi Hamiltonian Systems

2005 ◽  
Author(s):  
W. Q. Zhu
Keyword(s):  
Optimal Control ◽  
Hamiltonian Systems ◽  
Stochastic Optimal Control ◽  
Control Strategies ◽  
First Passage Time ◽  
Stochastic Averaging ◽  
Passage Time ◽  
Stochastic Averaging Method ◽  
Dynamic Programming Principle ◽  
Stochastic Dynamic

In recent years, a class of nonlinear stochastic optimal control strategies were developed by the present author and his co-workers for minimizing the response, stabilization and maximizing the reliability and mean first-passage time of quasi Hamiltonian systems based on the stochastic averaging method for quasi Hamiltonian systems and the stochastic dynamic programming principle. This review summaries the basic idea, procedures and applications of these strategies and pointes out necessary further work.

scholarly journals Determinants of Brain Rhythm Burst Statistics

2019 ◽  
Vol 9 (1) ◽  
Author(s):  
Arthur S. Powanwe ◽  
André Longtin
Keyword(s):  
Optimal Parameter ◽  
First Passage Time ◽  
Stochastic Averaging ◽  
Gamma Oscillations ◽  
Passage Time ◽  
Recurrent Network ◽  
Stochastic Averaging Method ◽  
Model Parameters ◽  
Brain Rhythm

AbstractBrain rhythms recorded in vivo, such as gamma oscillations, are notoriously variable both in amplitude and frequency. They are characterized by transient epochs of higher amplitude known as bursts. It has been suggested that, despite their short-life and random occurrence, bursts in gamma and other rhythms can efficiently contribute to working memory or communication tasks. Abnormalities in bursts have also been associated with e.g. motor and psychiatric disorders. It is thus crucial to understand how single cell and connectivity parameters influence burst statistics and the corresponding brain states. To address this problem, we consider a generic stochastic recurrent network of Pyramidal Interneuron Network Gamma (PING) type. Using the stochastic averaging method, we derive dynamics for the phase and envelope of the amplitude process, and find that they depend on only two meta-parameters that combine all the model parameters. This allows us to identify an optimal parameter regime of healthy variability with similar statistics to those seen in vivo; in this regime, oscillations and bursts are supported by synaptic noise. The probability density for the rhythm’s envelope as well as the mean burst duration are then derived using first passage time analysis. Our analysis enables us to link burst attributes, such as duration and frequency content, to system parameters. Our general approach can be extended to different frequency bands, network topologies and extra populations. It provides the much needed insight into the biophysical determinants of rhythm burst statistics, and into what needs to be changed to correct rhythms with pathological statistics.

scholarly journals First-Passage Failure of Quasi-Integrable Hamiltonian Systems

2002 ◽  
Vol 69 (3) ◽  
pp. 274-282 ◽  
Author(s):  
W. Q. Zhu ◽  
M. L. Deng ◽  
Z. L. Huang
Keyword(s):  
Hamiltonian Systems ◽  
First Passage Time ◽  
Digital Simulation ◽  
Reliability Function ◽  
Passage Time ◽  
Kolmogorov Equation ◽  
Conditional Probability Density ◽  
Integrable Hamiltonian Systems ◽  
First Passage ◽  
First Passage Failure

The first-passage failure of quasi-integrable Hamiltonian systems (multidegree-of-freedom integrable Hamiltonian systems subject to light dampings and weakly random excitations) is investigated. The motion equations of such a system are first reduced to a set of averaged Ito^ stochastic differential equations by using the stochastic averaging method for quasi-integrable Hamitonian systems. Then, a backward Kolmogorov equation governing the conditional reliability function and a set of generalized Pontryagin equations governing the conditional moments of first-passage time are established. Finally, the conditional reliability function, and the conditional probability density and moments of first-passage time are obtained by solving these equations with suitable initial and boundary conditions. Two examples are given to illustrate the proposed procedure and the results from digital simulation are obtained to verify the effectiveness of the procedure.

scholarly journals First passage time on pattern formation in a non-local Fisher population dynamics

Open Physics ◽  
2013 ◽  
Vol 11 (12) ◽  
Author(s):  
Miguel Fuentes ◽  
Manuel Cáceres
Keyword(s):  
Stochastic Dynamics ◽  
First Passage Time ◽  
Mode Selection ◽  
Passage Time ◽  
Unstable State ◽  
Perturbation Approach ◽  
Stochastic Field ◽  
Small Noise ◽  
Theoretical Predictions ◽  
Non Local

