scholarly journals Information Geometry of Spatially Periodic Stochastic Systems

Entropy ◽  
2019 ◽  
Vol 21 (7) ◽  
pp. 681
Author(s):  
Rainer Hollerbach ◽  
Eun-jin Kim
Keyword(s):  
Equilibrium Point ◽  
Stochastic Systems ◽  
Initial Conditions ◽  
Planck Equation ◽  
Information Geometry ◽  
Initial Position ◽  
Equilibrium Solutions ◽  
Gaussian Peak ◽  
Unstable Fixed Point ◽  
Spatially Periodic

We explore the effect of different spatially periodic, deterministic forces on the information geometry of stochastic processes. The three forces considered are f 0 = sin ( π x ) / π and f ± = sin ( π x ) / π ± sin ( 2 π x ) / 2 π , with f - chosen to be particularly flat (locally cubic) at the equilibrium point x = 0 , and f + particularly flat at the unstable fixed point x = 1 . We numerically solve the Fokker–Planck equation with an initial condition consisting of a periodically repeated Gaussian peak centred at x = μ , with μ in the range [ 0 , 1 ] . The strength D of the stochastic noise is in the range 10 - 4 – 10 - 6 . We study the details of how these initial conditions evolve toward the final equilibrium solutions and elucidate the important consequences of the interplay between an initial PDF and a force. For initial positions close to the equilibrium point x = 0 , the peaks largely maintain their shape while moving. In contrast, for initial positions sufficiently close to the unstable point x = 1 , there is a tendency for the peak to slump in place and broaden considerably before reconstituting itself at the equilibrium point. A consequence of this is that the information length L ∞ , the total number of statistically distinguishable states that the system evolves through, is smaller for initial positions closer to the unstable point than for more intermediate values. We find that L ∞ as a function of initial position μ is qualitatively similar to the force, including the differences between f 0 = sin ( π x ) / π and f ± = sin ( π x ) / π ± sin ( 2 π x ) / 2 π , illustrating the value of information length as a useful diagnostic of the underlying force in the system.

scholarly journals A Type of Time-Symmetric Stochastic System and Related Games

Symmetry ◽  
2021 ◽  
Vol 13 (1) ◽  
pp. 118
Author(s):  
Qingfeng Zhu ◽  
Yufeng Shi ◽  
Jiaqiang Wen ◽  
Hui Zhang
Keyword(s):  
Nash Equilibrium ◽  
Time Delay ◽  
Differential Equations ◽  
Equilibrium Point ◽  
Stochastic Differential Equations ◽  
Stochastic System ◽  
Stochastic Systems ◽  
Necessary Condition ◽  
Nash Equilibrium Point ◽  
Doubly Stochastic

This paper is concerned with a type of time-symmetric stochastic system, namely the so-called forward–backward doubly stochastic differential equations (FBDSDEs), in which the forward equations are delayed doubly stochastic differential equations (SDEs) and the backward equations are anticipated backward doubly SDEs. Under some monotonicity assumptions, the existence and uniqueness of measurable solutions to FBDSDEs are obtained. The future development of many processes depends on both their current state and historical state, and these processes can usually be represented by stochastic differential systems with time delay. Therefore, a class of nonzero sum differential game for doubly stochastic systems with time delay is studied in this paper. A necessary condition for the open-loop Nash equilibrium point of the Pontriagin-type maximum principle are established, and a sufficient condition for the Nash equilibrium point is obtained. Furthermore, the above results are applied to the study of nonzero sum differential games for linear quadratic backward doubly stochastic systems with delay. Based on the solution of FBDSDEs, an explicit expression of Nash equilibrium points for such game problems is established.

scholarly journals Fokker-Planck Models for Rotating Stellar Systems

1996 ◽  
Vol 174 ◽  
pp. 363-364 ◽  
Author(s):  
Christian Einsel ◽  
Rainer Spurzem
Keyword(s):  
Initial Conditions ◽  
Planck Equation ◽  
Rotational Energy ◽  
Momentum Transport ◽  
Core Collapse ◽  
Initial Model ◽  
Fundamental Parameters ◽  
Angular Momentum Transport ◽  
Collapse Phase ◽  
Fokker Planck

