Crossing Periodic Orbits via First Integrals

Llibre, Jaume; Tonon, Durval J.; Velter, Mariana Queiroz

doi:10.1142/s0218127420501631

Cited by 5 publications

(4 citation statements)

References 6 publications

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“…The limit cycles of a piecewise differential system are a highly challenging problem to analyze. The authors in [10], [11] studied the limit cycles of the piecewise differential systems, linear or nonlinear, using the first integrals. On the other hand, the authors in [12] studied the limit cycles bifurcating from a zero-Hopf equilibrium point using the averaging theory for Lipschitz differential systems.…”

Section: Introduction and The Main Resultsmentioning

confidence: 99%

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Limit cycles and Integrability of a continuous system with a line of equilibrium points

Abdulkareem

Amen

Hussein

2023

Preprint

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We focus on a chaotic differential system in 3-dimension, including an absolute term and a line of equilibrium points. Which describes in the following ẋ = y , ẏ = −ax + yz , ż = by − cxy − x2. This system has an implementation in electronic components. The first purpose of this paper is to provide sufficient conditions for the existence of a limit cycle bifurcating from the zero-Hopf equilibrium point located at the origin of the coordinates. The second aim is to study the integrability of each differential system, one defined in half–space y ≥ 0 and the other in half–space y < 0. We prove that these two systems have no polynomial, rational, or Darboux first integrals for any value of a, b, and c. Furthermore, we provide a formal series and an analytic first integral of these systems. We also classify Darboux polynomials and exponential factors.

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Section: Introduction and The Main Resultsmentioning

confidence: 99%

“…Also the cylindrical coordinates (r, θ, w) are defined as u = r cos(θ) and v = r sin(θ). System (10) becomes…”

Section: Proof Of Theoremmentioning

confidence: 99%

Limit cycles and Integrability of a continuous system with a line of equilibrium points

Abdulkareem

Amen

Hussein

2023

Preprint

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“…In the literature, one of the tools studied for this purpose is the Poincaré map, more precisely finding the fixed points; some of the works in this context are: [2], [8], and [15]. In the papers [16], and [17], the authors use the first integral to study the existence of crossing periodic orbits. Finally, the well-known theory of averaging was also used, for example, in the work [14].…”

Section: Introductionmentioning

confidence: 99%

Existence of periodic orbits for piecewise-smooth vector fields with sliding region via Conley theory

Romero,

Vieira

2021

Preprint

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The Conley theory has a tool to guarantee the existence of periodic trajectories in isolating neighborhoods of semi-dynamical systems; this tool is the main result of [18]. We prove that the positive trajectories generated by a piecewise-smooth vector field Z = (X, Y ) defined in a closed manifold of dimension three without the scape region produce a semi-dynamical system. Thus, supported by [18], we have a mechanism to warrant the existence of periodic orbits in this class of piecewise-smooth vector fields.

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“…For cases in R 2 , there are many works that determine the maximum number of limit cycles for a given class of vector fields and different separation manifolds (see [5], [18], [6]). There are also studies of piecewise smooth differential systems in R 3 (see [17], [12]), for which the discontinuity manifold is a plane.…”

Section: Introductionmentioning

confidence: 99%

Limit Cycles in Discontinuous Generalized Liénard Differential Equations

Diab

2020

Preprint

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In this paper, we study the maximum number of limit cycles for the piecewise smooth system of differential equations). Using the averaging method, we were able to generalize a previous result for Liénard systems. In our generalization, we consider g as a polynomial of degree m. We conclude that for sufficiently small values of |ϵ|, the number n 2 + m 2 + 1 serves as a lower bound for the maximum number of limit cycles in this system, which bifurcates from the periodic orbits of the linear center ẋ = y, ẏ = −x. Furthermore, we demonstrate that it is indeed possible to achieve such a number of limit cycles.

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Crossing Periodic Orbits via First Integrals

Cited by 5 publications

References 6 publications

Limit cycles and Integrability of a continuous system with a line of equilibrium points

Limit cycles and Integrability of a continuous system with a line of equilibrium points

Existence of periodic orbits for piecewise-smooth vector fields with sliding region via Conley theory

Limit Cycles in Discontinuous Generalized Liénard Differential Equations

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