Maximizing Sensitivity Vector Fields: A Parametric Study

2014 ◽  
Vol 9 (2) ◽  
Author(s):  
Andrew R. Sloboda ◽  
Bogdan I. Epureanu
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Vector Fields ◽  
Nonlinear Feedback ◽  
Baseline System ◽  
Nominal Parameter ◽  
Construction Play ◽  
Sensitivity Vector ◽  
System Attractor ◽  
Made In

Sensitivity vector fields (SVFs) have proven to be an effective method for identifying parametric variations in dynamical systems. These fields are constructed using information about how a dynamical system's attractor deforms under prescribed parametric variations. Once constructed, they can be used to quantify any additional variations from the nominal parameter set as they occur. Since SVFs are based on attractor deformations, the geometry and other qualities of the baseline system attractor impact how well a set of SVFs will perform. This paper examines the role attractor characteristics and the choices made in SVF construction play in determining the sensitivity of SVFs. The use of nonlinear feedback to change a dynamical system with the intent of improving SVF sensitivity is explored. These ideas are presented in the context of constructing SVFs for several dynamical systems.

High-Sensitivity Mass Sensing Based on Enhanced Nonlinear Dynamics and Attractor Morphing Modes

2006 ◽  
Author(s):  
Shih-Hsun Yin ◽  
Bogdan I. Epureanu
Keyword(s):  
Lead Zirconate Titanate ◽  
Vector Fields ◽  
Experimental System ◽  
High Sensitivity ◽  
Lead Zirconate ◽  
Chaotic Regime ◽  
Nonlinear Feedback ◽  
Regime Changes ◽  
Distribution Of Points ◽  
Sensitivity Vector

This paper demonstrates two novel methods for identifying small parametric variations in an experimental system based on the analysis of sensitivity vector fields (SVFs) and probability density functions (PDFs). The experimental system includes a smart sensing beam excited by a nonlinear feedback excitation through two PZT (lead zirconate titanate) patches symmetrically bonded on both sides at the root of the beam. The nonlinear feedback excitation requires the measurement of the dynamics (e.g. velocity of one point at the tip of the beam) and a nonlinear feedback loop, and is designed such that the beam vibrates in a chaotic regime. Changes in the state space attractor of the dynamics due to small parametric variations can be captured by SVFs which, in turn, are collected by applying point cloud averaging (PCA) to points distributed in the attractors for nominal and changed parameters. Also, the PDFs characterize statistically the distribution of points in the attractors. The differences between the PDFs of the attractors for different changed parameters and the baseline attractor can provide different attractor morphing modes for identifying variations in distinct parameters. The experimental results based on the proposed approaches show that very small amounts of added mass at different locations along the beam can be accurately identified.

Complex Dynamics of Bouncing Motions at Boundaries and Corners in a Discontinuous Dynamical System

2017 ◽  
Vol 12 (6) ◽  
Author(s):  
Jianzhe Huang ◽  
Albert C. J. Luo
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Vector Fields ◽  
Complex Dynamics ◽  
Singularity Theory ◽  
Local Theory ◽  
Periodic Motions ◽  
Border Collision Bifurcations ◽  
Discontinuous Dynamical System ◽  
Stability And Bifurcations

In this paper, from the local theory of flow at the corner in discontinuous dynamical systems, obtained are analytical conditions for switching impact-alike chatter at corners. The objective of this investigation is to find the dynamics mechanism of border-collision bifurcations in discontinuous dynamical systems. Multivalued linear vector fields are employed, and generic mappings are defined among boundaries and corners. From mapping structures, periodic motions switching at the boundaries and corners are determined, and the corresponding stability and bifurcations of periodic motions are investigated by eigenvalue analysis. However, the grazing and sliding bifurcations are determined by the local singularity theory of discontinuous dynamical systems. From such analytical conditions, the corresponding parameter map is developed for periodic motions in such a multivalued dynamical system in the single domain with corners. Numerical simulations of periodic motions are presented for illustrations of motions complexity and catastrophe in such a discontinuous dynamical system.

