Grazing Bifurcations of Initially Quasi-Periodic System Attractors

2001 ◽  
Author(s):  
Harry J. Dankowicz ◽  
Petri T. Piiroinen ◽  
Arne B. Nordmark
Keyword(s):  
Periodic System ◽  
Predictive Power ◽  
Legged Locomotion ◽  
Dimensional Model ◽  
Mapping Approach ◽  
Parameter Variations ◽  
Periodic Attractors ◽  
Discontinuity Mapping ◽  
Low Dimensional ◽  
Grazing Bifurcations

Abstract This paper discusses the dramatic changes in system dynamics that arise as parameter variations lead to the appearance of grazing intersections between periodic and quasi-periodic attractors and state-space discontinuities. In particular, a method based on the discontinuity-mapping approach is employed to predict the effects of near-grazing interactions with the discontinuity surface solely based on information about the discontinuity and the pre-grazing trajectory and requiring no knowledge of the impacting system. An example from the study of legged locomotion is used to illustrate the significance of such grazing bifurcations on the presence of sustained gait in simple anthropomorphic mechanisms. The predictive power of the discontinuity-mapping approach is illustrated on a low-dimensional model example.

Global Stability Analysis of the Wake behind a Circular Cylinder Based on the POD-Chiba Method

2011 ◽  
Vol 137 ◽  
pp. 72-76
Author(s):  
Wei Zhang ◽  
Xian Wen ◽  
Yan Qun Jiang
Keyword(s):  
Stability Analysis ◽  
Circular Cylinder ◽  
Global Stability ◽  
Dimensional Model ◽  
Global Stability Analysis ◽  
The Stability ◽  
Low Dimensional ◽  
Proper Orthogonal ◽  
Calculated Amount ◽  
Good Agreement

A proper orthogonal decomposition (POD) method is applied to study the global stability analysis for flow past a stationary circular cylinder. The flow database at Re=100 is obtained by CFD software, i.e. FLUENT, with which POD bases are constructed by a snapshot method. Based on the POD bases, a low-dimensional model is established for solving the two-dimensional incompressible NS equations. The stability of the flow solution is evaluated by a POD-Chiba method in the way of the eigensystem analysis for the velocity disturbance. The linear stability analysis shows that the first Hopf bifurcation takes place at Re=46.9, which is in good agreement with available results by other high-order accurate stability analysis methods. However, the calculated amount of POD is little, which shows the availability and advantage of the POD method.

scholarly journals Prosthetic Avian Vocal Organ Controlled by a Freely Behaving Bird Based on a Low Dimensional Model of the Biomechanical Periphery

2012 ◽  
Vol 8 (6) ◽  
pp. e1002546 ◽  
Author(s):  
Ezequiel M. Arneodo ◽  
Yonatan Sanz Perl ◽  
Franz Goller ◽  
Gabriel B. Mindlin
Keyword(s):  
Dimensional Model ◽  
Freely Behaving ◽  
Low Dimensional ◽  
Vocal Organ

scholarly journals Neuroimaging: into the Multiverse

2020 ◽  
Author(s):  
Jessica Dafflon ◽  
Pedro F. Da Costa ◽  
František Váša ◽  
Ricardo Pio Monti ◽  
Danilo Bzdok ◽  
...  
Keyword(s):  
Active Learning ◽  
Statistical Power ◽  
Predictive Power ◽  
Dimensional Space ◽  
Graph Theoretic ◽  
Processing Pipeline ◽  
Regression Approach ◽  
Low Dimensional ◽  
Sequential Analyses ◽  
Processing Steps

AbstractFor most neuroimaging questions the huge range of possible analytic choices leads to the possibility that conclusions from any single analytic approach may be misleading. Examples of possible choices include the motion regression approach used and smoothing and threshold factors applied during the processing pipeline. Although it is possible to perform a multiverse analysis that evaluates all possible analytic choices, this can be computationally challenging and repeated sequential analyses on the same data can compromise inferential and predictive power. Here, we establish how active learning on a low-dimensional space that captures the inter-relationships between analysis approaches can be used to efficiently approximate the whole multiverse of analyses. This approach balances the benefits of a multiverse analysis without the accompanying cost to statistical power, computational power and the integrity of inferences. We illustrate this approach with a functional MRI dataset of functional connectivity across adolescence, demonstrating how a multiverse of graph theoretic and simple pre-processing steps can be efficiently navigated using active learning. Our study shows how this approach can identify the subset of analysis techniques (i.e., pipelines) which are best able to predict participants’ ages, as well as allowing the performance of different approaches to be quantified.

