Abstract
In this chapter, by researching the algorithm of the formal series, and deducing the recursion formula of computing the nondegenerate and degenerate singular point quantities on center manifold, we investigate the Hopf bifurcation of high‐dimensional nonlinear dynamic systems. And more as applications, the singular point quantities for two classes of typical three‐ or four‐dimensional polynomial systems are obtained, the corresponding multiple limit cycles or Hopf cyclicity restricted to the center manifold are discussed.
Keywords
- high‐dimensional system
- center manifold
- Hopf bifurcation
- singular point quantities
1. Introduction
This chapter is concerned with Hopf bifurcation restricted to the center manifold from the equilibrium for three‐, four‐, and more higher‐dimensional nonlinear dynamical systems.
Let us first consider the generic real systems which take the form
where
where
Suppose that
The first equation in Eq. (3) is called the restriction of system (2) to its center manifold at the origin. The local center manifold, which is tangent to the
If
If
2. Case of the nondegenerate singular point
In this section, we consider Hopf bifurcation from the nondegenerate origin of system (1) restricted to the center manifold, in which the Jacobian matrix
2.1. The formal series method of computing nondegenerate singular point quantities on center manifold
Considering the Jacobian matrix
where
Here, we recall first the calculation method of the singular point quantities on center manifold for the above real three‐dimensional nonlinear dynamical systems. By means of transformation
system (4) is also transformed into the following complex system:
where
Based on the previous work in Ref. [16], we have developed the calculation method of the focal values on the center manifold for real four‐dimensional nonlinear dynamical systems in Ref. [35]. In fact, here Theorem 1 can be generalized in the
where
By means of transformation of Eq. (5), system (10) can be transformed into the following complex system
where the subscript “
2.2. An example of four‐dimensional system
Recently, the study of chaos has become a hot research topic, and the attention of many researchers is turning to 4D systems from 3D dynamical systems, for example, the authors of Ref. [36] investigated Hopf bifurcation of a 4D‐hyoerchaotic system by applying the normal form theory in 2012, but its multiple Hopf bifurcation on the center manifold have not been considered. Here, we will investigate the system further by computing the singular point quantities of its equilibrium point, which takes the following form
where
with the characteristic equation:
To guarantee that
Thus, we obtain the critical condition of Hopf bifurcation at
where
such that
Namely, we can use the nondegenerate transformation and the time rescaling:
where
where
According to Theorem 3, we obtain the recursive formulas of
By applying the above formulas in the Mathematica symbolic computation system, we figure out easily the first two singular point quantities of the origin of system (18):
where
and the above expression of
From Remark 1 and the singular point quantities (21), we have
Considering its Hopf bifurcation form of Theorem 6, we have the following:
The rigorous proof of the above theorem is very similar to the previous ones in [14, 16], namely, by calculating the Jacobian determinant with respect to the functions
3. Case of the degenerate singular point
Up till now, study on bifurcation of limit cycles from the degenerate singularity of higher dimensional nonlinear systems (1) is hardly seen in published references. Here, we will investigate the Hopf bifurcation problem from the high‐order critical point on the center manifold.
3.1. The formal series method of computing degenerate singular point quantities on center manifold
Let us consider the real
where the subscript “
In order to discuss the calculation method of the focal values on center manifold of the system (23), from the center manifold theorem [1], we take an approximation to the center manifold:
where
where
For system (25), some significant works have been done in Refs. [26] and [27]. Let us recall the related notions and results.
By means of transformation (5)
system (25) is transformed into following system:
where
For any positive integer
a homogeneous polynomial of degree
such that
Similar to Theorem 2, there also exists a equivalence between the
3.2. An example of three‐dimensional system
Now we consider an example for system (23) with
where
namely,
where
Hence,
Applying the powerful symbolic computation function of the Mathematica system and the recursive formulas in Theorem 10, and from Remark 5, we obtain the first three singular point quantities as follows
In the above expression of each
Thus, from Theorem 9 and Eqs. (35) and (31), we have
Now we consider Hopf bifurcation of limit cycles from the origin for perturbed system (30).
Considering the conditions (37) of Theorem 12 and substituting the group of critical values of Eq. (38) into Eq. (39), we obtain
and
hold, one must obtain that the succession function on the center manifold has three small real positive roots, just the system (30) has at least three limit cycles in the neighborhood of the origin. We can refer to references [16, 26, 27] for more details about the construction of limit cycles.
4. Conclusion and discussion
The two classes of methods for computing the nondegenerate and degenerate singular point quantities on center manifold of the three‐, four‐, and more higher dimensional polynomial systems are discussed here, and more as the applications of them, the multiple limit cycles or Hopf cyclicity of two typical nonlinear dynamic systems restricted to the corresponding center manifolds are investigated.
Appendix A
where
Acknowledgments
This work was supported by Natural Science Foundation of China grants (11461021, 11261013), Nature Science Foundation of Guangxi (2015GXNSFAA139011), Research Foundation of Hezhou University (No.HZUBS201302), and Guangxi Education Department Key Laboratory of Symbolic Computation and Engineering Data Processing.
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