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

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Introduction
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  2 is finite 1- Published Online First: February, 2024 https://doi.org/10.21123/bsj.2024.9306P-ISSN: 2078-8665 -E-ISSN: 2411-7986 Baghdad Science Journal an unlimited number of limit cycles 4 .The dynamic richness of quadratic 3-dimensional differential systems makes them incredibly valuable in various fields, including cryptography, image encryption, non-linear control systems, signal processing, and time series prediction.The Liu system, the Lorenz system, the Rossler system, the Chen system, and the Rikitake system are all examples of quadratic systems in  3 .
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.It can gain insights into the system's behavior through qualitative analysis.The study involves analyzing the qualitative properties of solutions, such as equilibrium point stability and limit cycle existence.
For this purpose, that is, the search for analytical results that give information about the existence and location of limit cycles, this quadratic threedimensional first-order differential system will be studied.
Various methodologies, including the classical projection method, computation of singular point quantities, and averaging methods, investigate limit cycles emanating from Hopf points.Some researchers have undertaken analytical studies of quadratic differential systems.They use the projection method to calculate Lyapunov coefficients related to Hopf points, focusing on specific systems with two or three limit cycles and analyzing their stability 8,9 .Others have explored limit cycles via singular quantities (focal values), providing a method to identify more limit cycles without addressing their stability 10 .The Averaging method represents another approach to analytically studying limit cycles, although it presents challenges in calculating high zeros in equations 11 .In contrast, some researchers opt for a numerical exploration of limit cycles, leveraging this strategy to uncover more limit cycles 11,12 .
In this note, the projection method is used to study and analyze the limit cycles related to the Hopf point.This method allowed us to calculate the first three Lyapunov coefficients, and in the neighborhood of Hopf point, granted the existence of three limit cycles.Our result deals with general differential systems of degree 2, which in this form were not considered before.The four families of parameter conditions drive the existence of three limit cycles.
Our system and conditions differ from particular systems in previous studies [8][9][10][11][12] .Moreover, the stability of limit cycles is investigated.All results in this study were verified and numerically studied by the Maplesoft program.
The present research is structured in the following manner.The concept of Hopf bifurcation is elucidated by using the projection technique.The definition and Lyapunov coefficients have been given in this paper.The main result of our investigation has been documented and verified.In the subsequent part, The quadratic differential system that encompasses -scroll chaotic attractors is examined, serving as an illustrative illustration of the main outcome.The main result of our study was used to make estimations concerning the presence and stability of the three limit cycles associated with the origin Hopf point.The numerical simulations are carried out using precise parameter values to validate and demonstrate the analytical findings.

Review on Hopf bifurcation
This section provides a comprehensive overview of the projection method 13 .This method is used to compute the first and second Lyapunov coefficients related to Hopf bifurcations.This method was additionally employed to compute the third and fourth Lyapunov coefficients 14 .
Consider the following general differential equation, Suppose that  is a class of  ∞ ∈  3 ×   , where vector  ∈  3 denotes phase variables and the vector  ∈   represents control parameters.Suppose  =  0 is an equilibrium point of system 2 at  =  0 .
Moving the equilibrium  0 to the origin of the coordinates by the linear change of the variable  −  0 .Also, by , it expresses as | =0       , are two multilinear functions.Suppose that at equilibrium (0,  0 ),  has an eigenvalue  1 ≠ 0 with a pair of complex eigenvalues on the imaginary axis:  2,3 = ± 0 , ( 0 > 0), and the eigenvalues  2,3 are the only eigenvalues with () = 0. Assuming that   is the generalized eigenspace of  that relates to  2,3 .Let , and  be vectors in  3 in a way that where   is the transposed of the matrix .Any vector  ∈   could be stated as  =  +  ̅  ̅, in which  = 〈, 〉 ∈  and  ̅ is the conjugate of .
By using an immersion of the form  = (,  ̅), it is possible to parameterize the 2-dimensional center manifold related to the eigenvalues  2,3 by  and  ̅, where :  →  3 has a Taylor series of the form with ℎ  ∈  3 and ℎ  = ℎ ̅  .Substituting the above expression into Eq 2, the following differential equation was obtained in which  is expressed by Eq 3. By solving the system of linear equations described by the coefficients in Eq 5, the complex vectors ℎ  are produced, taking the coefficients  into consideration, so that Eq 5, on the chart  for a central manifold, writes as follows: The explicit expressions for the Lyapunov coefficients can be found in Kuznetsov's book and Sotomayor et al. research 13,14 .
A transversal Hopf point is a Hopf point of codimension one for which the complex eigenvalues, depending on the parameters, cross the imaginary axis with a non-zero derivative.In a neighborhood of a transversal Hopf point with  1 ≠ 0 the dynamic behavior of the system 2, reduced to the family of https://doi.org/10.21123/bsj.2024.9306P-ISSN: 2078-8665 -E-ISSN: 2411-7986 Baghdad Science Journal parameter-dependent continuations of the center manifold, is orbitally topologically equivalent to the following complex normal form  ′ = ( 0 +  0 ) +  1 || 2 , where  ∈ ;  0 ,  0 , and  1 are real functions having derivatives of arbitrary higher order.As  1 < 0 ( 1 > 0) one family of stable (unstable) limit cycles can be found on the center manifold and its continuation, shrinking to the Hopf point.
A Hopf point of codimension 2 is a Hopf point where  1 vanishes.It is called transversal if  0 = 0 and  1 = 0 have transversal intersections.In a neighborhood of a transversal Hopf point of codimension 2 with  2 ≠ 0 the dynamic behavior of the system 2, reduced to the family of parameterdependent continuations of the center manifold, is orbitally topologically equivalent to  ′ = ( 0 +  0 ) +  1 || 2 +  2 || 4 , where  0 and  1 are unfolding parameters.

