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Abstract: By using the bifurcation theory of dynamical systems to modified nonlinear dispersive phi-four equation, we analysis all bifurcations and phase portraits in the parametric space, the existence of solitary wave solutions and uncountably infinite many smooth and non-smooth periodic wave solutions is obtained. Under different parametric conditions, various sufficient conditions to guarantee the existence of the above solutions are given. Some explicit exact solution formulas are acquired for some special cases. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, shock wave solutions of the compound Burgers–KdV equation is studied by using qualitative theory of differential equation and ansatz method. The existence and uniqueness of different types of travelling waves of the system are obtained by analyzing qualitatively the existence and uniqueness of the heteroclinic trajectories of the ordinary differential equation. We construct some travelling wave solutions and indicate their stability and bifurcation. [Copyright &y& Elsevier]

Abstract: By using bifurcation method of phase portraits and theory of dynamical systems to the nonlinear Klein–Gordon model, in the range , numbers of kink compactons, solitary wave, kink and anti-kink waves are given for each parameter condition. The parameter condition when the type of travelling wave changes is obtained. [Copyright &y& Elsevier]

Abstract: The dynamical behavior and special exact solutions of the generalized Degasperis–Procesi equation u t + c 0 u x − u txx + auu x =3u x u xx + uu xxx is analyzed by using bifurcation theory and the method of phase portraits analysis. As a result, the analytic expressions of peakon solutions, compacton solutions and periodic cusp wave solutions are obtained. And the condition under which peakon and compacton solutions appear are also given. In the meantime, the reason that solitary cusp wave and periodic cusp wave appear is given. [Copyright &y& Elsevier]

Abstract: By using bifurcation method to Whitham–Broer–Kaup shallow water equations, bifurcation parameter sets are shown. Numbers of solitary waves, kink waves and periodic waves are given. Under various parameter conditions, all explicit formulas of solitary wave solutions, kink wave solutions and periodic wave solutions are listed. [Copyright &y& Elsevier]

Abstract: By using the theory of bifucations of dynamical systems to a model of the Hydrogen-bonded systems, the existence of solitary wave, kink and anti-kink wave solutions and uncountably infinite many smooth and non-smooth periodic wave solutions is obtained. Under different parametric conditions, various sufficient conditions to guarantee the existence of the above solutions are given. In some simple parametric conditions, exact explicit and implicit solution formulas are listed. [Copyright &y& Elsevier]

By using the bifurcation theory of dynamical systems to the generalized Hirota–Satsuma coupled KdV system, bifurcation parameter sets are shown. Numbers of solitary waves, kink waves and periodic waves are given. Under various parameter conditions, all explicit formulas of solitary wave solutions and kink wave solutions and periodic wave solutions are listed. [Copyright &y& Elsevier]

• The approximate range of eigenvalue of the network matrix is shown. • The instability condition about diffusion and the connection probability in the network-organized system are derived. • The estimated range of connection probability is also deduced. • Mechanism of spiking of the neuron can be explained by using diffusion network and matrix theory. Although there is general agreement that the network plays an essential role in the biological system, how the connection probability of network affects the natural model(Especially neural network) is poorly understood. In this paper, we show the impact of the network on Turing instability in the FitzHugh-Nagumo(FN) model. Then we obtain the condition of how the Turing bifurcation, saddle-node bifurcation, and Turing instability occur. By using the Gershgorin circle theorem, we investigate the relationship between degree and eigenvalue of the network matrix, and obtain the approximate range of eigenvalue of the network matrix. Also, We derive the instability condition about diffusion and the connection probability in the network-organized system. And then we obtain the estimated range of connection probability. Meanwhile we apply these results to explaining the spiking of neuron and find this system has rich dynamics behavior. Finally, the numerical simulation verifies our analytical results. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, explicit and exact travelling wave solutions of B(m, n) equations are investigated by using a generalized algebraic method. A series of explicit exact solutions, such as solitary wave, triangular periodic, rational, Jacobi and Weierstrass elliptic function periodic solutions are obtained. Many new results are presented. [Copyright &y& Elsevier]

Abstract: By using the theory of bifurcations of dynamical systems to a class of the generalized Benjamin–Bona–Mahony (GBBM) equation, the existence of solitary wave solutions and uncountably infinite, many smooth and non-smooth periodic wave solutions are obtained. It can be shown that the existence of singular curves in a traveling wave system is the reason why smooth waves converge to cusp waves, finally. When parameters are varied, under different parametric conditions, various sufficient conditions to guarantee the existence of the above solutions are given. [Copyright &y& Elsevier]