HYPERBOLIC HOMOCLINIC POINTS OF ℤd-ACTIONS IN LATTICE DYNAMICAL SYSTEMS

1996 ◽  
Vol 06 (06) ◽  
pp. 1059-1075 ◽  
Author(s):  
V. S. AFRAIMOVICH ◽  
SHUI-NEE CHOW ◽  
WENXIAN SHEN
Keyword(s):  
Dynamical System ◽  
Homoclinic Solution ◽  
Homoclinic Point ◽  
Equilibrium Solutions ◽  
Homoclinic Points ◽  
Lattice Dynamical System ◽  
Spatial Coordinates ◽  
The Stability ◽  
Homoclinic And Heteroclinic Points ◽  
Heteroclinic Points

We study ℤd action on a set of equilibrium solutions of a lattice dynamical system, i.e., a system with discrete spatial variables, and the stability and hyperbolicity of the equilibrium solutions. Complicated behavior of ℤd-action corresponds to the existence of an infinite number of equilibrium solutions which are randomly situated along spatial coordinates. We prove that the existence of a homoclinic point of a ℤd-action implies complicated behavior, provided the hyperbolicity of the homoclinic solution with respect to the lattice dynamical system (this is a generalization of the previous work of the first two authors). Similar result holds for hyperbolic partially homoclinic and heteroclinic points. We show the equivalence of stability for any equilibrium solutions and the equivalence of hyperbolicity for homoclinic points under various norms.

scholarly journals Mathematical Modeling Analysis to Simulate the Dynamics of Immune Cells, HIV, and Tuberculosis

2017 ◽  
Vol 14 (1) ◽  
Author(s):  
Rumana Ahmed ◽  
Mahbubur Rahman
Keyword(s):  
Dynamical System ◽  
Immune Cells ◽  
Phase Plane ◽  
Disease Modeling ◽  
Jacobian Matrix ◽  
Equilibrium Solutions ◽  
Modeling Analysis ◽  
Starting Conditions ◽  
The Stability ◽  
Infectious Disease Modeling

The dynamics of immune cells, HIV, and tuberculosis can be described by a system of differential equations. We developed the formulations for this dynamical system. To evaluate the system as time goes to infinity, we investigated the equilibrium solutions. We established the criteria for stability based on the characteristics of the Jacobian matrix associated with the dynamical system. To investigate the bifurcation of the solution, we developed phase plane diagrams for the sets of assumed values of the parameters. we have investigated the curves for different values of the starting conditions of immune cells and the antigens. Along the curves, we observed the growth and decay processes. The stability of the system has been established by examining the phase plane diagrams as the solution approaches the equilibrium point. Based on phase diagrams, both stable and unstable systems have been simulated and examined in this study. Finally, we developed and evaluated the graphs for the unsteady variations of immune cells, HIV, and tuberculosis to see how the antigens grow because of the diminishing effects of immune cells in the system as time increases. KEYWORDS: Mathematical Biology; Infectious Disease Modeling; Dynamical System; Simulation of Immune Cells and Antigens

scholarly journals Stability of traveling wavefronts for a 2D lattice dynamical system arising in a diffusive population model

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Haiqin Zhao
Keyword(s):  
Dynamical System ◽  
Existence And Uniqueness ◽  
Population Model ◽  
Wave Speed ◽  
Exponential Convergence ◽  
Nonlinear Anal ◽  
Lattice Dynamical System ◽  
Two Component ◽  
Exponentially Stable ◽  
The Stability

Abstract This paper is concerned with the traveling wavefronts of a 2D two-component lattice dynamical system. This problem arises in the modeling of a species with mobile and stationary subpopulations in an environment in which the habitat is two-dimensional and divided into countable niches. The existence and uniqueness of the traveling wavefronts of this system have been studied in (Zhao and Wu in Nonlinear Anal., Real World Appl. 12: 1178–1191, 2011). However, the stability of the traveling wavefronts remains unsolved. In this paper, we show that all noncritical traveling wavefronts with given direction of propagation and wave speed are exponentially stable in time. In particular, we obtain the exponential convergence rate.

scholarly journals Upper semicontinuity of the attractor for lattice dynamical systems of partly dissipative reaction diffusion systems

2005 ◽  
Vol 2005 (3) ◽  
pp. 273-288 ◽  
Author(s):  
Ahmed Y. Abdallah
Keyword(s):  
Dynamical System ◽  
Hilbert Space ◽  
Upper Semicontinuity ◽  
Reaction Diffusion ◽  
Dimensional Lattice ◽  
Reaction Diffusion Systems ◽  
Infinite Dimensional ◽  
Diffusion Systems ◽  
Lattice Dynamical System ◽  
Lattice Dynamical Systems

We investigate the existence of a global attractor and its upper semicontinuity for the infinite-dimensional lattice dynamical system of a partly dissipative reaction diffusion system in the Hilbert spacel2×l2. Such a system is similar to the discretized FitzHugh-Nagumo system in neurobiology, which is an adequate justification for its study.

scholarly journals Stability-preserving model order reduction for linear stochastic Galerkin systems

2019 ◽  
Vol 9 (1) ◽  
Author(s):  
Roland Pulch
Keyword(s):  
Dynamical System ◽  
Model Order Reduction ◽  
Order Reduction ◽  
Electric Circuit ◽  
Physical Parameters ◽  
Science And Engineering ◽  
Model Order ◽  
Linear Dynamical Systems ◽  
Stochastic Galerkin ◽  
The Stability

