Bifurcation of limit cycles from quadratic isochrones
Carmen Chicone,Marc Q. Jacobs +1 more
TLDR
In this paper, the number and position of the local families of limit cycles which emerge from the periodic trajectories surrounding an isochronous (or linearizable) center were determined for a one parameter family of plane quadratic vector fields X (.,e) depending analytically on a small real parameter e.About:
This article is published in Journal of Differential Equations.The article was published on 1991-06-01 and is currently open access. It has received 140 citations till now. The article focuses on the topics: Limit (mathematics).read more
Citations
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Ordinary Differential Equations with Applications
TL;DR: In this paper, the authors introduce the notion of Ordinary Differential Equations (ODE) for linear systems and the stability of nonlinear systems and apply it to Hyperbolic Theory.
Journal ArticleDOI
Hilbert's 16th problem and bifurcations of planar polynomial vector fields
TL;DR: The progress of study on Hilbert's 16th problem is presented, and the relationship between Hilbert's 15th problem and bifurcations of planar vector fields is discussed.
Journal ArticleDOI
Averaging methods for finding periodic orbits via Brouwer degree
Adriana Buică,Jaume Llibre +1 more
TL;DR: In this article, the problem of finding T -periodic solutions for a differential system whose vector field depend on a small parameter e.g., the displacement function is considered and an answer to this problem can be given using the averaging method.
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On the nonexistence, existence and uniqueness of limit cycles
TL;DR: In this paper, two new criteria for studying the nonexistence, existence and uniqueness of limit cycles of planar vector fields are presented, and a method to derive the shape of the bifurcated limit cycles from a centre is presented.
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Invariant algebraic curves and conditions for a centre
TL;DR: In this article, conditions for the existence of a centre in two-dimensional systems are considered along the lines of Darboux, and they are used in the search for maximal numbers of bifurcating limit cycles.
References
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Partial Differential Equations
TL;DR: In this paper, the authors present a theory for linear PDEs: Sobolev spaces Second-order elliptic equations Linear evolution equations, Hamilton-Jacobi equations and systems of conservation laws.
Book
Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields
TL;DR: In this article, the authors introduce differential equations and dynamical systems, including hyperbolic sets, Sympolic Dynamics, and Strange Attractors, and global bifurcations.
A Reflection on Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields
J. Guckenheimer,P. J. Holmes +1 more
TL;DR: In this paper, the authors introduce differential equations and dynamical systems, including hyperbolic sets, Sympolic Dynamics, and Strange Attractors, and global bifurcations.
Book
Geometrical Methods in the Theory of Ordinary Differential Equations
TL;DR: In the first edition of this book, geometrical methods in the theory of ordinary differential equations have become very popular and some progress has Since the author explains basic ideas free from a number of 2nd order odes as discussed by the authors.
Related Papers (5)
Bifurcation of limit cycles from centers and separatrix cycles of planar analytic systems
T. R. Blows,L. M. Perko +1 more