In this paper, we study the dynamical behavior and exact parametric representations of the traveling wave solutions for the generalized Burger-Fisher equation and the Sharma-Tasso-Olver equation under different parametric conditions, the exact monotonic and non-monotonic kink wave solutions, two-peak solitary wave solutions, periodic wave solutions, as well as unbounded traveling wave solutions are obtained.

We use three different results of the averaging theory of first order for studying the existence of new periodic solutions in the two Duffing differential equations $\ddot y+ a \sin y= b \sin t$ and $\ddot y+a y-c y^3=b\sin t$, where $a$, $b$ and $c$ are real parameters.

A twisted heteroclinic cycle was proved to exist more than twenty-five years ago for the reaction-diffusion FitzHugh-Nagumo equations in their traveling wave moving frame. The result implies the existence of infinitely many traveling front waves and infinitely many traveling back waves for the system. However efforts to numerically render the twisted cycle were not fruitful for the main reason that such orbits are structurally unstable. Presented here is a bisectional search method for the primary types of traveling wave solutions for the type of bistable reaction-diffusion systems the FitzHugh-Nagumo equations represent. The algorithm converges at a geometric rate and the wave speed can be approximated to significant precision in principle. The method is then applied for a recently obtained axon model with the conclusion that twisted heteroclinic cycle maybe more of a theoretical artifact.

In this paper, we investigate a diffusive predator-prey model with fear effect. It is shown that, for the linear predator functional response case, the positive constant steady state is globally asymptotically stable if it exists. On the other hand, for the Holling type II predator functional response case, it is proved that there exist no nonconstant positive steady states for large conversion rate. Our results limit the parameters range where complex spatiotemporal pattern formation can occur.

Recent manipulations on vertebrates showed that the fear of predators, caused by prey after they perceived predation risk, could reduce the prey's reproduction greatly. And it's known that predator-prey systems with fear effect exhibit very rich dynamics. On the other hand, incorporating the time delay into predator-prey models could also induce instability and oscillations via Hopf bifurcation. In this paper, we are interested in studying the combined effects of the fear effect and time delay on the dynamics of the classic Lotka-Volterra predator-prey model. It's shown that the time delay can cause the stable equilibrium to become unstable, while the fear effect has a stabilizing effect on the equilibrium. In particular, the model loses stability when the delay varies and then regains its stability when the fear effect is stronger. At last, by using the normal form theory and center manifold argument, we derive explicit formulas which determine the stability and direction of periodic solutions bifurcating from Hopf bifurcation. Numerical simulations are carried out to explain the mathematical conclusions.

A modified Leslie-Gower predator-prey system with discrete and distributed delays is introduced. By analyzing the associated characteristic equation, stability and local Hopf bifurcation of the model are studied. It is found that the positive equilibrium is asymptotically stable when $\tau$ is less than a critical value and unstable when $\tau$ is greater than this critical value and the system can also undergo Hopf bifurcation at the positive equilibrium when $\tau$ crosses this critical value. Furthermore, using the normal form theory and center manifold theorem, the formulae for determining the direction of periodic solutions bifurcating from positive equilibrium are derived. Some numerical simulations are also carried out to illustrate our results.

In this paper, we first establish the separable $Hamiltonian$ system of rectangular cantilever thin plate bending problems by choosing proper dual vectors. Then using the characteristics of off-diagonal infinite-dimensional $Hamiltonian$ operator matrix, we derive the biorthogonal relationships of the eigenfunction systems and based on it we further obtain the complete biorthogonal expansion theorem. Finally, applying this theorem we obtain the general solutions of rectangular cantilever thin plate bending problems with two opposite edges slidingly supported.

In this paper, we study dynamics and bifurcation of limit cycles in a recently developed new chaotic system, called extended Lorenz system. A complete analysis is provided for the existence of limit cycles bifurcating from Hopf critical points. The system has three equilibrium solutions: a zero one at the origin and two non-zero ones at two symmetric points. It is shown that the system can either have one limit cycle around the origin, or three limit cycles enclosing each of the two symmetric equilibria, giving a total six limit cycles. It is not possible for the system to have limit cycles simultaneously bifurcating from all the three equilibria. Simulations are given to verify the analytical predictions.