Observations of Globular Cluster ellipticity distributions related to some fundamental parameters give strong evidence for a decay of rotational energy in these systems with time. In order to study the effectiveness of angular momentum transport (or loss, resp.) a code has been written which solves the Fokker-Planck equation in (E, Jz)-space and follows the evolution from some initial conditions through core collapse (and possibly gravothermal oscillations) up to the post-collapse phase. For the purpose of comparability with N-body simulations rotating initial model configurations according to the prescriptions of Lupton & Gunn (1987) have been constructed. These models are intended to continue previous work by Goodman (1983, Fokker-Planck) and Akiyama & Sugimoto (1989, N-Body). In this contribution the derivation of the flux coefficients is given.

scholarly journals Characterization of the Transient Response of Coupled Optimization in Multidisciplinary Design

2013 ◽  
Vol 2013 ◽  
pp. 1-15 ◽  
Author(s):  
Erich Devendorf ◽  
Kemper Lewis
Keyword(s):  
Optimization Problems ◽  
Initial Conditions ◽  
Systems Design ◽  
Discrete Systems ◽  
Product Development Process ◽  
Multidisciplinary Design ◽  
Distributed Design ◽  
Equilibrium Solutions ◽  
Acceptable Solution ◽  
Number Of Iterations

Time is an asset of critical importance in a multidisciplinary design process and it is desirable to reduce the amount of time spent designing products and systems. Design is an iterative activity and designers consume a significant portion of the product development process negotiating a mutually acceptable solution. The amount of time necessary to complete a design depends on the number and duration of design iterations. This paper focuses on accurately characterizing the number of iterations required for designers to converge to an equilibrium solution in distributed design processes. In distributed design, systems are decomposed into smaller, coupled design problems where individual designers have control over local design decisions and seek to achieve their own individual objectives. These smaller coupled design optimization problems can be modeled using coupled games and the number of iterations required to reach equilibrium solutions varies based on initial conditions and process architecture. In this paper, we leverage concepts from game theory, classical controls, and discrete systems theory to evaluate and approximate process architectures without carrying out any solution iterations. As a result, we develop an analogy between discrete decisions and a continuous time representation that we analyze using control theoretic techniques.

scholarly journals Three-Saddle-Foci Chaotic Behavior of a Modified Jerk Circuit with Chua’s Diode

Symmetry ◽  
2020 ◽  
Vol 12 (11) ◽  
pp. 1803
Author(s):  
Pattrawut Chansangiam
Keyword(s):  
Equilibrium Point ◽  
Chaotic Behavior ◽  
Initial Conditions ◽  
Piecewise Linear ◽  
Equilibrium Points ◽  
Initial Point ◽  
Electrical Circuit ◽  
Nonlinear Resistor ◽  
Voltage Current Characteristic ◽  
Time Waveform

This paper investigates the chaotic behavior of a modified jerk circuit with Chua’s diode. The Chua’s diode considered here is a nonlinear resistor having a symmetric piecewise linear voltage-current characteristic. To describe the system, we apply fundamental laws in electrical circuit theory to formulate a mathematical model in terms of a third-order (jerk) nonlinear differential equation, or equivalently, a system of three first-order differential equations. The analysis shows that this system has three collinear equilibrium points. The time waveform and the trajectories about each equilibrium point depend on its associated eigenvalues. We prove that all three equilibrium points are of type saddle focus, meaning that the trajectory of (x(t),y(t)) diverges in a spiral form but z(t) converges to the equilibrium point for any initial point (x(0),y(0),z(0)). Numerical simulation illustrates that the oscillations are dense, have no period, are highly sensitive to initial conditions, and have a chaotic hidden attractor.

scholarly journals Fractional Prabhakar Derivative in Diffusion Equation with Non-Static Stochastic Resetting

Physics ◽  
2019 ◽  
Vol 1 (1) ◽  
pp. 40-58 ◽  
Author(s):  
Maike dos Santos
Keyword(s):  
Diffusion Equation ◽  
Planck Equation ◽  
Fractional Diffusion ◽  
Initial Position ◽  
Quadratic Potential ◽  
Generalized Model ◽  
Integration Kernel ◽  
The Standard Model ◽  
New Perspective ◽  
Diffusive Processes

