Studies On Bifurcations Of Infinite-dimensional Dynamical Systems Via Traveling Wave Transformation

Bifurcation analysis and its control of infinite-dimensional dynamical system is a important branch in researching chemistry and biology, including fluid mechanics, solid mechanics, fracture mechanics, atmospheric dynamics, chemical reaction systems, biological evolution system and so on. This paper based on the travlling wave transformation, central manifold and perturbation method, studied the bifurcation and control problem of several kinds of classical infinite-dimensional dynamical system.Introduce the research situation of bifurcation and control problem of infinite-dimensional dynamical system.Based on auxiliary differential equation method, the travelling wave solution of nonlinear evolution eqautions are investigated. The main steps of the auxiliary differential equation method are introduced and analysised. Accroding to a discuss of auxiliary differential equation and utilizing the extended hyperbolic function expansion method, the exactly solution of (2+1)-demensional Nizhnik-Novikov-Veselov eqzution and generalized (2+1)-demensional Nizhnik-Novikov-Veselov eqzution are obtained. The method presented in this paper can be applied in solving other equations’solitons, triangle periodic solution and elliptical solution if this equations have exactly solution through the travlling wave transformation. Based on this exactly solutons, the bifurcation behaviors can be investigated directly.Based in the travlling wave transformation and integral method, the exactly solution of Burgers equation and (2+1)-dimensional Burgers equation can be obtained. Then, the static bifurcation of these equations are invrstigated, which find that the Burgers equation and (2+1)-dimensional Burgers equation have the typical behavior of transcritical bifurcation. The reserch on bifurcation control of infinite-dimensional dynamical system are seldom. In this paper, the Burgers equation and (2+1)-dimensional Burgers equation are reansformed into a one dimensional systems. Then, the transcritical bifurcation these two equatons are discussed. This method can be also extended to research other kind of static bifurcation like pitchfork bifurcation and saddle node bifurcation.Discuss that nonlinear partial differential equations also have the pitchfork bifurcation, saddle node bifurcation and transcritical bifurcation like the ordinary differential equations. Three kinds of equations are transformed into one dimensionalordinary equation and the bifurcation behaviors are investigated. Burgersequation,(2+1)-demensional Burgers equation and (2+1)-demensional Burgers-KP equationhave transcritical bifurcation.(2+1)-demensional Kadomtsev-Pet-viashvili equationhas saddle node bifurcation. By constructing a new partial differential equation, thepitchfork bifurcation of partial differential equation is also studied. By utilizing the exactly transformation, the partial differential equation can betransformed into ordinary differential equation. Untilizing the central manifoldthroem, the ordinary differential equation can be transformed into one dimensionalequation. Then the pitchfork bifurcation, saddle node bifurcation and transcriticalbifurcation of the transformed equation are investigated. And the stable region ofsolutons in parameter plane are obtained. Unitlizing the feedback control thorem, thelinear controller, nonlinear controller and the hybrid controller are designed to controlthe pitchfork bifurcation, saddle node bifurcation and transcritical bifurcationrespectively. The feedback controllers not change the bifurcation traits of the systems,but shift the bifurcation point and the stable region. The saddle node bifurcation of forced Burgers-KdV equaton is studied. Thissytem has the phenomenon of jump and lag in the frequency response. Based theperturbation method, the frequency response curve can be obtained. Then thebifurcation graph of this system can be also drawn. To achieve the control ofbifurcation, a feedback controller is designed. The linear control, nonlinear controland the hybrid control are untilized to shift the unstable region and the nonlinear traitof the system. Then method presented in this paper can be also extended to researchother evlution euations.