BIMODAL PIECEWISE LINEAR DYNAMICAL SYSTEMS: REDUCED FORMS

2010 ◽  
Vol 20 (09) ◽  
pp. 2795-2808 ◽  
Author(s):  
JOSEP FERRER ◽  
M. DOLORS MAGRET ◽  
MARTA PEÑA
Keyword(s):  
Dynamical Systems ◽  
Linear Systems ◽  
Piecewise Linear ◽  
Equivalence Classes ◽  
State Variables ◽  
Linear Dynamical Systems ◽  
Piecewise Linear Systems ◽  
Wide Range ◽  
General Systems ◽  
Reduced Forms

Piecewise linear systems constitute a class of nonlinear systems which have recently attracted the interest of researchers because of their interesting properties and the wide range of applications from which they arise. Different authors have used reduced forms when studying these systems, mostly in the case where they are observable. In this work, we focus on bimodal continuous dynamical systems (those consisting of two linear systems on each side of a given hyperplane, having continuous dynamics along that hyperplane) depending on two or three state variables, which are the most common piecewise linear systems found in practice. Reduced forms are obtained for general systems, not necessarily observable. As an application, we calculate the dimension of the equivalence classes.

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.

Limit Cycles in Discontinuous Planar Piecewise Linear Systems Separated by a Nonregular Line of Center–Center Type

2021 ◽  
Vol 31 (09) ◽  
pp. 2150136
Author(s):  
Qianqian Zhao ◽  
Cheng Wang ◽  
Jiang Yu
Keyword(s):  
Dynamical Systems ◽  
Linear Systems ◽  
Limit Cycles ◽  
Piecewise Linear ◽  
The Other ◽  
Natural Phenomena ◽  
Separation Line ◽  
Piecewise Linear Systems

Many natural phenomena can be modeled as discontinuous dynamical systems separated by a nonregular line. The number and distribution of limit cycles in discontinuous linear systems are important topics for research. In this paper, we focus on the limit cycles created by discontinuous planar piecewise linear systems separated by a nonregular line of center–center type, and prove that such systems have at most two limit cycles, which can be reached. Furthermore, the two limit cycles are nested and intersect the separation line at two points or four points, that is, either both intersect the separation line at two points or one intersects the separation line at two points and the other one at four points.

scholarly journals Designing the mesoscopic approach of an autonomous linear dynamical system by a quantum formulation

2016 ◽  
Author(s):  
Juan Carlos Micó Ruiz
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Linear Systems ◽  
Second Order ◽  
Linear Dynamical System ◽  
Linear Dynamical Systems ◽  
General Systems ◽  
Non Linear ◽  
Quantum Formulation

The work presents a mesoscopic approach to general systems modelled by dynamical systems. The quantum formulation is possible to be obtained by their quantum formulation from a second order Hamiltonian. However, only autonomous linear systems are proved to obtain a Hamiltonian like this. Some application cases are presented, and a discussion about how to generalize the formalism to non-linear dynamical systems is sketched.DOI: http://dx.doi.org/10.4995/IFDP.2016.2795

Piecewise Linear Systems With Time Varying Coefficients

1997 ◽  
Author(s):  
S. Natsiavas ◽  
S. Theodossiades
Keyword(s):  
Linear Systems ◽  
Piecewise Linear ◽  
Duffing Oscillator ◽  
Constant Load ◽  
Time Varying ◽  
Gear Pair ◽  
Linear Dynamical Systems ◽  
Varying Coefficients ◽  
Piecewise Linear Systems ◽  
Time Varying Coefficients

Abstract A new method is presented for determining periodic steady state response of piecewise linear dynamical systems with time varying coefficients. As an example mechanical model, a gear-pair system with backlash is examined, under the action of a constant torque. Originally, some useful insight is gained on the type of motions expected by investigating the response of a weakly nonlinear Mathieu-Duffing oscillator, subjected to a constant external load. The information obtained is then used in seeking the appropriate form of approximate periodic solutions of the piecewise linear system. Finally, these solutions are determined by developing a new analytical method. This method combines elements from approaches applied for piecewise linear systems with constant coefficients as well as classical perturbation techniques applied for systems with time varying coefficients. The validity and accuracy of the approach is verified by numerical results. In addition, response diagrams are presented, illustrating the effect of the constant load and the damping on the gear-pair response.

