Assessing the Local Stability Properties of Discrete Three-Dimensional Dynamical Systems: A Geometrical Approach with Triangles and Planes and an Application with Some Cones

2016 ◽  
pp. 15-39
Author(s):  
Jean-Paul Barinci ◽  
Jean-Pierre Drugeon
Keyword(s):  
Dynamical Systems ◽  
Local Stability ◽  
Three Dimensional ◽  
Geometrical Approach ◽  
Stability Properties

scholarly journals Local Stability in 3D Discrete Dynamical Systems: Application to a Ricker Competition Model

2017 ◽  
Vol 2017 ◽  
pp. 1-16
Author(s):  
Rafael Luís ◽  
Elias Rodrigues
Keyword(s):  
Population Dynamics ◽  
Dynamical Systems ◽  
Fixed Points ◽  
Local Stability ◽  
Three Dimensional ◽  
Discrete Dynamical Systems ◽  
Competition Model ◽  
Centre Manifold ◽  
The Stability ◽  
Centre Manifold Theorem

A survey on the conditions of local stability of fixed points of three-dimensional discrete dynamical systems or difference equations is provided. In particular, the techniques for studying the stability of nonhyperbolic fixed points via the centre manifold theorem are presented. A nonlinear model in population dynamics is studied, namely, the Ricker competition model of three species. In addition, a conjecture about the global stability of the nontrivial fixed points of the Ricker competition model is presented.

scholarly journals Data-Informed Decomposition for Localized Uncertainty Quantification of Dynamical Systems

Vibration ◽  
2020 ◽  
Vol 4 (1) ◽  
pp. 49-63
Author(s):  
Waad Subber ◽  
Sayan Ghosh ◽  
Piyush Pandita ◽  
Yiming Zhang ◽  
Liping Wang
Keyword(s):  
Dynamical Systems ◽  
Computational Cost ◽  
Three Dimensional ◽  
Region Of Interest ◽  
System Response ◽  
Computational Domain ◽  
User Interest ◽  
Numerical Resolution ◽  
Multi Scale ◽  
Operation Conditions

Industrial dynamical systems often exhibit multi-scale responses due to material heterogeneity and complex operation conditions. The smallest length-scale of the systems dynamics controls the numerical resolution required to resolve the embedded physics. In practice however, high numerical resolution is only required in a confined region of the domain where fast dynamics or localized material variability is exhibited, whereas a coarser discretization can be sufficient in the rest majority of the domain. Partitioning the complex dynamical system into smaller easier-to-solve problems based on the localized dynamics and material variability can reduce the overall computational cost. The region of interest can be specified based on the localized features of the solution, user interest, and correlation length of the material properties. For problems where a region of interest is not evident, Bayesian inference can provide a feasible solution. In this work, we employ a Bayesian framework to update the prior knowledge of the localized region of interest using measurements of the system response. Once, the region of interest is identified, the localized uncertainty is propagate forward through the computational domain. We demonstrate our framework using numerical experiments on a three-dimensional elastodynamic problem.

Development of mixing and isotropy in inviscid homogeneous turbulence

1982 ◽  
Vol 120 ◽  
pp. 155-183 ◽  
Author(s):  
Jon Lee
Keyword(s):  
Dynamical Systems ◽  
Lower Order ◽  
Stokes System ◽  
Stokes Equations ◽  
Three Dimensional ◽  
Navier Stokes ◽  
Fourier Space ◽  
Kolmogorov Entropy ◽  
Navier Stokes Equations ◽  
Constant Energy

We have investigated a sequence of dynamical systems corresponding to spherical truncations of the incompressible three-dimensional Navier-Stokes equations in Fourier space. For lower-order truncated systems up to the spherical truncation of wavenumber radius 4, it is concluded that the inviscid Navier-Stokes system will develop mixing (and a fortiori ergodicity) on the constant energy-helicity surface, and also isotropy of the covariance spectral tensor. This conclusion is, however, drawn not directly from the mixing definition but from the observation that one cannot evolve the trajectory numerically much beyond several characteristic corre- lation times of the smallest eddy owing to the accumulation of round-off errors. The limited evolution time is a manifestation of trajectory instability (exponential orbit separation) which underlies not only mixing, but also the stronger dynamical charac- terization of positive Kolmogorov entropy (K-system).

scholarly journals Three-dimensional forced-damped dynamical systems with rich dynamics: Bifurcations, chaos and unbounded solutions

2015 ◽  
Vol 311-312 ◽  
pp. 25-36 ◽  
Author(s):  
Tomoyuki Miyaji ◽  
Hisashi Okamoto ◽  
Alex D.D. Craik
Keyword(s):  
Dynamical Systems ◽  
Three Dimensional ◽  
Unbounded Solutions

Shape of attractors for three-dimensional dissipative dynamical systems

2000 ◽  
Vol 61 (5) ◽  
pp. 5098-5107 ◽  
Author(s):  
S. Neukirch ◽  
H. Giacomini
Keyword(s):  
Dynamical Systems ◽  
Three Dimensional ◽  
Dissipative Dynamical Systems

HYPERBOLIC PLYKIN ATTRACTOR CAN EXIST IN NEURON MODELS

2005 ◽  
Vol 15 (11) ◽  
pp. 3567-3578 ◽  
Author(s):  
VLADIMIR BELYKH ◽  
IGOR BELYKH ◽  
ERIK MOSEKILDE
Keyword(s):  
Neuron Model ◽  
Three Dimensional ◽  
Geometrical Approach ◽  
Hyperbolic Attractor ◽  
Neuron Models ◽  
Neuron System ◽  
Physical Systems ◽  
System A ◽  
Bursting Neurons ◽  
Hyperbolic Attractors

Strange hyperbolic attractors are hard to find in real physical systems. This paper provides the first example of a realistic system, a canonical three-dimensional (3D) model of bursting neurons, that is likely to have a strange hyperbolic attractor. Using a geometrical approach to the study of the neuron model, we derive a flow-defined Poincaré map giving an accurate account of the system's dynamics. In a parameter region where the neuron system undergoes bifurcations causing transitions between tonic spiking and bursting, this two-dimensional map becomes a map of a disk with several periodic holes. A particular case is the map of a disk with three holes, matching the Plykin example of a planar hyperbolic attractor. The corresponding attractor of the 3D neuron model appears to be hyperbolic (this property is not verified in the present paper) and arises as a result of a two-loop (secondary) homoclinic bifurcation of a saddle. This type of bifurcation, and the complex behavior it can produce, have not been previously examined.

Extraction of three-dimensional soil pore space from microtomography images using a geometrical approach

Geoderma ◽  
2011 ◽  
Vol 163 (1-2) ◽  
pp. 127-134 ◽  
Author(s):  
Ndeye Fatou Ngom ◽  
Patricia Garnier ◽  
Olivier Monga ◽  
Stephan Peth
Keyword(s):  
Pore Space ◽  
Three Dimensional ◽  
Geometrical Approach ◽  
Soil Pore Space
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