BIFURCATION SETS OF SYMMETRICAL CONTINUOUS PIECEWISE LINEAR SYSTEMS WITH THREE ZONES

Planar symmetrical continuous piecewise linear vector fields with three zones are thoroughly analyzed. Special emphasis is placed on their oscillatory behavior. The study is made by using a Van der Pol canonical form which captures the most interesting dynamics and minimizes the number of parameters to be dealt with. This work is a continuation of a previous paper [Freire et al., 1998] and uses the same approach and techniques.

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Bifurcation Sets of Continuous Piecewise Linear Systems with Two Zones

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127498001728 ◽

1998 ◽

Vol 08 (11) ◽

pp. 2073-2097 ◽

Cited By ~ 128

Author(s):

Emilio Freire ◽

Enrique Ponce ◽

Francisco Rodrigo ◽

Francisco Torres

Keyword(s):

Linear Systems ◽

Periodic Orbits ◽

Canonical Form ◽

Vector Fields ◽

Piecewise Linear ◽

Oscillatory Behavior ◽

Linear Vector ◽

Piecewise Linear Systems ◽

The Creation

Planar continuous piecewise linear vector fields with two zones are considered. A canonical form which captures the most interesting oscillatory behavior is obtained and their bifurcation sets are drawn. Different mechanisms for the creation of periodic orbits are detected, and their main characteristics are emphasized.

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

Proceedings of The Royal Society A Mathematical Physical and Engineering Sciences ◽

10.1098/rspa.2012.0603 ◽

2013 ◽

Vol 469 (2154) ◽

pp. 20120603 ◽

Cited By ~ 20

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

Volume 7A: 17th Biennial Conference on Mechanical Vibration and Noise ◽

10.1115/detc99/vib-8037 ◽

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.

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Hopf bifurcation at infinity in 3D symmetric piecewise linear systems. Application to a Bonhoeffer–van der Pol oscillator

Nonlinear Analysis Real World Applications ◽

10.1016/j.nonrwa.2020.103112 ◽

2020 ◽

Vol 54 ◽

pp. 103112 ◽

Cited By ~ 1

Author(s):

E. Freire ◽

E. Ponce ◽

J. Ros ◽

E. Vela ◽

A. Amador

Keyword(s):

Hopf Bifurcation ◽

Linear Systems ◽

Piecewise Linear ◽

Van Der Pol Oscillator ◽

Van Der Pol ◽

Piecewise Linear Systems

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Analysis of Piecewise Linear Systems With Boundaries in Displacement and Velocity Using Vector Fields

Volume 6B: 18th Biennial Conference on Mechanical Vibration and Noise ◽

10.1115/detc2001/vib-21459 ◽

2001 ◽

Author(s):

Mikio Nakai ◽

Seiji Hagio ◽

Kazuki Mizutani ◽

Hossain Md. Zahid

Keyword(s):

Linear Systems ◽

Present Method ◽

Vector Fields ◽

Piecewise Linear ◽

Generalized Solution ◽

Dynamic Responses ◽

Spring Stiffness ◽

Piecewise Linear Systems ◽

Solution Methodology ◽

Machinery System

Abstract A generalized solution methodology based on piecewise linear vector fields has been proposed for piecewise linear systems with boundaries for both displacement and time. In this paper, this methodology is applied in analyzing dynamic responses of torsional vibration through bilinear spring stiffness with changing hysteretic torque in the rotating machinery system, which has boundaries for both displacement and velocity. For asymmetric torque with hysteresis, two cases exist when restoring torque changes smoothly and discontinuously. Period-one and period-two solutions are sought in two cases and compared with ones obtained by the numerical method. Our present method can be applied to dynamic responses of piecewise linear systems having velocity and displacement boundaries.

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An Analytical Method for Piecewise Linear Systems by Vector Fields.

TRANSACTIONS OF THE JAPAN SOCIETY OF MECHANICAL ENGINEERS Series C ◽

10.1299/kikaic.65.4618 ◽

1999 ◽

Vol 65 (640) ◽

pp. 4618-4625

Author(s):

Mikio NAKAI ◽

Hiroki ETOH ◽

Shinji MURATA

Keyword(s):

Linear Systems ◽

Analytical Method ◽

Vector Fields ◽

Piecewise Linear ◽

Piecewise Linear Systems

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Analysis of Piecewise Linear Systems With Boundaries in Displacement and Time

Journal of Vibration and Acoustics ◽

10.1115/1.1502668 ◽

2002 ◽

Vol 124 (4) ◽

pp. 527-536 ◽

Cited By ~ 2

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.

Download Full-text