Error estimate for influence of model reduction of nonlinear dissipative autonomous dynamical system on long-term behaviours

2005 ◽  
Vol 26 (7) ◽  
pp. 938-943 ◽  
Author(s):  
Zhang Jia-zhong ◽  
Liu Yan ◽  
Chen Dang-min
Keyword(s):  
Dynamical System ◽  
Error Estimate ◽  
Model Reduction ◽  
Autonomous Dynamical System

scholarly journals Geosystemics and Earthquakes

2018 ◽  
Author(s):  
Angelo De Santis ◽  
Gianfranco Cianchini ◽  
Rita Di Giovambattista ◽  
Cristoforo Abbattista ◽  
Lucilla Alfonsi ◽  
...  
Keyword(s):  
Dynamical System ◽  
Solid Earth ◽  
Case Studies ◽  
Central Italy ◽  
Point Of View ◽  
Earth System ◽  
Complex Dynamical System ◽  
The Earth ◽  
Earth’S Interior

Abstract. Geosystemics (De Santis 2009, 2014) studies the Earth system as a whole focusing on the possible coupling among the Earth layers (the so called geo-layers), and using universal tools to integrate different methods that can be applied to multi-parameter data, often taken on different platforms. Its main objective is to understand the particular phenomenon of interest from a holistic point of view. In this paper we will deal with earthquakes, considered as a long term chain of processes involving, not only the interaction between different components of the Earth’s interior, but also the coupling of the solid earth with the above neutral and ionized atmosphere, and finally culminating with the main rupture along the fault of concern (De Santis et al., 2015a). Some case studies (particular emphasis is given to recent central Italy earthquakes) will be discussed in the frame of the geosystemic approach for better understanding the physics of the underlying complex dynamical system.

scholarly journals Autonomous dynamical system description of de Sitter evolution in scalar assisted f(R) − ϕ gravity

2018 ◽  
Vol 15 (12) ◽  
pp. 1850212 ◽  
Author(s):  
K. Kleidis ◽  
V. K. Oikonomou
Keyword(s):  
Dynamical System ◽  
Numerical Analysis ◽  
Scalar Field ◽  
Analysis Approach ◽  
Single Variable ◽  
De Sitter ◽  
Exponential Potential ◽  
System Description ◽  
Variable Formula ◽  
Autonomous Dynamical System

In this paper we will study the cosmological dynamical system of an [Formula: see text] gravity in the presence of a canonical scalar field [Formula: see text] with an exponential potential by constructing the dynamical system in a way that it is rendered autonomous. This feature is controlled by a single variable [Formula: see text], which when it is constant, the dynamical system is autonomous. We focus on the [Formula: see text] case which, as we demonstrate by using a numerical analysis approach, leads to an unstable de Sitter attractor, which occurs after [Formula: see text] [Formula: see text]-foldings. This instability can be viewed as a graceful exit from inflation, which is inherent to the dynamics of de Sitter attractors.

scholarly journals A model reduction approach for simulation of long-term voltage and frequency dynamics

2017 ◽  
Author(s):  
Tilman Weckesser ◽  
Valeri Franz ◽  
Eckhard Grebe ◽  
Thierry Van Cutsem
Keyword(s):  
Model Reduction ◽  
Reduction Approach

scholarly journals Complex Canard Explosion in a Fractional-Order FitzHugh–Nagumo Model

2019 ◽  
Vol 29 (08) ◽  
pp. 1950111 ◽  
Author(s):  
Mohammed-Salah Abdelouahab ◽  
René Lozi ◽  
Guanrong Chen
Keyword(s):  
Dynamical System ◽  
Fractional Order ◽  
Complex Dynamics ◽  
Search Algorithm ◽  
Two Dimensional ◽  
Mixed Mode Oscillations ◽  
The Stability ◽  
Two Parameters ◽  
Complex Phenomena ◽  
Autonomous Dynamical System

This article investigates the complex phenomena of canard explosion with mixed-mode oscillations, observed from a fractional-order FitzHugh–Nagumo (FFHN) model. To rigorously analyze the dynamics of the FFHN model, a new mathematical notion, referred to as Hopf-like bifurcation (HLB), is introduced. HLB provides a precise definition for the change between a fixed point and an [Formula: see text]-asymptotically [Formula: see text]-periodic solution of the fractional-order dynamical system, as well as the stability of the FFHN model and the appearance of the HLB. The existence of canard oscillations in the neighborhoods of such HLB points are numerically investigated. Using a new algorithm, referred to as the global-local canard explosion search algorithm, the appearance of various patterns of solutions is revealed, with an increasing number of small-amplitude oscillations when two key parameters of the FFHN model are varied. The numbers of such oscillations versus the two parameters, respectively, are perfectly fitted using exponential functions. Finally, it is conjectured that chaos could occur in a two-dimensional fractional-order autonomous dynamical system, with the fractional order close to one. After all, the article demonstrates that the FFHN model is a very simple two-dimensional model with an incredible ability to present the complex dynamics of neurons.

