Number of Limit Cycles for Planar Systems with Invariant Algebraic Curves

Gasull, Armengol; Giacomini, Héctor

doi:10.1007/s12346-023-00746-7

Cited by 1 publication

(1 citation statement)

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“…The number of limit cycles that a particular differential system can have is one of the significant challenges in the qualitative theory of differential systems. The number of limit cycles in 𝑅 2 is finite 1-The analytical solution to the quadratic 3dimensional differential system is elusive 5 . However, several alternative approaches and areas of study can be explored: numerical methods 6 , computer simulations 7 , and qualitative analysis.…”

Section: Introductionmentioning

confidence: 99%

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

Abddulkareem,

Amen,

Hussein

2024

Baghdad Sci.J

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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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