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Bifurcations in continuous piecewise linear differential systems : applications to low-dimensional electronic oscillators / Enrique Ponce, Javier Ros, Elísabet Vela.
Ponce, Enrique, author.; Ros, Javier, author.; Vela, Elísabet, author.
2022
QA372
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Title
Bifurcations in continuous piecewise linear differential systems : applications to low-dimensional electronic oscillators / Enrique Ponce, Javier Ros, Elísabet Vela.
Author
Ponce, Enrique, author.
ISBN
9783031211355 (electronic bk.)
3031211359 (electronic bk.)
9783031211348
3031211340
DOI
https://doi.org/10.1007/978-3-031-21135-5
Published
Cham : Springer, [2022]
Copyright
©2022
Language
English
Description
1 online resource (xv, 311 pages).
Item Number
10.1007/978-3-031-21135-5 doi
Call Number
QA372
Dewey Decimal Classification
515/.354
Summary
The book is devoted to the qualitative study of differential equations defined by piecewise linear (PWL) vector fields, mainly continuous, and presenting two or three regions of linearity. The study focuses on the more common bifurcations that PWL differential systems can undergo, with emphasis on those leading to limit cycles. Similarities and differences with respect to their smooth counterparts are considered and highlighted. Regarding the dimensionality of the addressed problems, some general results in arbitrary dimensions are included. The manuscript mainly addresses specific aspects in PWL differential systems of dimensions 2 and 3, which are sufficinet for the analysis of basic electronic oscillators. The work is divided into three parts. The first part motivates the study of PWL differential systems as the natural next step towards dynamic complexity when starting from linear differential systems. The nomenclature and some general results for PWL systems in arbitrary dimensions are introduced. In particular, a minimal representation of PWL systems, called canonical form, is presented, as well as the closing equations, which are fundamental tools for the subsequent study of periodic orbits. The second part contains some results on PWL systems in dimension 2, both continuous and discontinuous, and both with two or three regions of linearity. In particular, the focus-center-limit cycle bifurcation and the Hopf-like bifurcation are completely described. The results obtained are then applied to the study of different electronic devices. In the third part, several results on PWL differential systems in dimension 3 are presented. In particular, the focus-center-limit cycle bifurcation is studied in systems with two and three linear regions, in the latter case with symmetry. Finally, the piecewise linear version of the Hopf-pitchfork bifurcation is introduced. The analysis also includes the study of degenerate situations. Again, the above results are applied to the study of different electronic oscillators.
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Includes bibliographical references.
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Access limited to authorized users.
Source of Description
Online resource; title from PDF title page (SpringerLink, viewed January 3, 2023).
Added Author
Ros, Javier, author.
Vela, Elísabet, author.
Series
RSME Springer series ; volume 7. 2509-8896
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Bifurcations in Continuous Piecewise Linear Differential Systems
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1.7 Limit Cycles in a Continuous Piecewise Linear Worked Example
1.7.1 The Right Half-Return Map
1.7.2 The Left Half-Return Map
1.7.3 A Bifurcation Analysis
2 Preliminary Results
2.1 A Unified Liénard Form for Continuous Planar Piecewise Linear Systems
2.2 Observable Piecewise Linear Luré Systems
2.3 Some Generic Results About Equilibria
2.3.1 Observable Continuous Piecewise Linear Systems with Two Zones
2.3.2 Observable Symmetric Continuous Piecewise Linear Systems with Three Zones
2.4 Analysis of Periodic Orbits Through Their Closing Equations

2.4.1 Closing Equations for Observable Continuous Piecewise Linear Systems with Two Zones
2.4.2 Closing Equations for Symmetric Continuous Piecewise Linear Systems with Three Zones
2.5 Periodic Orbits and Poincaré Maps in Piecewise Linear Systems
2.5.1 Derivatives of Transition Maps
2.5.2 Poincaré Maps in Continuous Piecewise Linear Systems with Two Zones
2.5.3 Poincaré Maps in Continuous Piecewise Linear Systems with Three Zones
Part II Planar Piecewise Linear Differential Systems
3 Analysis of Planar Continuous Systems with Two Zones

3.1 Equilibria in Continuous Planar Piecewise Linear Systems with Two Zones
3.2 Some Preliminary Results on Limit Cycles
3.3 The Massera's Method for Uniqueness of Limit Cycles
3.4 General Results About Limit Cycles
3.5 Refracting Systems
3.6 The Bizonal Focus-Center-Limit Cycle Bifurcation
4 First Results for Planar Continuous Systems with Three Zones
4.1 Limit Cycle Existence and Uniqueness
4.2 The Focus-Center-Limit Cycle Bifurcation for Symmetric Continuous Planar Piecewise Linear Systems with Three Zones
5 Boundary Equilibrium Bifurcations and Limit Cycles

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