Zero-Hopf Bifurcation in a Generalized Genesio Differential Equation

Diab, Zouhair; Guirao, Juan Luis García; Vera, Juan A.

doi:10.3390/math9040354

Cited by 3 publications

(2 citation statements)

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“…Braun and Mereu [2] obtained a zero-Hopf bifurcation in a chaotic jerk system. Diab et al [6] investigated zero-Hopf bifurcations of the generalized Genesio differential equation. In their work, they characterized the existence of a zero-Hopf equilibrium point and showed that at most 6 periodic solutions bifurcate from the origin of the system.…”

Section: Introductionmentioning

confidence: 99%

Bifurcation analysis with self-excited and hidden attractors for a chaotic jerk system

Rasul,

Salih

2024

J. Math. Computer Sci.

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This paper is devoted to investigating the local bifurcation of a chaotic jerk system. The local stability of equilibrium points is analyzed, as well as the existence of transcritical bifurcation at the origin. For the proposed jerk system, the Hopf and Zero-Hopf bifurcations are investigated at the origin. Moreover, a zero-Hopf equilibrium point at the origin is characterized for the system. By using the averaging theory of first order, a limit cycle can be bifurcated from the zero-Hopf equilibrium located at the origin. Liapunov quantities techniques are used to investigate the cyclicity of the system. It is shown that three limit cycles can be bifurcated from the origin. Finally, both self-excited chaotic attractors and hidden chaotic attractors are studied for special cases of the chaotic jerk systems using bifurcation diagrams, Lyapunov exponents, and cross-sections. All results reported in this study have been obtained using Maple software.

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Section: Introductionmentioning

confidence: 99%

Bifurcation analysis with self-excited and hidden attractors for a chaotic jerk system

Rasul,

Salih

2024

J. Math. Computer Sci.

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“…Otherwise, Appendix: Averaging theory of first order for limit cycles Now we'll go through the basic averaging theory for Lipschitz differential systems which we'll need to prove result of isolated limit cycle bifurcate from zero-Hopf point. The following theorem offers a first-order of the averaging theory for differential system which founded in [25], [26] and used in [12], [27], and [28]. For more information and the proof see previous references.…”

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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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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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