Center Manifold of Fractional Dynamical System

2015 ◽  
Vol 11 (2) ◽  
Author(s):  
Li Ma ◽  
Changpin Li
Keyword(s):  
Dynamical System ◽  
Center Manifold ◽  
High Dimensional ◽  
Dimensional System ◽  
Infinite Dimensional ◽  
Infinite Dimensional Systems ◽  
Finite Dimensional ◽  
Reduction Methods ◽  
Fractional Dynamical System ◽  
Lower Dimensional

Dimension reduction of dynamical system is a significant issue for technical applications, as regards both finite dimensional system and infinite dimensional systems emerging from either science or engineering. Center manifold method is one of the main reduction methods for ordinary differential systems (ODSs). Does there exists a similar method for fractional ODSs (FODSs)? In other words, does there exists a method for reducing the high-dimensional FODS into a lower-dimensional FODS? In this study, we establish a local fractional center manifold for a finite dimensional FODS. Several examples are given to illustrate the theoretical analysis.

Approximating dynamics of a number-conserving cellular automaton by a finite-dimensional dynamical system

2020 ◽  
Vol 31 (12) ◽  
pp. 2050172
Author(s):  
Henryk Fukś ◽  
Yucen Jin
Keyword(s):  
Dynamical System ◽  
Local Structure ◽  
Structure Theory ◽  
Dimensional System ◽  
Perfect Agreement ◽  
Infinite Dimensional ◽  
Finite Dimensional Approximation ◽  
Finite Dimensional ◽  
Dimensional Approximation ◽  
Dimensional Dynamical System

The local structure theory for cellular automata (CA) can be viewed as an finite-dimensional approximation of infinitely dimensional system. While it is well known that this approximation works surprisingly well for some CA, it is still not clear why it is the case, and which CA rules have this property. In order to shed some light on this problem, we present an example of a four input CA for which probabilities of occurrence of short blocks of symbols can be computed exactly. This rule is number conserving and possesses a blocking word. Its local structure approximation correctly predicts steady-state probabilities of small length blocks, and we present a rigorous proof of this fact, without resorting to numerical simulations. We conjecture that the number-conserving property together with the existence of the blocking word are responsible for the observed perfect agreement between the finite-dimensional approximation and the actual infinite-dimensional dynamical system.

The Description of a Distributed Parameter Process through a Finite Discrete Dimensional System

2014 ◽  
Vol 657 ◽  
pp. 874-878
Author(s):  
Sever Şerban ◽  
Doina Corina Şerban
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Industrial Process ◽  
Time Discretization ◽  
Distributed Parameter ◽  
Geometrical Parameters ◽  
Dimensional System ◽  
Infinite Dimensional ◽  
Infinite Dimensional Systems ◽  
Distributed Parameters

This article analyses the process of warming a metal by using a walking beam furnace. This process is meant to offer the technologist objective information that may allow him to produce eventual modifications of the temperature references from the furnaces zones. Thus making the metals temperature at the furnaces exit to have an imposed distribution, within precise limits, according to the technological requests. This industrial process has a geometrical parameters distribution, more precisely it can be described through a partial differential equation, by being attached to dynamic infinite dimensional systems (or with distributed parameters). Using a procedure called geometric-time discretization (in the condition of the solutions convergence), we have managed to obtain a representation under the form of a finite discrete dimensional linear system for a process with distributed parameters.

Modeling and boundary control of infinite dimensional systems in the Brayton–Moser framework

2017 ◽  
Vol 36 (2) ◽  
pp. 485-513
Author(s):  
Krishna Chaitanya Kosaraju ◽  
Ramkrishna Pasumarthy ◽  
Dimitri Jeltsema
Keyword(s):  
Boundary Control ◽  
Zero Energy ◽  
Line System ◽  
Infinite Dimensional ◽  
Infinite Dimensional Systems ◽  
Finite Dimensional ◽  
Boundary Control Problem ◽  
Energy Flows ◽  
Admissible Pairs ◽  
The Stability

Abstract It is well documented that shaping the energy of finite-dimensional port-Hamiltonian systems by interconnection is severely restricted due to the presence of dissipation. This phenomenon is usually referred to as the dissipation obstacle. In this paper, we show the existence of dissipation obstacle in infinite dimensional systems. Motivated by this, we present the Brayton–Moser formulation, together with its equivalent Dirac structure. Analogous to finite dimensional systems, identifying the underlying gradient structure is crucial in presenting the stability analysis. We elucidate this through an example of Maxwell’s equations with zero energy flows through the boundary. In the case of mixed-finite and infinite-dimensional systems, we find admissible pairs for all the subsystems while preserving the overall structure. We illustrate this using a transmission line system interconnected to finite dimensional systems through its boundary. This ultimately leads to a new passive map, using this we solve a boundary control problem, circumventing the dissipation obstacle.

Introduction to anti-control of discrete chaos: theory and applications

2006 ◽  
Vol 364 (1846) ◽  
pp. 2433-2447 ◽  
Author(s):  
Guanrong Chen ◽  
Yuming Shi
Keyword(s):  
Dynamical System ◽  
Feedback Control ◽  
Chaos Theory ◽  
Control Of Chaos ◽  
Main Interest ◽  
Chaotic Dynamical System ◽  
Infinite Dimensional ◽  
Control Techniques ◽  
Finite Dimensional ◽  
Discrete Chaos

In this paper, the notion of anti-control of chaos (or chaotification) is introduced, which means to make an originally non-chaotic dynamical system chaotic or enhance the existing chaos of a chaotic system. The main interest in this paper is to employ the classical feedback control techniques. Only the discrete case is discussed in detail, including both finite-dimensional and infinite-dimensional settings.

Data assimilation for a multiscale stochastic dynamical system with Gaussian noise

2019 ◽  
Vol 19 (03) ◽  
pp. 1950019
Author(s):  
Yanjie Zhang ◽  
Jian Ren
Keyword(s):  
Dynamical System ◽  
Data Assimilation ◽  
Energy Method ◽  
Gaussian Noise ◽  
Original System ◽  
Stochastic Averaging ◽  
Dimensional System ◽  
Mean Square ◽  
Stochastic Dynamical System ◽  
Lower Dimensional

This paper is devoted to studying dimensional reduction for slow-fast data assimilation driven by Gaussian noise via stochastic averaging. We apply an energy method to show that the probability density for the reduced lower-dimensional system approximates that for the original system in mean square. In other words, the reduced system filter thus effectively captures the filter of the original system.

Sign in / Sign up

Export Citation Format

Share Document