scholarly journals Feature-based data assimilation in geophysics

2017 ◽  
Author(s):  
Matthias Morzfeld ◽  
Jesse Adams ◽  
Spencer Lunderman ◽  
Rafael Orozco
Keyword(s):  
Computational Models ◽  
Chaotic Systems ◽  
Efficient Solutions ◽  
Effective Dimension ◽  
Dimensional Model ◽  
Computationally Efficient ◽  
Fundamental Understanding ◽  
Feature Based ◽  
Low Dimensional ◽  
Noise Models

Abstract. Many applications in science require that computational models and data be combined. In a Bayesian framework, this is usually done by defining likelihoods based on the mismatch of model outputs and data. However, matching model outputs and data in this way can be unnecessary or impossible. For example, using large amounts of steady state data is unnecessary because these data are redundant, it is numerically difficult to assimilate data in chaotic systems, and it is often impossible to assimilate data of a complex system into a low-dimensional model. These issues can be addressed by selecting features of the data, and defining likelihoods based on the features, rather than by the usual mismatch of model output and data. Our goal is to contribute to a fundamental understanding of such a feature-based approach that allows us to assimilate selected aspects of data into models. Specifically, we explain how the feature-based approach can be interpreted as a method for reducing an effective dimension, and derive new noise models, based on perturbed observations, that lead to computationally efficient solutions. Numerical implementations of our ideas are illustrated in four examples.

A low-dimensional model for describing the oxygen storage capacity and transient behavior of a three-way catalytic converter

2012 ◽  
Vol 73 ◽  
pp. 373-387 ◽  
Author(s):  
Pankaj Kumar ◽  
Imad Makki ◽  
James Kerns ◽  
Karolos Grigoriadis ◽  
Matthew Franchek ◽  
...  
Keyword(s):  
Storage Capacity ◽  
Catalytic Converter ◽  
Transient Behavior ◽  
Oxygen Storage Capacity ◽  
Oxygen Storage ◽  
Dimensional Model ◽  
Low Dimensional

scholarly journals Combining trajectory optimization, supervised machine learning, and model structure for mitigating the curse of dimensionality in the control of bipedal robots

2019 ◽  
Vol 38 (9) ◽  
pp. 1063-1097 ◽  
Author(s):  
Xingye Da ◽  
Jessy Grizzle
Keyword(s):  
Machine Learning ◽  
Trajectory Optimization ◽  
Open Loop ◽  
Supervised Machine Learning ◽  
High Dimensional ◽  
Dimensional Model ◽  
Bipedal Robots ◽  
State Variable ◽  
Walking Motion ◽  
Low Dimensional

To overcome the obstructions imposed by high-dimensional bipedal models, we embed a stable walking motion in an attractive low-dimensional surface of the system’s state space. The process begins with trajectory optimization to design an open-loop periodic walking motion of the high-dimensional model and then adding to this solution a carefully selected set of additional open-loop trajectories of the model that steer toward the nominal motion. A drawback of trajectories is that they provide little information on how to respond to a disturbance. To address this shortcoming, supervised machine learning is used to extract a low-dimensional state-variable realization of the open-loop trajectories. The periodic orbit is now an attractor of the low-dimensional state-variable model but is not attractive in the full-order system. We then use the special structure of mechanical models associated with bipedal robots to embed the low-dimensional model in the original model in such a manner that the desired walking motions are locally exponentially stable. The design procedure is first developed for ordinary differential equations and illustrated on a simple model. The methods are subsequently extended to a class of hybrid models and then realized experimentally on an Atrias-series 3D bipedal robot.

scholarly journals The manifold structure of limb coordination in walking Drosophila

eLife ◽  
2019 ◽  
Vol 8 ◽  
Author(s):  
Brian D DeAngelis ◽  
Jacob A Zavatone-Veth ◽  
Damon A Clark
Keyword(s):  
Genetic Model ◽  
Model Organism ◽  
Legged Locomotion ◽  
Neural Activation ◽  
Symmetric Pattern ◽  
Multiple Dimensions ◽  
Limb Coordination ◽  
Spontaneous Behavior ◽  
Low Dimensional ◽  
Coordination Patterns

Terrestrial locomotion requires animals to coordinate their limb movements to efficiently traverse their environment. While previous studies in hexapods have reported that limb coordination patterns can vary substantially, the structure of this variability is not yet well understood. Here, we characterized the symmetric and asymmetric components of variation in walking kinematics in the genetic model organism Drosophila. We found that Drosophila use a single continuum of coordination patterns without evidence for preferred configurations. Spontaneous symmetric variability was associated with modulation of a single control parameter—stance duration—while asymmetric variability consisted of small, limb-specific modulations along multiple dimensions of the underlying symmetric pattern. Commands that modulated walking speed, originating from artificial neural activation or from the visual system, evoked modulations consistent with spontaneous behavior. Our findings suggest that Drosophila employ a low-dimensional control architecture, which provides a framework for understanding the neural circuits that regulate hexapod legged locomotion.

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