Theory of Limit Cycles

Yan-qian, YE; Lo, Chi

doi:10.1090/mmono/066

Cited by 90 publications

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“…Its behavior runs from near-harmonic oscillations when ǫ → 0 to relaxation oscillations when ǫ → ∞, making it a good model for many practical situations [3]. Other partial results on the number and form of limit cycles in Liénard systems are scattered in the literature [4]. When f (x) is a polynomial of degree N = 2n + 1 or 2n, with n a natural number, Lins, Melo and Pugh have conjectured (LMP-conjecture) that the maximum number of limit cycles allowed is just n [5].…”

Section: Introductionmentioning

confidence: 99%

The homotopy analysis method and the Liénard equation

Abbasbandy

López

López‐Ruiz

2011

International Journal of Computer Mathematics

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Section: Introductionmentioning

confidence: 99%

The homotopy analysis method and the Liénard equation

Abbasbandy

López

López‐Ruiz

2011

International Journal of Computer Mathematics

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“…They are important subjects in the bifurcation theory and also in the study of the Hilbert's 16th problem [6,7,15,17].…”

Section: Background and Statement Of The Main Resultsmentioning

confidence: 99%

Inverse Jacobian multipliers and Hopf bifurcation on center manifolds

Zhang

2014

Journal of Differential Equations

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Abstract. In this paper we consider a class of higher dimensional differential systems in R n which have a two dimensional center manifold at the origin with a pair of pure imaginary eigenvalues. First we characterize the existence of either analytic or C ∞ inverse Jacobian multipliers of the systems around the origin, which is either a center or a focus on the center manifold. Later we study the cyclicity of the system at the origin through Hopf bifurcation by using the vanishing multiplicity of the inverse Jacobian multiplier. Background and statement of the main resultsFor real planar differential systems, the problems on center-focus and Hopf bifurcation are classical and related. They are important subjects in the bifurcation theory and also in the study of the Hilbert's 16th problem [6,7,15,17].For planar non-degenerate center, Poincaré provided an equivalent characterization. Poincaré center Theorem. For a real planar analytic differential system with the origin as a singularity having a pair of pure imaginary eigenvalues, then the origin is a center if and only if the system has a local analytic first integral, and if and only if the system is analytically equivalent tȯwith g(uv) without constant terms, where we have used the conjugate complex coordinates instead of the two real ones.This result has a higher dimensional version, see for instance [13,18,20], which characterizes the equivalence between the analytic integrability and the existence of analytic normalization of analytic differential systems to its Poincaré-Dulac normal form of a special type.Reeb [14] in 1952 provided another characterization on planar centers via inverse integrating factor. Recall that a function V is an inverse integrating factor of a planar differential system if 1/V is an integrating factor of the

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“…To do this, we use Dulac theorem [10] to rule out the existence of limit cycles and homoclinic loops. Proposition 2.10.…”

Section: Global Dynamicsmentioning

confidence: 99%

Bifurcation analysis and global dynamics of a mathematical model of antibiotic resistance in hospitals

et al. 2017

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Antibiotic-resistant bacteria has posed a grave threat to public health by causing a number of nosocomial infections in hospitals. Mathematical models have been used to study the transmission of antibiotic-resistant bacteria within a hospital and the measures to control antibiotic resistance in nosocomial pathogens. Studies presented in [4, 5] have shown great value in understanding the transmission of antibiotic-resistant bacteria in a hospital. However, their results are limited to numerical simulations of a few different scenarios without analytical analysis of the models in all biologically feasible parameter regions. Bifurcation analysis and identification of the global stability conditions are necessary to assess the interventions which are proposed to limit nosocomial infection and stem the spread of antibiotic-resistant bacteria. In this paper we study the global dynamics of the mathematical model of antibiotic resistance in hospitals in [4, 5]. The invasion reproduction number R ar of antibiotic-resistant bacteria is introduced. We give the relationship of R ar and two control reproduction numbers of sensitive bacteria and resistant bacteria (R sc and R rc ). More importantly, we prove that a backward bifurcation may occur at R ar = 1 when the model includes superinfection which is not mentioned in [5]. That is, there exists a new threshold R c ar , and if R c ar < R ar < 1, then the system can have two interior equilibria and it supports an interesting bistable phenomenon. This provides critical information on controlling the antibiotic-resistance in a hospital.

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Theory of Limit Cycles

Cited by 90 publications

References 0 publications

The homotopy analysis method and the Liénard equation

The homotopy analysis method and the Liénard equation

Inverse Jacobian multipliers and Hopf bifurcation on center manifolds

Bifurcation analysis and global dynamics of a mathematical model of antibiotic resistance in hospitals

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