scholarly journals Birkhoff Normal Forms and KAM Theory for Gumowski-Mira Equation

2014 ◽  
Vol 2014 ◽  
pp. 1-8 ◽  
Author(s):  
M. R. S. Kulenović ◽  
Z. Nurkanović ◽  
E. Pilav
Keyword(s):  
Normal Forms ◽  
Kam Theory ◽  
Equilibrium Solutions ◽  
Stability Of Equilibrium ◽  
The Stability

By using the KAM theory we investigate the stability of equilibrium solutions of the Gumowski-Mira equation:xn+1=(2axn)/(1+xn2)-xn-1,  n=0,1,…, wherex-1,x0∈(-∞,∞), and we obtain the Birkhoff normal forms for this equation for different equilibrium solutions.

A Note on the Stability of Equilibrium Solutions of Production-Constrained Spatial-Interaction Models

1981 ◽  
Vol 13 (5) ◽  
pp. 601-604 ◽  
Author(s):  
M Clarke
Keyword(s):  
Spatial Interaction ◽  
State Variables ◽  
Equilibrium Solutions ◽  
Interaction Models ◽  
Spatial Interaction Models ◽  
Stability Of Equilibrium ◽  
The Stability ◽  
Fold Catastrophe

In a recent paper an analysis of the stability of equilibrium solutions of production-constrained retailing models was undertaken by Harris and Wilson. It was argued that by examining equilibrium solutions in response to changes in a parameter, κ, jumps in the value of state variables, analogous to the fold catastrophe, could be expected. In this note it is shown that this is not necessarily always the case, and two different conditions are identified, one that gives rise to jump behaviours, and another that does not. The distinction arises from the nature of the κ parameter.

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.

Bifurcation Analysis of a Diffusive Activator–Inhibitor Model in Vascular Mesenchymal Cells

2015 ◽  
Vol 25 (10) ◽  
pp. 1530026 ◽  
Author(s):  
Rui Yang ◽  
Yongli Song
Keyword(s):  
Steady State ◽  
Center Manifold ◽  
Normal Forms ◽  
Mesenchymal Cells ◽  
Positive Equilibrium ◽  
Normal Form Theory ◽  
Form Theory ◽  
Spatially Homogeneous ◽  
The Stability ◽  
Activator Inhibitor

In this paper, a diffusive activator–inhibitor model in vascular mesenchymal cells is considered. On one hand, we investigate the stability of the equilibria of the system without diffusion. On the other hand, for the unique positive equilibrium of the system with diffusion the conditions ensuring stability, existence of Hopf and steady state bifurcations are given. By applying the center manifold and normal form theory, the normal forms corresponding to Hopf bifurcation and steady state bifurcation are derived explicitly. Numerical simulations are employed to illustrate where the spatially homogeneous and nonhomogeneous periodic solutions and the steady states can emerge. The numerical results verify the obtained theoretical conclusions.

scholarly journals The Lyapunov Stability for the Linear and Nonlinear Damped Oscillator with Time-Periodic Parameters

2010 ◽  
Vol 2010 ◽  
pp. 1-12 ◽  
Author(s):  
Jifeng Chu ◽  
Ting Xia
Keyword(s):  
Periodic Function ◽  
Lyapunov Stability ◽  
Stability Criterion ◽  
Normal Forms ◽  
Sufficient Condition ◽  
Periodic Functions ◽  
Linear And Nonlinear ◽  
The Stability ◽  
Time Periodic ◽  
Damped Oscillator

Leta(t),b(t)be continuousT-periodic functions with∫0Tb(t)dt=0. We establish one stability criterion for the linear damped oscillatorx′′+b(t)x′+a(t)x=0. Moreover, based on the computation of the corresponding Birkhoff normal forms, we present a sufficient condition for the stability of the equilibrium of the nonlinear damped oscillatorx′′+b(t)x′+a(t)x+c(t)x2n-1+e(t,x)=0, wheren≥2,c(t)is a continuousT-periodic function,e(t,x)is continuousT-periodic intand dominated by the powerx2nin a neighborhood ofx=0.

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 Numerical Exploration of Kaldorian Interregional Macrodynamics: Enhanced Stability and Predominance of Period Doubling under Flexible Exchange Rates

2010 ◽  
Vol 2010 ◽  
pp. 1-29 ◽  
Author(s):  
Toichiro Asada ◽  
Christos Douskos ◽  
Vassilis Kalantonis ◽  
Panagiotis Markellos
Keyword(s):  
Exchange Rates ◽  
Government Expenditure ◽  
Period Doubling ◽  
Numerical Approach ◽  
Capital Movement ◽  
Dimensional Parameter ◽  
Stability Of Equilibrium ◽  
The Stability ◽  
Flexible Exchange Rates ◽  
Enhanced Stability

We present a discrete two-regional Kaldorian macrodynamic model with flexible exchange rates and explore numerically the stability of equilibrium and the possibility of generation of business cycles. We use a grid search method in two-dimensional parameter subspaces, and coefficient criteria for the flip and Hopf bifurcation curves, to determine the stability region and its boundary curves in several parameter ranges. The model is characterized by enhanced stability of equilibrium, while its predominant asymptotic behavior when equilibrium is unstable is period doubling. Cycles are scarce and short-lived in parameter space, occurring at large values of the degree of capital movementβ. By contrast to the corresponding fixed exchange rates system, for cycles to occur sufficient amount of trade is requiredtogetherwith high levels of capital movement. Rapid changes in exchange rate expectations and decreased government expenditure are factors contributing to the creation of interregional cycles. Examples of bifurcation and Lyapunov exponent diagrams illustrating period doubling or cycles, and their development into chaotic attractors, are given. The paper illustrates the feasibility and effectiveness of the numerical approach for dynamical systems of moderately high dimensionality and several parameters.

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