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

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 DYNAMICAL SYSTEMS ON THREE MANIFOLDS PART II: THREE-MANIFOLDS, HEEGAARD SPLITTINGS AND THREE-DIMENSIONAL SYSTEMS

2007 ◽  
Vol 17 (06) ◽  
pp. 2085-2095 ◽  
Author(s):  
YI SONG ◽  
STEPHEN P. BANKS
Keyword(s):  
Dynamical Systems ◽  
Nonlinear Systems ◽  
Systems Theory ◽  
Topological Structure ◽  
Three Dimensional ◽  
Local System ◽  
Operating Point ◽  
Global Behavior ◽  
Heegaard Splittings

The global behavior of nonlinear systems is extremely important in control and systems theory since the usual local theories will only provide information about a system in some neighborhood of an operating point. Away from that point, the system may have totally different behavior and so the theory developed for the local system will be useless for the global one. In this paper we shall consider the analytical and topological structure of systems on two- and three-manifolds and show that it is possible to obtain systems with "arbitrarily strange" behavior, i.e. arbitrary numbers of chaotic regimes which are knotted and linked in arbitrary ways. We shall do this by considering Heegaard Splittings of these manifolds and the resulting systems defined on the boundaries.

scholarly journals Coherent structures and chaotic advection in three dimensions

2010 ◽  
Vol 654 ◽  
pp. 1-4 ◽  
Author(s):  
STEPHEN WIGGINS
Keyword(s):  
Dynamical Systems ◽  
Fluid Mechanics ◽  
Coherent Structures ◽  
Three Dimensional ◽  
Point Of View ◽  
Three Dimensions ◽  
Chaotic Advection ◽  
Periodic Flow ◽  
Two Dimensional ◽  
Time Periodic

In the 1980s the incorporation of ideas from dynamical systems theory into theoretical fluid mechanics, reinforced by elegant experiments, fundamentally changed the way in which we view and analyse Lagrangian transport. The majority of work along these lines was restricted to two-dimensional flows and the generalization of the dynamical systems point of view to fully three-dimensional flows has seen less progress. This situation may now change with the work of Pouransari et al. (J. Fluid Mech., this issue, vol. 654, 2010, pp. 5–34) who study transport in a three-dimensional time-periodic flow and show that completely new types of dynamical systems structures and consequently, coherent structures, form a geometrical template governing transport.

A DYNAMICAL SYSTEMS APPROACH TO MEMBRANE PHENOMENA UNDERLYING CARDIAC ARRHYTHMIAS

1995 ◽  
Vol 05 (01) ◽  
pp. 75-88 ◽  
Author(s):  
RICHARD P. KLINE ◽  
B. MITCHELL BAKER
Keyword(s):  
Dynamical Systems ◽  
Action Potential ◽  
Systems Approach ◽  
Chaotic Behavior ◽  
Three Dimensional ◽  
Quantitative Agreement ◽  
Current Activation ◽  
Patterns Of Behavior ◽  
On Line ◽  
Considerable Detail

A model is constructed for cardiac rhythmic response to stimulation via a family of continuous time dynamical systems. Starting with experimentally observed properties common to the kinetics of both repolarizing membrane currents and cardiac action potential responses to sudden changes in cycle length, extremely elementary dynamical assumptions are made concerning current activation and decay, and repolarization threshold. A two-parameter family of one-dimensional dynamical systems emerges. The resulting systems are analytically tractable in considerable detail generating restitution curves, bifurcation schemes, rhythmic responses and chaotic behavior for a representative cardiac cell. The excellent qualitative and quantitative agreement with experimental data reported for several cardiac preparations is discussed. The two-dimensional analog produces unexpected basin behavior which could be of clinical significance in explaining how a single extra beat or a pause could alter subsequent action potential behavior and cause dispersion of refractoriness across the ventricle increasing the risks for arrhythmias. By having a manageable number of parameters, analytically defined patterns of behavior, and computational ease, this dynamical system has the potential to be used in computer simulations to study the effects of antiarrhythmic drugs on complex two- and three-dimensional reentrant substrates, or used on line by an interactive pacemaker.

scholarly journals Construction of the destination set of a dynamic system in $\mathbb{R}^3$

2021 ◽  
Vol 25 (2(36)) ◽  
pp. 83-94
Author(s):  
D. G. Kartashov ◽  
M. S. Tairova
Keyword(s):  
Dynamical Systems ◽  
Convex Hull ◽  
Dynamic System ◽  
Differential Inclusion ◽  
Support Function ◽  
Dimensional Space ◽  
Three Dimensional ◽  
Arbitrary Point ◽  
Intersection Method ◽  
Three Dimensional Space

The article proposes two algorithms for the numerical construction of the convex hull of a set in three-dimensional space using its support function. The first uses the hyperplane intersection method to find the pivot points of a set. The second one is based on the deformation function and allows you to find an arbitrary point of the convex hull of a set, which is convenient in many applications. The algorithms are compared, and asymptotic complexities are found. The application of the proposed apparatus to finding the destination set of dynamical systems is shown. The dynamic system will be based on differential inclusion.

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

2020 ◽  
Author(s):  
Waad Subber ◽  
Sayan Ghosh ◽  
Piyush Pandita ◽  
Yiming Zhang ◽  
Liping Wang
Keyword(s):  
Bayesian Inference ◽  
Dynamical Systems ◽  
Uncertainty Quantification ◽  
Computational Cost ◽  
Three Dimensional ◽  
Region Of Interest ◽  
System Response ◽  
User Interest ◽  
Operation Conditions ◽  
Random Material Properties

Industrial dynamical systems often exhibit multi-scale response due to material heterogeneities, operation conditions and complex environmental loadings. In such problems, it is the case that the smallest length-scale of the systems dynamics controls the numerical resolution required to effectively resolve the embedded physics. In practice however, high numerical resolutions is only required in a confined region of the system where fast dynamics or localized material variability are exhibited, whereas a coarser discretization can be sufficient in the rest majority of the system. To this end, a unified computational scheme with uniform spatio-temporal resolutions for uncertainty quantification can be very computationally demanding. Partitioning the complex dynamical system into smaller easier-to-solve problems based of the localized dynamics and material variability can reduce the overall computational cost. However, identifying the region of interest for high-resolution and intensive uncertainty quantification can be a problem dependent. The region of interest can be specified based on the localization features of the solution, user interest, and correlation length of the random 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 our prior knowledge on the localized region of interest using measurements and system response. To address the computational cost of the Bayesian inference, we construct a Gaussian process surrogate for the forward model. Once, the localized region of interest is identified, we use polynomial chaos expansion to propagate the localization uncertainty. We demonstrate our framework through numerical experiments on a three-dimensional elastodynamic problem

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