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In this paper, bounded traveling waves of a generalized Burgers–Fisher equation are investigated. By studying some complicated local and nonlocal bifurcations such as the Hopf bifurcation, homoclinic bifurcation, heteroclinic bifurcation and Poincaré bifurcation of the corresponding traveling wave system, we obtain sufficient conditions to guarantee the existence of different kinds of bounded traveling waves, including solitary waves, kink waves and periodic waves. Further, we discuss the existence of various oscillatory bounded traveling waves.

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This considers the problem of H ∞ controller design for singular stochastic systems. The systems have Markovian jump and time-varying delay. We aim to design a controller ensuring the closed-loop system is stochastically admissible and satisfies a prescribed H ∞ performance index γ . Based on the linear matrix inequalities (LMIS) techniques, the mode-dependent singular matrices E (r t ) and the stochastic Lyapunov function method, a sufficient condition for the desired H ∞ controller for the system under consideration is given in terms of LMIs. Moreover, to clarify the proposed results, some illustrative examples are presented.INDEX TERMS Singular stochastic system, linear matrix inequality, H ∞ control, time-varying delays, Lyapunov-Krasovskii functional.

The main purpose of this article is to consider a Lotka-Volterra predator-prey system with ratio-dependent functional responses and feedback controls. By using a comparison theorem and constructing a suitable Lyapunov function as well as developing some new analysis techniques, we establish a set of easily verifiable sufficient conditions which guarantee the permanence of the system and the global attractivity of a positive solution for the predator-prey system. Furthermore, some conditions for the existence, uniqueness, and stability of a positive periodic solution for the corresponding periodic system are obtained by using the fixed point theory and some new analysis method. In additional, some numerical solutions of the equations describing the system are given to verify that the obtained criteria are new, general, and easily verifiable.

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The bifurcation method of dynamical system and numerical simulation method of differential equation are employed to investigate the (2+1)-dimensional Zoomeron equation. We obtain the parameter bifurcation sets that divide the parameter space into different regions which correspond to qualitatively different phase portraits. According to these phase portraits, all bounded traveling waves are identified and simulated, including solitary wave solutions, shock wave solutions, and periodic wave solutions. Furthermore, all exact expressions of these bounded traveling waves are given. Among them, the elliptic function periodic wave solutions are new solutions.