# Limit Cycles of the Three-dimensional Quadratic Differential System via Hopf Bifurcation

## Authors

• Aram A. Abddulkareem Abddulkareem Department of Mathematics, Faculty of Education, Soran University, Soran, Iraq. https://orcid.org/0009-0001-0063-2281
• Azad I. Amen Department of Mathematics, College of Basic Education, Salahaddin University-Erbil, Erbil, Iraq. &Department of Mathematics, Basic Education College, Raparin University, Ranya, Iraq. &Department of Mathematics, College of Science, Duhok University, Duhok, Iraq. https://orcid.org/0000-0002-9310-6474
• Niazy H. Hussein Department of Mathematics, Faculty of Science, Soran University, Soran, Iraq. & Department of Mathematics, College of Education, Salahaddin University-Erbil, Erbil, Iraq. https://orcid.org/0000-0002-3526-4933

## Keywords:

Hopf bifurcation, Limit cycles, Lyapunov coefficient, n-scroll chaotic attractors, Quadratic 3-dimensional differential systems

## Abstract

In this study, the quadratic 3-dimensional differential system is considered, in which the origin of the coordinate becomes the Hopf equilibrium point. The existence and stability of limit cycles that emerge from the Hopf point are being investigated. The Lyapunov coefficients connected to the Hopf point are calculated using the projection method. First, four families of parameter conditions are driven by which the quadratic 3-dimensional differential system can exhibit codimension three of the Hopf bifurcation. The analytical proof of each parameter family of conditions is given by calculating the Lyapunov coefficients, the vanishing of the first, sconed Lyapunov, and the non-zero of the third Lyapunov coefficients. The explicit conditions are presented for the existence and stability of three limit cycles arising from each family of Hopf bifurcations. The output of existence displays a stable (unstable) Hopf point, accompanied by the emergence of two stable (unstable) limit cycles alongside one unstable (stable) limit cycle in the neighborhood of the unstable (stable) origin of the coordinated 3-dimensional quadratic system. In addition, the outcome is utilized for exploring the limit cycles of the n-scroll chaotic attractor system, which has many practical uses, including secure communication, encryption, random number generation, and autonomous mobile robots. The conditions are derived under which the origin point of this system becomes the Hopf point, and three limit cycles can exist around the Hopf point. Finally, the numerical demonstrations show that the system undergoes a supercritical Hopf bifurcation, resulting in two stable and one unstable limit cycle. Furthermore, all results are verified.

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