Some Algebraic Properties of Continuous Time Dynamical Systems

1997 ◽  
Vol 12 (01) ◽  
pp. 137-142
Author(s):  
İnanç Birol ◽  
Avadis Hacinliyan
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Lyapunov Exponents ◽  
Algebraic Structure ◽  
Continuous Time ◽  
Attractor Dimension ◽  
Algebraic Properties ◽  
System Attractor

Number of zero Lyapunov exponents of a system is directly related to the dimension of the manifold of the system attractor. Moreover, this attractor dimension is governed by the algebraic structure of the manifold it lives on. In this work we try to establish a basis for the description of this manifold, which we aim to use in determining zero Lyapunov exponents of a continuous time dynamical system.

scholarly journals Local Bifurcations of Reversible Piecewise Smooth Planar Dynamical Systems

2020 ◽  
pp. 1-15
Author(s):  
V. Sh. Roitenberg
Keyword(s):  
Dynamical System ◽  
Vector Field ◽  
Dynamical Systems ◽  
Singular Point ◽  
Vector Fields ◽  
Singular Points ◽  
Piecewise Smooth ◽  
Smooth Vector ◽  
Local Bifurcations ◽  
Piecewise Smooth Vector Fields

There are quite a few works, which consider local bifurcations of piecewise-smooth vector fields on the plane. A number of papers also studied  the local bifurcations of smooth vector fields on the plane that are reversible with respect to involution. In the paper, we introduce reversible dynamical systems defined by piecewise-smooth vector fields on the coordinate plane (x, y) for which the discontinuity line y = 0 coincides with the set of fixed points of the system involution. We consider the generic one-parameter perturbations of such a vector field. The bifurcations of the singular point O lying on this line are described in two cases. In the first case, the point O is a rough saddle of the smooth vector fields that coincide with a piecewise smooth vector field in the half-planes y > 0 and y < 0. The parameter can be chosen so that for parameter values less than or equal to zero, the dynamical system has a unique singular point with four hyperbolic sectors in a vicinity of the point O. For positive values of the parameter in the vicinity of the point O, there are three singular points, a quasi-centre and two saddles, the separatrixes of which form a simple closed contour that bounds the cell from closed trajectories. In the second case, O is a rough node of the corresponding vector fields. The parameter can be chosen so that for values of the parameter less than or equal to zero, the dynamical system has a unique singular point in a vicinity of the point O, and all other trajectories are closed. For positive values of the parameter in the vicinity of the point O, there are three singular points, two nodes and a quasi-saddle, whose two separatrixes go to the nodes.

scholarly journals ONE–PARAMETRIC SEMIGROUPS OF DIFFEOMORPHISMS AND ONE‐SIDED VECTOR FIELDS

2010 ◽  
Vol 15 (2) ◽  
pp. 235-244 ◽  
Author(s):  
Paulius Miškinis
Keyword(s):  
Dynamical System ◽  
Vector Field ◽  
Dynamical Systems ◽  
Autonomous System ◽  
Vector Fields ◽  
Phase Flow ◽  
System Phase

The fractional generalization of dynamical systems is considered. For this purpose, the concepts of fractional phase semi‐flow, fractional autonomous system and the generalized exponent of the vector field are introduced. Two examples of their application are explained in detail. Keywords: dynamical system, phase flow, diffeomorphism.

Periodic Orbits in a Second-Order Discontinuous System with an Elliptic Boundary

2016 ◽  
Vol 26 (13) ◽  
pp. 1650224 ◽  
Author(s):  
Liping Li ◽  
Albert C. J. Luo
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Periodic Orbits ◽  
Vector Fields ◽  
Sliding Mode ◽  
Elliptic Boundary ◽  
Second Order ◽  
Discontinuous System ◽  
System A ◽  
Discontinuous Dynamical System

This paper develops the analytical conditions for the onset and disappearance of motion passability and sliding along an elliptic boundary in a second-order discontinuous system. A periodically forced system, described by two different linear subsystems, is considered mainly to demonstrate the methodology. The passable, sliding and grazing conditions of a flow to the elliptic boundary in the discontinuous dynamical system are provided through the analysis of the corresponding vector fields and [Formula: see text]-functions. Moreover, by constructing appropriate generic mappings, periodic orbits in such a discontinuous system are predicted analytically. Finally, three different cases are discussed to illustrate the existence of periodic orbits with passable and/or sliding flows. The results obtained in this paper can be applied to the sliding mode control in discontinuous dynamical systems.

