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This paper focuses on a generalized two-component Hunter-Saxton system. From a dynamic point of view, the existence of different kinds of periodic wave, solitary wave, and blow-up wave is proved and the sufficient conditions to guarantee the existence of the above solutions in different regions of the parametric space are given. Also, some exact parametric representations of the travelling waves are presented.

The Hunter-Saxton equation,

The two-component Hunter-Saxton system is as follows:

Very recently, the generalized two-component Hunter-Saxton system

In this paper, we investigated the following generalized two-component Hunter-Saxton system:

Using the following independent variable transformation:

Integrating equations of (

Substituting (

Letting

The rest of this paper is organized as follows. In Section

Using the transformation

For a fixed

Write

Let

For an equilibrium point

Since both systems (

By using the properties of equilibrium points and bifurcation method of dynamical systems, we can show that bifurcation sets and phase portraits of system (

Bifurcation sets and phase portraits of system (

Denote that

From Figures

If introducing a new parametric variable

Substituting (

From Figures

Substituting (

Substituting (

From Figures

Substituting (

Substituting (

From Figure

Substituting (

Substituting (

From Figure

Substituting (

Substituting (

From Figure

Substituting (

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Substituting (

From Figure

Substituting (

Substituting (

From Figure

Substituting (

Substituting (

From Figure

Substituting (

From Figure

Substituting (

From Figure

Substituting (

Substituting (

From Figure

Substituting (

Substituting (

In this paper, we studied the bifurcations of travelling wave solutions of a generalized two-component Hunter-Saxton system and obtained different kinds of periodic wave solutions, which concluded periodic blow-up wave and periodic loop solutions and so forth. Some solitary wave and blow-up wave solutions are also obtained. The results of this paper have enriched the results of [

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

The authors thank the referees very much for their perceptive comments and suggestions. This work is supported by the Natural Science Foundation of Yunnan Province, China (no. 2013FZ117), and the National Natural Science Foundation of China (no. 11161020).