Main results
First, take into consideration System 1 undergoes Hopf bifurcation at the origin (0,0,0) if the conditions in ℋ hold.When  = 0, the origin of system 1 is a Hopf point with two purely imaginary complex eigenvalues ± and nonzero eigenvalue  < 0. The classical projection method calculates the Lyapunov coefficients related to the Hopf point.This helps us ensure that, under certain parameter conditions, the system 1 has three limit cycles by examining its third Lyapunov coefficient.Now, consider the following families of parameters of the system 1: The following is the main result, which gives the Lyapunov coefficient for system 1: Theorem 1 ).

Proof of Theorem 1:
For parameters in ℋ, the eigenvalues of the Jacobian matrix of system 1 at the origin are  < 0, ± 0 , where  0 =  > 0. The eigenvectors  and  defined in Eq 4 are To verify the transversal condition of the Hopf bifurcation.Consider system 1, which is dependent on parameter . = () denotes the real part of the pair of complex eigenvalues at the critical parameter  =   = 0. Which satisfies where  denotes the Jacobian matrix of system 1 at the origin.Consequently, the transversal condition for the Hopf point holds.Moreover, each case of Theorem 1 is proved independently.
(a) Consider the parameters in the set  1 , The system's bilinear vector function is computed as follows.
Then, the complex vectors are given by ℎ 11 = (0, 0, 0), 2 )( 2 4 −  2 ), 0).Now, the first and second Lyapounv coefficients vanish. 1 ( 1 ) = 0,  2 ( 1 ) = 0, respectively.With more calculations, it obtained that The complex number  43 is given by According to the above computations and the analysis, the third Lyapunov coefficient  3 ( 1 ) as follows: If  3 ( 1 ) ≠ 0, then, at the origin, system 1 possesses a transversal Hopf point with codimension three.The sign of the numerator in Eq 12 determines the sign of the third Lyapunov coefficient  3 ( 1 ).Therefore, if  11  12 > 0 then  3 ( 1 ) > 0, the Hopf point I s unstable (weak repelling focus) at the origin, and for each  <   , close to   , three limit cycles exist, two unstable and one stable near the origin equilibrium point.If  11  12 < 0 then  3 ( 1 ) < 0, the Hopf point is stable (weak attractor focus) at the origin, and for each  >   , close to   , three limit cycles exist, two stable and one unstable, near the origin equilibrium point.Theorem 1 case (a) has been successfully proved.In the same way, as described in case (a), the third Lyapunov coefficient is obtained as follows: In case  3 ( 2 ) ≠ 0, transversality of codimension three of a Hopf point at the origin for system 1 hold.
For  2 ≠ 0, the sign of the third Lyapunov coefficient Eq 13, is opposite to the sign of  3  21 .If  3  21 < 0 then  3 ( 2 ) > 0. The Hopf point is unstable and for  <   close to   , three limit cycles exist: two unstable and one stable near the origin equilibrium point.If  3  21 > 0, then  3 ( 2 ) < 0.
The Hopf point is stable, and for  >   close to   , three limit cycles exist: two stable and one unstable near the origin equilibrium point.
(c) Consider the parameter in the set  3 .The bilinear vector function is written as follows: In the same way, as described in case (a), the third Lyapunov coefficient is given by Since  < 0, the opposite of the sign of  4  31 determines the sign of the third Lyapunov coefficient, given by Eq 14.If  4  31 < 0, then  3 ( 3 ) > 0. The Hopf point is unstable and for  <   close to   , three limit cycles exist: one stable and two unstable near the origin equilibrium point.If  4  31 > 0, then  3 ( 3 ) < 0. The Hopf point is stable, and for  >   close to   , three limit cycles exist: one unstable and two stable, near the origin equilibrium point.
(d) Consider the parameter conditions in the set  4 .
The bilinear vector function is written as follows: In the same way, as described in case (a), the third Lyapunov coefficient obtained as follows: For  2 ≠ 0, the sign of the third Lyapunov coefficient, given by Eq 15, is determined by the sign of  6  41 .If  6  41 > 0, then  3 ( 4 ) > 0. The Hopf point is unstable and for  <   close to   , three limit cycles exist: one stable and two unstable near the origin equilibrium point.If  6  41 < 0, then https://doi.org/10.21123/bsj.2024.9306P-ISSN: 2078-8665 -E-ISSN: 2411-7986 Baghdad Science Journal  3 ( 4 ) < 0. The Hopf point is stable, and for  >   close to   , there exist three limit cycles: one unstable and two stable, near the origin equilibrium point .The proof of Theorem 1 is now finished.