AbstractWe use stochastic dynamics to develop the patterned attractor of a non-local extended system. This is done analytically using the stochastic path perturbation approach scheme, where a theory of perturbation in the small noise parameter is introduced to analyze the random escape of the stochastic field from the unstable state. Emphasis is placed on the specific mode selection that these types of systems exhibit. Concerning the stochastic propagation of the front we have carried out Monte Carlo simulations which coincide with our theoretical predictions.

scholarly journals Dynamic Analysis of a Tumor-Immune System under Allee Effect

2020 ◽  
Vol 2020 ◽  
pp. 1-11
Author(s):  
Chunmei Zeng ◽  
Shaojuan Ma
Keyword(s):  
Immune System ◽  
Equilibrium Point ◽  
Allee Effect ◽  
Deterministic Model ◽  
First Passage Time ◽  
Stochastic Averaging ◽  
Threshold Value ◽  
Passage Time ◽  
Steady State Probability ◽  
Coexistence Equilibrium

In this paper, we develop a definite tumor-immune model considering Allee effect. The deterministic model is studied qualitatively by mathematical analysis method, including the positivity, boundness, and local stability of the solution. In addition, we explore the effect of random factors on the transition of the tumor-immune system from a stable coexistence equilibrium point to a stable tumor-free equilibrium point. Based on the method of stochastic averaging, we obtain the expressions of the steady-state probability density and the mean first-passage time. And we find that the Allee effect has the greatest impact on the number of cells in the system when the Allee threshold value is within a certain range; the intensity of random factors could affect the likelihood of the system crossing from the coexistence equilibrium to the tumor-free equilibrium.

Noise-Induced Chaos in a Piecewise Linear System

2017 ◽  
Vol 27 (09) ◽  
pp. 1750137 ◽  
Author(s):  
Chen Kong ◽  
Xian-Bin Liu
Keyword(s):  
Linear System ◽  
White Noise ◽  
First Passage Time ◽  
Piecewise Linear ◽  
Gaussian White Noise ◽  
Stochastic Averaging ◽  
Passage Time ◽  
Singular Perturbation Method ◽  
Generalized Cell Mapping ◽  
Piecewise Linear System

In the present paper, the phenomenon of noise-induced chaos in a piecewise linear system that is excited by Gaussian white noise is investigated. Firstly, the global dynamical behaviors of the deterministic piecewise linear system are investigated numerically in advance by using the generalized cell-mapping digraph (GCMD) method. Then, based on these global properties, the system that is excited by Gaussian white noise is introduced. Then, it is simplified by the stochastic averaging method, through which, a four-dimensional averaged Itô system is finally obtained. In order to reveal the phenomenon of noise-induced chaos quantitatively, MFPT (the mean first-passage time) is selected as the measure. The expression for MFPT is formulated by using the singular perturbation method and then a rather simple representation is obtained via the Laplace approximation, and within which, the concept of quasi-potential is introduced. Furthermore, with the rays method, the MFPT under a certain set of parameters is estimated. However, within the process of analysis, the authors had to face a difficult problem concerning the ill-conditioned matrix, which is the obstacle for the estimation of MFPT, which was then solved by applying one more approximation. Finally, the result is compared with the numerical one that is obtained by the Monte Carlo simulation.

Stochastic dynamics and fractional optimal control of quasi integrable Hamiltonian systems with fractional derivative damping

2013 ◽  
Vol 16 (1) ◽  
Author(s):  
Lincong Chen ◽  
Fang Hu ◽  
Weiqiu Zhu
Keyword(s):  
Optimal Control ◽  
Fractional Derivative ◽  
Hamiltonian Systems ◽  
Stochastic Dynamics ◽  
Averaging Method ◽  
Stochastic Averaging ◽  
Stochastic Averaging Method ◽  
Integrable Hamiltonian Systems ◽  
Fractional Optimal Control ◽  
Fractional Derivative Damping

AbstractIn the present survey, some progress in the stochastic dynamics and fractional optimal control of quasi integrable Hamiltonian systems with fractional derivative damping is reviewed. First, the stochastic averaging method for quasi integrable Hamiltonian systems with fractional derivative damping under various random excitations is briefly introduced. Then, the stochastic stability, stochastic bifurcation, first passage time and reliability, and stochastic fractional optimal control of the systems studied by using the stochastic averaging method are summarized. The focus is placed on the effects of fractional derivative order on the dynamics and control of the systems. Finally, some possible extensions are pointed out.

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