A 3D Autonomous System with Infinitely Many Chaotic Attractors

2019 ◽  
Vol 29 (12) ◽  
pp. 1950166 ◽  
Author(s):  
Ting Yang ◽  
Qigui Yang
Keyword(s):  
Dynamical System ◽  
Differential Equations ◽  
Autonomous System ◽  
Three Dimensional ◽  
Chaotic Attractors ◽  
Double Zero ◽  
Finite Dimensional ◽  
Periodic Attractors ◽  
The Stability ◽  
Autonomous Dynamical System

Intuitively, a finite-dimensional autonomous system of ordinary differential equations can only generate finitely many chaotic attractors. Amazingly, however, this paper finds a three-dimensional autonomous dynamical system that can generate infinitely many chaotic attractors. Specifically, this system can generate infinitely many coexisting chaotic attractors and infinitely many coexisting periodic attractors in the following three cases: (i) no equilibria, (ii) only infinitely many nonhyperbolic double-zero equilibria, and (iii) both infinitely many hyperbolic saddles and nonhyperbolic pure-imaginary equilibria. By analyzing the stability of double-zero and pure-imaginary equilibria, it is shown that the classic Shil’nikov criteria fail in verifying the existence of chaos in the above three cases.

Markov cellular automata models for chronic disease progression

2015 ◽  
Vol 08 (06) ◽  
pp. 1550085 ◽  
Author(s):  
Jane Hawkins ◽  
Donna Molinek
Keyword(s):  
Dynamical System ◽  
Human Immunodeficiency Virus ◽  
Hepatitis C ◽  
Chronic Conditions ◽  
Chronic Condition ◽  
Physical Significance ◽  
Markov Shifts ◽  
Immunodeficiency Virus ◽  
Long Term Prognosis

We analyze a Markov cellular automaton that models the spread of viruses that often progress to a chronic condition, such as human immunodeficiency virus (HIV) or hepatitis C virus (HCV). We show that the complex dynamical system produces a Markov process at the later stages, whose eigenvectors corresponding to the eigenvalue 1 have physical significance for the long-term prognosis of the virus. Moreover we show that drug treatment leads to chronic conditions that can be modeled by Markov shifts with more optimal eigenvectors.

scholarly journals On the Dynamics of Coupled Parametrically Forced Oscillators

2008 ◽  
Author(s):  
Jeff Moehlis
Keyword(s):  
Dynamical System ◽  
Hopf Bifurcation ◽  
Periodic Orbit ◽  
Weak Coupling ◽  
Weak Coupling Limit ◽  
Stable Periodic Orbit ◽  
Weakly Coupled ◽  
Forced Oscillators ◽  
Stable Periodic ◽  
Autonomous Dynamical System

It is well known that an autonomous dynamical system can have a stable periodic orbit, arising for example through a Hopf bifurcation. When a collection of such oscillators is coupled together, the system can display a number of phase-locked solutions which can be understood in the weak coupling limit by using a phase model. It is also well known that a stable periodic orbit can be found for a parametrically forced dynamical system, with the phase of the periodic orbit being locked to the forcing. Here we discuss the periodic solutions which occur for a collection of such parametrically forced oscillators that are weakly coupled together.

scholarly journals Appropriate initial conditions for asymptotic descriptions of the long term evolution of dynamical systems

1989 ◽  
Vol 31 (1) ◽  
pp. 48-75 ◽  
Author(s):  
A. J. Roberts
Keyword(s):  
Dynamical System ◽  
Invariant Manifold ◽  
Initial Conditions ◽  
Long Term Evolution ◽  
Equation Model ◽  
Starting Position ◽  
System A ◽  
Term Evolution ◽  
Shear Dispersion

AbstractA centre manifold or invariant manifold description of the evolution of a dynamical system provides a simplified view of the long term evolution of the system. In this work, I describe a procedure to estimate the appropriate starting position on the manifold which best matches an initial condition off the manifold. I apply the procedure to three examples: a simple dynamical system, a five-equation model of quasi-geostrophic flow, and shear dispersion in a channel. The analysis is also relevant to determining how best to account, within the invariant manifold description, for a small forcing in the full system.

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