Experimental Enhanced Nonlinear Dynamics and Identification of Attractor Morphing Modes for Damage Detection

2007 ◽  
Vol 129 (6) ◽  
pp. 763-770 ◽  
Author(s):  
Shih-Hsun Yin ◽  
Bogdan I. Epureanu
Keyword(s):  
Lead Zirconate Titanate ◽  
Vector Fields ◽  
Experimental System ◽  
Lead Zirconate ◽  
Chaotic Regime ◽  
Base Line ◽  
Nonlinear Feedback ◽  
Regime Changes ◽  
Distribution Of Points ◽  
Sensitivity Vector

This paper demonstrates two novel methods for identifying small parametric variations in an experimental system based on the analysis of sensitivity vector fields (SVFs) and probability density functions (PDFs). The experimental system includes a smart sensing beam excited by a nonlinear feedback excitation through two lead zirconate titanate patches symmetrically bonded on both sides at the root of the beam. The nonlinear feedback excitation requires the measurement of the dynamics (e.g., velocity of one point at the tip of the beam) and a nonlinear feedback loop, and is designed such that the beam vibrates in a chaotic regime. Changes in the state space attractor of the dynamics due to small parametric variations can be captured by SVFs, which, in turn, are collected by applying point cloud averaging to points distributed in the attractors for nominal and changed parameters. Also, the PDFs characterize statistically the distribution of points in the attractors. The differences between the PDFs of the attractors for different changed parameters and the base line attractor can provide different attractor morphing modes for identifying variations in distinct parameters. Experimental results based on the proposed approaches show that very small amounts of added mass at different locations along the beam can be accurately identified.2

Integrable two-dimensional dynamical systems and the characteristic Lie algebras

2007 ◽  
Vol 5 ◽  
pp. 195-200
Author(s):  
A.V. Zhiber ◽  
O.S. Kostrigina
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Second Order ◽  
System Of Equations ◽  
Two Dimensional ◽  
Finite Dimensional ◽  
Dimensional Dynamical System

In the paper it is shown that the two-dimensional dynamical system of equations is Darboux integrable if and only if its characteristic Lie algebra is finite-dimensional. The class of systems having a full set of fist and second order integrals is described.

scholarly journals Potential Well in Poincaré Recurrence

Entropy ◽  
2021 ◽  
Vol 23 (3) ◽  
pp. 379
Author(s):  
Miguel Abadi ◽  
Vitor Amorim ◽  
Sandro Gallo
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Stochastic Processes ◽  
Recurrence Time ◽  
Potential Well ◽  
Exponential Approximation ◽  
Mixing Processes ◽  
System Perspective ◽  
Return Times ◽  
Error Terms

From a physical/dynamical system perspective, the potential well represents the proportional mass of points that escape the neighbourhood of a given point. In the last 20 years, several works have shown the importance of this quantity to obtain precise approximations for several recurrence time distributions in mixing stochastic processes and dynamical systems. Besides providing a review of the different scaling factors used in the literature in recurrence times, the present work contributes two new results: (1) For ϕ-mixing and ψ-mixing processes, we give a new exponential approximation for hitting and return times using the potential well as the scaling parameter. The error terms are explicit and sharp. (2) We analyse the uniform positivity of the potential well. Our results apply to processes on countable alphabets and do not assume a complete grammar.

A DYNAMICAL SYSTEM WITH CHAOTIC BEHAVIOR

1989 ◽  
Vol 03 (15) ◽  
pp. 1185-1188 ◽  
Author(s):  
J. SEIMENIS
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Infinite Number ◽  
Chaotic Behavior ◽  
Equations Of Motion ◽  
Hamiltonian Dynamical Systems

We develop a method to find solutions of the equations of motion in Hamiltonian Dynamical Systems. We apply this method to the system [Formula: see text] We study the case a → 0 and we find that in this case the system has an infinite number of period dubling bifurcations.

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