The integrability and the zero-Hopf bifurcation of the three dimensional Lotka-Volterra systems

Salih, Rizgar

doi:10.1063/1.5020487

Cited by 3 publications

(3 citation statements)

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“…The Hopf bifurcation is a common phenomenon associated with the appearance or disappearance of a limit cycle near an equilibrium point. It can be investigated using various techniques, such as bifurcation formulas [8], Lyapunov quantities [23], and focus quantities [26]. In this subsection, we examine the cyclicity of the system (2.1) by employing Lyapunov quantities techniques to determine the number of limit cycles that can bifurcate from the Hopf points.…”

Section: Multiple Hopf Bifurcationmentioning

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: Multiple Hopf Bifurcationmentioning

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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“…There are many articles that investigate the number of limit cycles using various methods. For instance, the focus quantities method [2,3], the Lyapunov quantities [4,5] and the averaging method [6,7] are employed for this purpose.…”

Section: Introductionmentioning

confidence: 99%

Bifurcation analysis with chaotic attractor for a special case of jerk system

Rasul,

Salih

2024

Phys. Scr.

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This paper is focused on investigating the local bifurcations of a special type of chaotic jerk system. The occurrence and non-occurrence of saddle-node, transcritical and pitchfork bifurcations at the origin, as well as the Hopf and Zero-Hopf bifurcations when n = 3, are examined. For the proposed system, the parameters that lead to a zero-Hopf equilibrium point at the origin are characterized. Additionally, it has been demonstrated that employing the ﬁrst-order averaging theory results in the emergence of a single periodic solution branching out from the zero-Hopf equi-librium located at the origin. Furthermore, the focus quantities method is utilized to explore the periodicity of the system. This method helps to determine the number of periodic solutions that can emerge from the Hopf point. Due to the computational load for computing singular quantities, only three singular quantities were found. Under speciﬁc conditions, it is shown that three periodic solutions can emerge from the origin of the system. Finally, the chaotic attractors of the system are also studied.

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“…In 2018, Li et al considered the existence of zero-Hopf bifurcation and periodic solutions for the improved Chua system by applying the averaging theory [16]. In 2018, Salih studied the zero-Hopf bifurcation of the three-dimensional Lotka-Volterra systems [17]. In 2018, Candido et al studied the zero-Hopf bifurcation of 16 three-dimensional differential systems without equilibrium by using the averaging theory [18].…”

Section: Introductionmentioning

confidence: 99%