scholarly journals Representing Higher-Order Singularities in Vector Fields on Piecewise Linear Surfaces

2006 ◽  
Vol 12 (5) ◽  
pp. 1315-1322 ◽  
Author(s):  
Wan-chiu Li ◽  
Bruno Vallet ◽  
Nicolas Ray ◽  
Bruno Levy
Keyword(s):  
Vector Fields ◽  
Piecewise Linear ◽  
Higher Order

scholarly journals Higher Order Singularities in Piecewise Linear Vector Fields

2000 ◽  
pp. 99-113 ◽  
Author(s):  
Xavier Tricoche ◽  
Gerik Scheuermann ◽  
Hans Hagen
Keyword(s):  
Vector Fields ◽  
Piecewise Linear ◽  
Higher Order ◽  
Linear Vector

scholarly journals Higher order Melnikov analysis for planar piecewise linear vector fields with nonlinear switching curve

2021 ◽  
Vol 287 ◽  
pp. 1-36
Author(s):  
Kamila da S. Andrade ◽  
Oscar A.R. Cespedes ◽  
Dayane R. Cruz ◽  
Douglas D. Novaes
Keyword(s):  
Vector Fields ◽  
Piecewise Linear ◽  
Higher Order ◽  
Linear Vector ◽  
Switching Curve ◽  
Melnikov Analysis ◽  
Nonlinear Switching

One variation of R. Miron's spray theory in OsckM

2003 ◽  
Vol 40 (4) ◽  
pp. 443-462
Author(s):  
Irena Čomić
Keyword(s):  
Vector Fields ◽  
Transformation Group ◽  
Higher Order

Lately a big attention has been payed on the higher order geometry. Some relevant papers are mentioned in the references. R. Miron and Gh. Atanasiu in [16], [17] studied the geometry of OsckM. R. Miron in [19] gave the comprehend theory of higher order geometry and its application. The whole theory of sprays in OsckM M is established. Here, using R. Miron's method, a variation of this theory is given. The transformation group is slightly different from that used in [19] and it will change the geometry. The adapted basis, the Liouville vector fields, the equation of sprays, will have different form. We give the relations between coefficients of S and the Liouville vector fields.

CHUA’S CIRCUIT: AN OVERVIEW TEN YEARS LATER

1994 ◽  
Vol 04 (02) ◽  
pp. 117-159 ◽  
Author(s):  
LEON O. CHUA
Keyword(s):  
Secure Communication ◽  
Vector Fields ◽  
Piecewise Linear ◽  
International Workshop ◽  
Noise Spectrum ◽  
Chaotic Regime ◽  
Practical Significance ◽  
Chua’S Circuit ◽  
Chua's Circuit ◽  
New Phenomena

More than 200 papers, two special issues (Journal of Circuits, Systems, and Computers, March, June, 1993, and IEEE Trans. on Circuits and Systems, vol. 40, no. 10, October, 1993), an International Workshop on Chua’s Circuit: chaotic phenomena and applica tions at NOLTA’93, and a book (edited by R.N. Madan, World Scientific, 1993) on Chua’s circuit have been published since its inception a decade ago. This review paper attempts to present an overview of these timely publications, almost all within the last six months, and to identify four milestones of this very active research area. An important milestone is the recent fabrication of a monolithic Chua’s circuit. The robustness of this IC chip demonstrates that an array of Chua’s circuits can also be fabricated into a monolithic chip, thereby opening the floodgate to many unconventional applications in information technology, synergetics, and even music. The second milestone is the recent global unfolding of Chua’s circuit, obtained by adding a linear resistor in series with the inductor to obtain a canonical Chua’s circuit— now generally referred to as Chua’s oscillator. This circuit is most significant because it is structurally the simplest (it contains only 6 circuit elements) but dynamically the most complex among all nonlinear circuits and systems described by a 21-parameter family of continuous odd-symmetric piecewise-linear vector fields. The third milestone is the recent discovery of several important new phenomena in Chua’s circuits, e.g., stochastic resonance, chaos-chaos type intermittency, 1/f noise spectrum, etc. These new phenomena could have far-reaching theoretical and practical significance. The fourth milestone is the theoretical and experimental demonstration that Chua’s circuit can be easily controlled from a chaotic regime to a prescribed periodic or constant orbit, or it can be synchronized with 2 or more identical Chua’s circuits, operating in an oscillatory, or a chaotic regime. These recent breakthroughs have ushered in a new era where chaos is deliberately created and exploited for unconventional applications, e.g. secure communication.

