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The stability and bifurcations of multiple limit cycles for the physical model of thermonuclear reaction in Tokamak are investigated in this paper. The one-dimensional Ginzburg-Landau type perturbed diffusion equations for the density of the plasma and the radial electric field near the plasma edge in Tokamak are established. First, the equations are transformed to the average equations with the method of multiple scales and the average equations turn to be a

Periodic solution theory is mainly about the existence and stability of periodic solution of dynamical systems. The bifurcation theory of periodic solution, as the main method to study the periodic solution, reveals the connection between the topology of the solutions and the parameters. The investigation and application of bifurcations and chaos of nonlinear dynamical systems are frontier topics in the world.

Recently, a large number of important results of multiple limit cycles of polynomial planar vector fields have been achieved. Arnol’d [

The symmetry of dynamical systems is also widely studied by many researchers and a lot of results have been achieved. Li et al. [

In recent years, theories are widely used in mechanical systems. An important issue is the bifurcation of multiple limit cycles for Tokamak system. The Tokamak is the most promising device so far to attain the conditions for fusion. It is a toroidal device (shaped like a car tire) in which a vacuum vessel contains a plasma ring confined by twisting magnetic fields. In the past 20 years, researches and applications of the Tokamak have great achievements, especially with the implementation of the construction named EAST (Experimental Advanced Superconducting Tokamak). EAST is one of Chinese national fusion projects.

In this paper, the mechanism of the transition from L-mode to H-mode in Tokamak is an important and difficult problem. We mainly discussed the transition phenomenon between low confinement mode (L-mode) and high confinement mode (H-mode) observed in Tokamak. S. Itoh and K. Itoh [

This paper focuses on the bifurcations of multiple limit cycles for a Ginzburg-Landau type perturbed transport equation which can describe the L-mode to H-mode transition near the plasma edge in Tokamak. The average equation of Tokamak system turns out to be a

We get the nondimensional formulations by using the method in [

In order to analyze the diffusion of the particle and the stability and bifurcations of the normalized radial electric field near the plasma edge in Tokamak, some transformations may be introduced as follows:

Eliminate

We assume that the uniform solution of (

Then, the differential operators are given as

The main work of this paper focuses on the

Then (

Let

In Section

In this section, the method of detection functions will be described briefly based on references [

Consider the Abelian integral

We define the function

We have the following three statements on the local and global bifurcations.

If

If

The total number of isolated zeros of the Abelian integral is an upper bound for the number of limit cycles of system ((

We consider the unperturbed system of ((

Equation ((

Let

It is easily seen that there exist 25 singular points of system (

In this paper, we only consider a special case that

Under the conditions of

The diagram of the Hamiltonian function (

Families of closed orbits defined by system ((

Figure

Different schemes of ovals defined by ((

Notice that as

Based on the results obtained above, we can analyze the qualitative nonlinear characteristics of the perturbed system ((

Denote that

It follows that, under the parameter conditions of

Graphs of detection curves of system ((

We can see from Figure

When

Configuration of 22 limit cycles of system ((

Based on the local and global bifurcation theory and the results of paper [

Suppose that, as

Suppose that, as

From the theorems above, we can also prove the following results.

If

If the curve

If

On the basis of the method of the theorems above, we have the following analyses of stability.

If the period orbit

If the period orbit

If the period orbit

If the period orbit

If the period orbit

It follows that, under the parameter conditions of

The stability of 22 limit cycles of system ((

The relationship between the actual parameters of the physical equations and the parameter conditions

One has the following parameters relationship between ((

Based on Proposition

With the help of Propositions

With the help of algebraic method and Lingo mathematical software, one can get that

This paper focuses on the bifurcations of multiple limit cycles for a Tokamak system. First, the method of multiple scales and normal form theory are employed to obtain the average equation in the Tokamak system, which has the form of a

One control condition of parameters is given to obtain 22 limit cycles of the Tokamak system. Ten of them are stable and the others are unstable. The Hopf bifurcation and limit cycles in averaged equation ((

The coefficients given in ((

The coefficients given in (

The coefficients given in ((

The authors declare that there is no conflict of interests regarding the publication of this paper.

The authors gratefully acknowledge the support of the National Natural Science Foundation of China (NNSFC) through Grant nos. 11072007, 11372014, 11072008, and 11290152 and the Natural Science Foundation of Beijing (NSFB) through Grant no. 1122001.

_{2}polynomial differential systems

_{2}-equivariant planar vector fields of degree 5