Limit cycles for a class of second order differential equations

Llibre, Jaume; Pérez‐Chavela, Ernesto

doi:10.1016/j.physleta.2011.01.011

Cited by 8 publications

(5 citation statements)

References 10 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The first works in determining the number of limit cycles of a given vector field can be traced back to Liénard [9] and Andronov [1]. After these works, the detection of the number of limit cycles of a polynomial differential system, intrinsically related with the so-called 16th Hilbert problem [7], has been extensively studied in the mathematical community, see for instance the books [3,14] and the papers [5,6,10,11].…”

Section: Introduction and Statement Of The Main Resultsmentioning

confidence: 99%

On the Bifurcation of Limit Cycles Due to Polynomial Perturbations of Hamiltonian Centers

2017

Self Cite

View full text Add to dashboard Cite

show abstract

Section: Introduction and Statement Of The Main Resultsmentioning

confidence: 99%

On the Bifurcation of Limit Cycles Due to Polynomial Perturbations of Hamiltonian Centers

2017

Self Cite

View full text Add to dashboard Cite

show abstract

“…Many applications of ordinary differential equations (ODEs) of different orders can be found in the mathematical modeling of real-life problems. Second-and third-order DEs can be found in Refs [11][12][13][14], and fourth-order DEs often arise in many fields of applied science such as mechanics, quantum chemistry, electronic and control engineering and also beam theory [15], fluid dynamics [16,17], ship dynamics [18] and neural networks [19]. Numerically and analytically numerous approximations to solve such DEs of various orders have is studied in the literature.…”

Section: Introductionmentioning

confidence: 99%

Periodic solutions for a class of perturbed sixth-order autonomous differential equations

Berrehail

Makhlouf

2022

AJMS

View full text Add to dashboard Cite

PurposeThe objective of this work is to study the periodic solutions for a class of sixth-order autonomous ordinary differential equations x(6)+(1+p2+q2)x… .+(p2+q2+p2q2)x¨+p2q2x=εF(x,ẋ,x¨,x…,x… .,x(5)), where p and q are rational numbers different from 1, 0, −1 and p ≠ q, ε is a small enough parameter and F ∈ C2 is a nonlinear autonomous function.Design/methodology/approachThe authors shall use the averaging theory to study the periodic solutions for a class of perturbed sixth-order autonomous differential equations (DEs). The averaging theory is a classical tool for the study of the dynamics of nonlinear differential systems with periodic forcing. The averaging theory has a long history that begins with the classical work of Lagrange and Laplace. The averaging theory is used to the study of periodic solutions for second and higher order DEs.FindingsAll the main results for the periodic solutions for a class of perturbed sixth-order autonomous DEs are presenting in the Theorem 1. The authors present some applications to illustrate the main results.Originality/valueThe authors studied Equation 1 which depends explicitly on the independent variable t. Here, the authors studied the autonomous case using a different approach.

show abstract

“…where C is a real constant usually taken as positive, and the dot stays for the time derivative,ẋ = dx/dt of the x-coordinate, is a ubiquitous nonlinear non-autonomous ordinary differential equation with many applications, in particular in problems related to the timedependent harmonic oscillator or in connection with exact solutions of the one-dimensional time-independent Schrödinger equation. In more generality, applications of the EMP equation appear in cosmological models [4][5][6], Bose-Einstein condensates [7,8], photonic lattices [9], accelerator dynamics [10,11], gravitational wave propagation [12], higher order spin models [13], quantum plasmas [14], limit cycles [15], dynamical symmetries [16], magneto-gasdynamics [17], time-dependent non-Hermitian quantum system [18,19], supersymmetric systems [20], noncommutative quantum mechanics [21], etc. Historical notes can be found e.g., in [22,23].…”

Section: Introductionmentioning

confidence: 99%

Relativistic Ermakov–Milne–Pinney Systems and First Integrals

Haas

2021

Physics

View full text Add to dashboard Cite

The Ermakov–Milne–Pinney equation is ubiquitous in many areas of physics that have an explicit time-dependence, including quantum systems with time-dependent Hamiltonian, cosmology, time-dependent harmonic oscillators, accelerator dynamics, etc. The Eliezer and Gray physical interpretation of the Ermakov–Lewis invariant is applied as a guiding principle for the derivation of the special relativistic analog of the Ermakov–Milne–Pinney equation and associated first integral. The special relativistic extension of the Ray–Reid system and invariant is obtained. General properties of the relativistic Ermakov–Milne–Pinney are analyzed. The conservative case of the relativistic Ermakov–Milne–Pinney equation is described in terms of a pseudo-potential, reducing the problem to an effective Newtonian form. The non-relativistic limit is considered to be well. A relativistic nonlinear superposition law for relativistic Ermakov systems is identified. The generalized Ermakov–Milne–Pinney equation has additional nonlinearities, due to the relativistic effects.

show abstract

Limit cycles for a class of second order differential equations

Cited by 8 publications

References 10 publications

On the Bifurcation of Limit Cycles Due to Polynomial Perturbations of Hamiltonian Centers

On the Bifurcation of Limit Cycles Due to Polynomial Perturbations of Hamiltonian Centers

Periodic solutions for a class of perturbed sixth-order autonomous differential equations

Relativistic Ermakov–Milne–Pinney Systems and First Integrals

Contact Info

Product

Resources

About