Analysis of Piecewise Linear Systems With Boundaries in Displacement and Time

2002 ◽  
Vol 124 (4) ◽  
pp. 527-536 ◽  
Author(s):  
Mikio Nakai ◽  
Shinji Murata ◽  
Seiji Hagio
Keyword(s):  
Linear Systems ◽  
Analytical Solutions ◽  
Vector Fields ◽  
Matrix Representation ◽  
Piecewise Linear ◽  
Generalized Solution ◽  
State Variables ◽  
Piecewise Linear Systems ◽  
Restoring Forces ◽  
Solution Methodology

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 when the orbit hits 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 Limit cycles from a monodromic infinity in planar piecewise linear systems

2021 ◽  
Vol 496 (2) ◽  
pp. 124818
Author(s):  
Emilio Freire ◽  
Enrique Ponce ◽  
Joan Torregrosa ◽  
Francisco Torres
Keyword(s):  
Linear Systems ◽  
Limit Cycles ◽  
Piecewise Linear ◽  
Piecewise Linear Systems

Nondegenerate Piecewise Linear Systems: A Finite Newton Algorithm and Applications in Machine Learning

2012 ◽  
Vol 24 (4) ◽  
pp. 1047-1084 ◽  
Author(s):  
Xiao-Tong Yuan ◽  
Shuicheng Yan
Keyword(s):  
Linear Systems ◽  
Optimization Problems ◽  
Piecewise Linear ◽  
Optimization Methods ◽  
Coefficient Matrix ◽  
Learning Problems ◽  
Support Vector ◽  
Data Sets ◽  
Piecewise Linear Systems ◽  
Vector Machines

We investigate Newton-type optimization methods for solving piecewise linear systems (PLSs) with nondegenerate coefficient matrix. Such systems arise, for example, from the numerical solution of linear complementarity problem, which is useful to model several learning and optimization problems. In this letter, we propose an effective damped Newton method, PLS-DN, to find the exact (up to machine precision) solution of nondegenerate PLSs. PLS-DN exhibits provable semiiterative property, that is, the algorithm converges globally to the exact solution in a finite number of iterations. The rate of convergence is shown to be at least linear before termination. We emphasize the applications of our method in modeling, from a novel perspective of PLSs, some statistical learning problems such as box-constrained least squares, elitist Lasso (Kowalski & Torreesani, 2008 ), and support vector machines (Cortes & Vapnik, 1995 ). Numerical results on synthetic and benchmark data sets are presented to demonstrate the effectiveness and efficiency of PLS-DN on these problems.

scholarly journals Canards in piecewise-linear systems: explosions and super-explosions

2013 ◽  
Vol 469 (2154) ◽  
pp. 20120603 ◽  
Author(s):  
Mathieu Desroches ◽  
Emilio Freire ◽  
S. John Hogan ◽  
Enrique Ponce ◽  
Phanikrishna Thota
Keyword(s):  
Phase Space ◽  
Linear Systems ◽  
Limit Cycles ◽  
Piecewise Linear ◽  
Single Parameter ◽  
Van Der Pol ◽  
Piecewise Linear Systems ◽  
Homoclinic Connections ◽  
Limiting Case

We show that a planar slow–fast piecewise-linear (PWL) system with three zones admits limit cycles that share a lot of similarity with van der Pol canards, in particular an explosive growth. Using phase-space compactification, we show that these quasi-canard cycles are strongly related to a bifurcation at infinity. Furthermore, we investigate a limiting case in which we show the existence of a continuum of canard homoclinic connections that coexist for a single-parameter value and with amplitude ranging from an order of ε to an order of 1, a phenomenon truly associated with the non-smooth character of this system and which we call super-explosion .

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