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We consider the nonlinear dynamics in a double-chain model of DNA which consists of two long elastic homogeneous strands connected with each other by an elastic membrane. By using the method of dynamical systems, the bounded traveling wave solutions such as bell-shaped solitary waves and periodic waves for the coupled nonlinear dynamical equations of DNA model are obtained and simulated numerically. For the same wave speed, bell-shaped solitary waves of different heights are found to coexist.

In 1953, Waston and Crick discovered the structure of deoxyribonucleic acid (DNA) double helix [

To study DNA structure from the view of nonlinear science, it is necessary to look for the right nonlinear mathematical models. A number of researchers have tried to establish mathematical models to describe DNA system. At the beginning, it is difficult to use a specific mathematical model to simulate DNA system due to its complex structure and the presence of various movements [

In [

In [

Reference [

In this paper, we use the bifurcation method of dynamical systems [

In fact, system (

This paper is organized as follows. In Section

Firstly, we compute the equilibrium points of system (

When

When

Define

Secondly, we compute the intersection points of phase portraits of (

If

If

If

The bifurcation phase portraits of system (

The bifurcation phase portraits of system (

The bifurcation phase portraits of system (

Finally, from the above analysis about system (

The case

Corresponding to the homoclinic orbit of (

The case

Similarly, in Figure

The case

Smooth solitary wave and periodic wave solutions.

In this case, on the

The case

In this case, on the

The case

In this case, there exist two homoclinic orbits and their expressions are (

The case

In this case, on the

The case

Similarly, corresponding to the periodic orbit of (

In this paper, we employ the method of dynamical systems to study the nonlinear dynamical equations of DNA model. The bifurcation phase portraits of the DNA under some parametric conditions are drawn, and explicit exact expressions of bell-shaped solitary waves and periodic waves are obtained via some special homoclinic and periodic orbits; their planar graphs are simulated. These solutions are new and different from those in [

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

This work was supported by the National Natural Science Foundation of China (no. 11226303) and Guangdong Province (no. 2013KJCX0189). The authors thank the editors for their hard working and also gratefully acknowledge helpful comments and suggestions by reviewers.