A symmetry-preserving difference scheme and analytical solutions of a generalized higher-order beam equation

2021 ◽  
Vol 477 (2255) ◽  
Author(s):  
Shou-Fu Tian ◽  
Mei-Juan Xu ◽  
Tian-Tian Zhang
Keyword(s):  
Power Series ◽  
Vector Fields ◽  
Higher Order ◽  
Beam Equation ◽  
Series Solutions ◽  
Local Conservation ◽  
Power Series Method ◽  
Potential System ◽  
Power Series Solutions ◽  
The Difference

Under investigation in this work is a generalized higher-order beam equation, which is an important physical model and describes the vibrations of a rod. By considering Lie symmetry analysis, and using the power series method, we derive the geometric vector fields, symmetry reductions, group invariant solutions and power series solutions of the equation, respectively. The convergence analysis of the power series solutions are also provided with detailed proof. Furthermore, by virtue of the multipliers, the local conservation laws of the equation are derived as well. Furthermore, an effective and direct approach is proposed to study the symmetry-preserving discretization for the equation via its potential system. Finally, the invariant difference models of the generalized beam equation are successfully constructed. Our results show that it is very useful to construct the difference models of the potential system instead of the original equation.

Unique Normal Form and the Associated Coefficients for a Class of Three-Dimensional Nilpotent Vector Fields

2017 ◽  
Vol 27 (14) ◽  
pp. 1750224
Author(s):  
Jing Li ◽  
Liying Kou ◽  
Duo Wang ◽  
Wei Zhang
Keyword(s):  
Normal Form ◽  
Vector Fields ◽  
Three Dimensional ◽  
Recursive Formula ◽  
Higher Order ◽  
Dimensional Vector ◽  
Symbolic Calculations

In this paper, we mainly focus on the unique normal form for a class of three-dimensional vector fields via the method of transformation with parameters. A general explicit recursive formula is derived to compute the higher order normal form and the associated coefficients, which can be achieved easily by symbolic calculations. To illustrate the efficiency of the approach, a comparison of our result with others is also presented.

Analysis of Piecewise Linear Systems With Displacement and Time Regions by Vector Fields

1999 ◽  
Author(s):  
Mikio Nakai ◽  
Shinji Murata ◽  
Seiji Hagio
Keyword(s):  
Linear Systems ◽  
Vector Fields ◽  
Matrix Representation ◽  
Piecewise Linear ◽  
Generalized Solution ◽  
State Variables ◽  
Piecewise Linear Systems ◽  
Restoring Forces ◽  
Pass Through ◽  
Solution Methodology

Abstract A generalized solution methodology based on piecewise linear vector fields is proposed for piecewise linear systems with singular regions or asymmetric restoring forces which vary spatially and temporally. In matrix representation for these systems, state variables in each region can be explicitly expressed as a function of the time the orbit spends between two boundaries or the time the orbit takes to pass through the boundary. The time can be determined by the Brent method, and periodic solutions can then be obtained. Analytical solutions are validated on a system with 3-regions of displacement and 2-regions of time, a circumferential vibration of gear meshing system, by using the newly developed numerical method.

scholarly journals Zeno breaking, the Contact effect and sensitive behaviour in piecewise-linear systems

2018 ◽  
Vol 29 (5) ◽  
pp. 826-844
Author(s):  
R. EDWARDS
Keyword(s):  
Vector Fields ◽  
Infinite Sequence ◽  
Piecewise Linear ◽  
Need For Care ◽  
Time Intervals ◽  
Finite Time Convergence ◽  
Contact Effect ◽  
Qualitative Models ◽  
Pass Through ◽  
Smooth Approximations

Non-smooth approximations of steep sigmoidal switching networks, such as those used as qualitative models of gene regulation, lead to analytic and computational challenges that arise as a result of the discontinuities in the vector fields. In order to highlight the need for care in dealing with such systems, several particular phenomena are presented here through illustrative examples, including ‘Zeno breaking’, or computing beyond the finite time convergence of an infinite sequence of threshold transitions; the ‘Contact’ effect, in which in the discontinuous limit, trajectories can pass through a ‘saddle point’ without stopping, though these solutions are not unique and other solutions stop for arbitrary time intervals; and sensitive behaviour that arises from exotic dynamics within switching regions.

scholarly journals Modeling n -Symmetry Vector Fields using Higher-Order Energies

2018 ◽  
Vol 37 (2) ◽  
pp. 1-18 ◽  
Author(s):  
Christopher Brandt ◽  
Leonardo Scandolo ◽  
Elmar Eisemann ◽  
Klaus Hildebrandt
Keyword(s):  
Vector Fields ◽  
Higher Order
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