In this work, we investigate a series of mathematical aspects for the fractional diffusion equation with stochastic resetting. The stochastic resetting process in Evans–Majumdar sense has several applications in science, with a particular emphasis on non-equilibrium physics and biological systems. We propose a version of the stochastic resetting theory for systems in which the reset point is in motion, so the walker does not return to the initial position as in the standard model, but returns to a point that moves in space. In addition, we investigate the proposed stochastic resetting model for diffusion with the fractional operator of Prabhakar. The derivative of Prabhakar consists of an integro-differential operator that has a Mittag–Leffler function with three parameters in the integration kernel, so it generalizes a series of fractional operators such as Riemann–Liouville–Caputo. We present how the generalized model of stochastic resetting for fractional diffusion implies a rich class of anomalous diffusive processes, i.e., ⟨ ( Δ x ) 2 ⟩ ∝ t α , which includes sub-super-hyper-diffusive regimes. In the sequence, we generalize these ideas to the fractional Fokker–Planck equation for quadratic potential U ( x ) = a x 2 + b x + c . This work aims to present the generalized model of Evans–Majumdar’s theory for stochastic resetting under a new perspective of non-static restart points.

scholarly journals Bifurcation in Mean Phase Portraits for Stochastic Dynamical Systems with Multiplicative Gaussian Noise

2020 ◽  
Vol 30 (11) ◽  
pp. 2050216
Author(s):  
Hui Wang ◽  
Athanasios Tsiairis ◽  
Jinqiao Duan
Keyword(s):  
Dynamical Systems ◽  
Gaussian Noise ◽  
Stochastic Systems ◽  
Planck Equation ◽  
Phase Portraits ◽  
Stochastic Bifurcation ◽  
Stochastic Dynamical Systems ◽  
Solution Processes ◽  
Bifurcation Phenomena ◽  
Stability Type

We investigate the bifurcation phenomena for stochastic systems with multiplicative Gaussian noise, by examining qualitative changes in mean phase portraits. Starting from the Fokker–Planck equation for the probability density function of solution processes, we compute the mean orbits and mean equilibrium states. A change in the number or stability type, when a parameter varies, indicates a stochastic bifurcation. Specifically, we study stochastic bifurcation for three prototypical dynamical systems (i.e. saddle-node, transcritical, and pitchfork systems) under multiplicative Gaussian noise, and have found some interesting phenomena in contrast to the corresponding deterministic counterparts.

scholarly journals On prediction and chaos in stochastic systems

1994 ◽  
Vol 348 (1688) ◽  
pp. 357-369 ◽  
Keyword(s):  
Statistical Estimation ◽  
Stochastic Systems ◽  
Initial Conditions ◽  
Stochastic Environment ◽  
Nonlinear Prediction ◽  
Sensitivity To Initial Conditions

We propose a new measure of sensitivity to initial conditions within a stochastic environment and explore its connection with nonlinear prediction and statistical estimation. We use modern statistical developments to construct and illustrate pointwise predictors and predictive intervals/distributions.

Global Dynamics of an Autoparametric System With Multiple Pendulums

2005 ◽  
Vol 1 (1) ◽  
pp. 35-46 ◽  
Author(s):  
Ashwin Vyas ◽  
Anil K. Bajaj
Keyword(s):  
Initial Conditions ◽  
Hamiltonian Formulation ◽  
Forced Response ◽  
Global Dynamics ◽  
Equilibrium Solutions ◽  
External Excitation ◽  
Poincaré Sections ◽  
Poincare Sections ◽  
Hamilton's Equations ◽  
Hamilton’S Equations

The Hamiltonian dynamics of a resonantly excited linear spring-mass-damper system coupled to an array of pendulums is investigated in this study under 1:1:1:…:2 internal resonance between the pendulums and the linear oscillator. To study the small-amplitude global dynamics, a Hamiltonian formulation is introduced using generalized coordinates and momenta, and action-angle coordinates. The Hamilton’s equations are averaged to obtain equations for the first-order approximations to free and forced response of the system. Equilibrium solutions of the averaged Hamilton’s equations in action-angle or comoving variables are determined and studied for their stability characteristics. The system with one pendulum is known to be integrable in the absence of damping and external excitation. Exciting the system with even a small harmonic forcing near a saddle point leads to stochastic response, as clearly demonstrated by the Poincaré sections of motion. Poincaré sections are also computed for motions started with initial conditions near center-center, center-saddle and saddle-saddle-type equilibria for systems with two, three and four pendulums. In case of the system with more than one pendulum, even the free undamped dynamics exhibits irregular exchange of energy between the pendulums and the block. The increase in complexity is also demonstrated as the number of pendulums is increased, and when external excitation is present.

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