Limit Cycles of 𝒏-Scroll Chaotic Attractor
This section presents a specific differential system as an example of Theorem 1.This theorem is used to examine and analyze both the Hopf bifurcation and the limit cycle of the system.Additionally, numerical simulations are provided to validate our findings.
The existence of differential systems capable of producing -scroll chaotic attractors is a significant open topic with a challenging solution 16 .-scroll chaotic attractors have been extensively studied throughout history 17 .Compared to double-scroll oscillators like Lorenz, they offer richer dynamics and a larger maximum Lyapunov exponent 18,19 .This type of system has many practical uses, such as secure communications 20 , encryption 21,22 , random number generating 23 , autonomous mobile robots 24 , and other technical fields.Now, consider the simplest family of systems with scroll chaotic attractor 25 .Which is described as follows: ̇=  It set 14 parameters, and tweaking one resulted in an uncommon three-scroll odd attractor 25 .This system is a subsystem of system 1.Under certain conditions, it has been shown that system 16 undergoes Hopf bifurcation and gives rise to three limit cycles emerging from the Hopf equilibrium point.The following theorem describes the third Lyapunov coefficient associated with a Hopf equilibrium point of the system 16.
It noted that when  = 0, the origin of the coordinate of the system 18 is a Hopf point with eigenvalues  and ± .Now, if  =  2 < 0, and  = √− 3  1 , where  3  1 < 0, then the coefficients of the system 18 satisfy the family of parameter conditions  1 .Thus, by Theorem 1 case (a), the first and second Lyapunov coefficients are vanish.From Eq 12, the third Lyapunov coefficient is given by Eq 17.When the conditions  1 or  2 are satisfied.It follows that,  3 > 0, then the origin is a weak repelling focus for the flow of the system 16 restricted to the attracting center manifold.Consequently, for  < 0, three limit cycles exist, one stable and two unstable, for appropriate values of the parameters.On the other hand, when the conditions  1 or  2 holds.It follows that,  3 < 0, then the origin is a weak attracting focus for the flow of the system 16 restricted to the attracting center manifold.Consequently, for  > 0, three limit cycles exist, one unstable and two stable, near the Hopf equilibrium for appropriate values of the parameters.Then, the third Lyapunov coefficient is  3 = −35.556< 0. The Hopf point, which is located at the origin, is asymptotically stable and for a suitable  > 0 close to   = 0, three limit cycles exist, two stable and one unstable.

If ℎ =
Using the MatCont 26 continuing numerical bifurcation program to comprehend system dynamics changes and their influence on parameters.

Results and Discussion
The present study delves into the dynamics of a quadratic 3-dimensional differential system, focusing on the Hopf equilibrium point at the coordinate origin.Our primary objective is to investigate the existence and stability of limit cycles stemming from this key Hopf point.To achieve this, the classical projection method is employed to calculate the Lyapunov coefficients associated with the Hopf point.
Our investigation uncovers four distinct families of parameter conditions, each leading to a codimensionthree Hopf bifurcation in the quadratic 3dimensional differential system.This exploration is significant for theoretical considerations and has practical implications.Notably, Our findings are leveraged to elucidate the intricacies of the n-scroll chaotic attractor system, a system with broad applications in secure communication and engineering.Precise conditions are established under which the origin point of the  -scroll chaotic attractor system serves as a Hopf point, allowing for the existence of three limit cycles around it.Numerical demonstrations are meticulously conducted to provide empirical support for our theoretical framework, and the robustness of our results is rigorously verified.This comprehensive approach contributes to a deeper understanding of the intricate dynamics within the quadratic 3dimensional differential system and its practical relevance in real-world applications.
Instead of adding a non-linear perturbation part, studying high-order Lyapunov coefficients for future work is more suitable.Some researchers focus on the numerical calculation of their studied system and may use the projection method to get more results.

Conclusion
In conclusion, our exploration of the quadratic 3dimensional differential system centered on the Hopf equilibrium point has provided valuable insights into the existence and stability of limit cycles arising from this critical point.Using the classical projection method to compute the Lyapunov coefficients associated with the Hopf point reveals four distinct families of parameter conditions leading to the system's codimension of three Hopf bifurcations.Furthermore, our findings extend to the n-scroll chaotic attractor system, a versatile model with secure communication and engineering applications.The conditions are identified under which the origin point of this system serves as a Hopf point, allowing for three limit cycles around it.The practical implications of these results underscore the significance of our study for real-world applications.The numerical demonstrations validate our theoretical contributions, confirming the robustness and applicability of our derived conditions.