Abstract Mathematical modeling often yields linear dynamical systems in science and engineering. We change physical parameters of the system into random variables to perform an uncertainty quantification. The stochastic Galerkin method yields a larger linear dynamical system, whose solution represents an approximation of random processes. A model order reduction (MOR) of the Galerkin system is advantageous due to the high dimensionality. However, asymptotic stability may be lost in some MOR techniques. In Galerkin-type MOR methods, the stability can be guaranteed by a transformation to a dissipative form. Either the original dynamical system or the stochastic Galerkin system can be transformed. We investigate the two variants of this stability-preserving approach. Both techniques are feasible, while featuring different properties in numerical methods. Results of numerical computations are demonstrated for two test examples modeling a mechanical application and an electric circuit, respectively.

scholarly journals Dynamical system analysis of three-form field dark energy model with baryonic matter

2020 ◽  
Vol 80 (9) ◽  
Author(s):  
Soumya Chakraborty ◽  
Sudip Mishra ◽  
Subenoy Chakraborty
Keyword(s):  
Dynamical System ◽  
Dark Energy ◽  
Cold Dark Matter ◽  
System Analysis ◽  
Evolution Equations ◽  
Matter Field ◽  
Baryonic Matter ◽  
Form Field ◽  
The Stability ◽  
Manifold Theory

AbstractA cosmological model having matter field as (non) interacting dark energy (DE) and baryonic matter and minimally coupled to gravity is considered in the present work with flat FLRW space time. The DE is chosen in the form of a three-form field while radiation and dust (i.e; cold dark matter) are the baryonic part. The cosmic evolution is studied through dynamical system analysis of the autonomous system so formed from the evolution equations by suitable choice of the dimensionless variables. The stability of the non-hyperbolic critical points are examined by Center manifold theory and possible bifurcation scenarios have been examined.

scholarly journals Fully localized post-buckling states of cylindrical shells under axial compression

2017 ◽  
Vol 473 (2205) ◽  
pp. 20170177 ◽  
Author(s):  
Tobias Kreilos ◽  
Tobias M. Schneider
Keyword(s):  
Axial Compression ◽  
Stability Boundary ◽  
Axial Direction ◽  
Edge State ◽  
Equilibrium Solutions ◽  
Thin Cylindrical Shell ◽  
Nonlinear Force ◽  
Post Buckling ◽  
The Stability ◽  
Nonlinear Solutions

We compute nonlinear force equilibrium solutions for a clamped thin cylindrical shell under axial compression. The equilibrium solutions are dynamically unstable and located on the stability boundary of the unbuckled state. A fully localized single dimple deformation is identified as the edge state —the attractor for the dynamics restricted to the stability boundary. Under variation of the axial load, the single dimple undergoes homoclinic snaking in the azimuthal direction, creating states with multiple dimples arranged around the central circumference. Once the circumference is completely filled with a ring of dimples, snaking in the axial direction leads to further growth of the dimple pattern. These fully nonlinear solutions embedded in the stability boundary of the unbuckled state constitute critical shape deformations. The solutions may thus be a step towards explaining when the buckling and subsequent collapse of an axially loaded cylinder shell is triggered.

On the stability of superlubricity in dynamical system

2020 ◽  
Vol 2020.11 (0) ◽  
pp. 26A3-MN2-4
Author(s):  
Haruka Yamanashi ◽  
Yuasku Kubota ◽  
Shinya Hasebe ◽  
Motohisa Hirano
Keyword(s):  
Dynamical System ◽  
The Stability

External Thought and Talk about Fiction

Fiction ◽  
2020 ◽  
pp. 150-183
Author(s):  
Catharine Abell
Keyword(s):  
Good Reason ◽  
Equilibrium Solutions ◽  
Coordination Problems ◽  
Fictional Entities ◽  
The Stability

This chapter examines the ontological implications of the various ways in which we can think and talk about fictional entities and examines the roles that external thought and talk about fiction can play in the institution of fiction. It argues that those who deny the existence of fictional entities are unable to accommodate the ways in which we think and talk about fictional entities from an external perspective, and that this gives us good reason to accept fictional entities into our ontology. It argues that external thought and talk about fiction are important to the identification of interpretative fictive content. It also argues that such thought and talk can play an important role in improving the stability of the content-determining rules of fiction institutions, and that they can help participants in fiction institutions to coordinate on rules that provide equilibrium solutions to novel coordination problems of communicating imaginings.

scholarly journals Fast and Deep Graph Neural Networks

2020 ◽  
Vol 34 (04) ◽  
pp. 3898-3905 ◽  
Author(s):  
Claudio Gallicchio ◽  
Alessio Micheli
Keyword(s):  
Neural Network ◽  
Dynamical System ◽  
Neural Networks ◽  
State Of The Art ◽  
Input Graph ◽  
Sparse Networks ◽  
Set Up ◽  
The Stability ◽  
Graph Neural Networks ◽  
Architectural Organization

We address the efficiency issue for the construction of a deep graph neural network (GNN). The approach exploits the idea of representing each input graph as a fixed point of a dynamical system (implemented through a recurrent neural network), and leverages a deep architectural organization of the recurrent units. Efficiency is gained by many aspects, including the use of small and very sparse networks, where the weights of the recurrent units are left untrained under the stability condition introduced in this work. This can be viewed as a way to study the intrinsic power of the architecture of a deep GNN, and also to provide insights for the set-up of more complex fully-trained models. Through experimental results, we show that even without training of the recurrent connections, the architecture of small deep GNN is surprisingly able to achieve or improve the state-of-the-art performance on a significant set of tasks in the field of graphs classification.

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