Bifurcation equations of continuous piecewise-linear vector fields

In this paper we present an n-dimensional canonical piecewise-linear electrical circuit. It contains 2n two-terminal elements: n linear dynamic elements (capacitors and inductors), n - 1 linear resistors and one nonlinear (piecewise-linear) resistor. This circuit can realize any prescribed eigenvalue pattern, except for a set of measure zero, associated with (i) any n-dimensional two-region continuous piecewise-linear vector fields and (ii) any n-dimensional three-region symmetric (with respect to the origin) piecewise-linear continuous vector fields. We also proved a theorem that specifies the conditions for a vector field, realized with our canonical circuit, to have two or three equilibrium points.

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Limit Cycles Bifurcating from a Period Annulus in Continuous Piecewise Linear Differential Systems with Three Zones

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127417500225 ◽

2017 ◽

Vol 27 (02) ◽

pp. 1750022 ◽

Cited By ~ 2

Author(s):

Maurício Firmino Silva Lima ◽

Claudio Pessoa ◽

Weber F. Pereira

Keyword(s):

Limit Cycles ◽

Vector Fields ◽

Poincaré Map ◽

Piecewise Linear ◽

Differential Systems ◽

Linear Vector ◽

Period Annulus ◽

Linear Differential Systems ◽

Linear Differential ◽

Two Parameter

We study a class of planar continuous piecewise linear vector fields with three zones. Using the Poincaré map and some techniques for proving the existence of limit cycles for smooth differential systems, we prove that this class admits at least two limit cycles that appear by perturbations of a period annulus. Moreover, we describe the bifurcation of the limit cycles for this class through two examples of two-parameter families of piecewise linear vector fields with three zones.

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LIMIT CYCLES FOR A CLASS OF CONTINUOUS PIECEWISE LINEAR DIFFERENTIAL SYSTEMS WITH THREE ZONES

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127412501386 ◽

2012 ◽

Vol 22 (06) ◽

pp. 1250138 ◽

Cited By ~ 3

Author(s):

MAURÍCIO FIRMINO SILVA LIMA ◽

JAUME LLIBRE

Keyword(s):

Limit Cycle ◽

Limit Cycles ◽

Vector Fields ◽

Poincaré Map ◽

Piecewise Linear ◽

Differential Systems ◽

Linear Vector ◽

Linear Differential Systems ◽

Linear Differential ◽

Unique Limit

In this paper, we consider a class of planar continuous piecewise linear vector fields with three zones. Using the Poincaré map, we show that these systems admit always a unique limit cycle, which is hyperbolic.

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BIFURCATION SETS OF SYMMETRICAL CONTINUOUS PIECEWISE LINEAR SYSTEMS WITH THREE ZONES

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127402005509 ◽

2002 ◽

Vol 12 (08) ◽

pp. 1675-1702 ◽

Cited By ~ 21

Author(s):

EMILIO FREIRE ◽

ENRIQUE PONCE ◽

FRANCISCO RODRIGO ◽

FRANCISCO TORRES

Keyword(s):

Linear Systems ◽

Canonical Form ◽

Vector Fields ◽

Piecewise Linear ◽

Oscillatory Behavior ◽

Van Der Pol ◽

Linear Vector ◽

Piecewise Linear Systems

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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