Numerical Mountain Pass Periodic Solutions Of A Duffing Equation

The circulatory and repeated period movement universally exists inthe natural world,the process and phenomena of which is generallydescribed by differential equations. Many problem in astromechanics,mechanical vibration, electric power system, ecological system, economicfield and engineer technology can be belong to find periodic solution ofmathematic model for differential equation. Many people have alwaysbeen interested in the existence of the periodic solutions of differentialequations. With the constant development of modern mathematical theory,people put forward a number of means and tools for proving the existenceof periodic solution in differential equations, such as KAM theory,nonlinear functional analysis, critical point theory, coincidence degreetheory, optimal control theory, variational method etc. Many basic problems in mechanics,which studies the law of objects'motion, can be summed up into periodic orbital problems in resonance andnonresonant differential equations. The simplest pattern is duffingequationx″+cx′+g( x)=p(t),where c is constant, which indicates spring vibration and Newton motion.Token spring undamped vibration duffing equationx?? ?? +g( x)=p(t).if g(x) satisfy condition0 < <() 1 such thatg (u )?ü c1+c2u|á;H3) There exist constants |è ?ê(0,12) and M such that( ) ()()G u= ?ò 0 u gsds?ü|èugu , ? u ?YM;H4) There exist real T such that lui ?úm 0 g(uu)?üT;H5) ul i?úm ?T g(uu)=+?TDefined on the Hilbert space H 21|D:,(0)(2), (0)(2), ((0, 2);):[0, 2] where is absolutely continuous and212 ??H |D= ????? u|D?úRu =uu|Du=u|Du?êL|DR???the equationu ???? +g(u )=0in space H 21|D correspond to functional :I ( u)= ?ò 02|D(12u2 ?G(u))dt,where G (u )= ?ò 0 u g(s)ds, it is clear that I (u)is C1 functional in H 21|D.As known us, the critical points of the functional I(u)correspond to 2|D-periodic solution of the equation. Accordingmountain pass theorem , we can prove the existence of 2|D-periodicsolutions for the equation, the following result:Theorem 2.1 Assume g(u) satisfy: H1) --H5), then equationu ???? +g(u )=0admits a 2|D-periodic solution.The main steps of Mountain Pass algorithm:step1: Take an initial w0 in E such that I(w0)?ü0?￡step2: Find t * ?ê(0,1) such that I (t * w0)= mt?ê[a0,x1]I(tw0)and set w1 = t*w030step3: Compute ? I( w1)and set v = ?I( w1)step4: If v ?ü|? then output w1 , and stop.step5: Let w = ?v+w1and find t * ?ê(0,1) such that I (t * w)= tm?ê(a0,x1)I(tw).step6: If I (t * w)< I(w1), and set w1 = t*wand goto step3, elseset vv2= 1 and goto step5.Example1. Find the solution of equationu ???? +u3 =0, u ( 0)= u(2|D), u ??( 0)=u??(2|D)Thus, the periodic solutions of the equation can be found byMountain Pass algorithm.picture1: curve of solution picture 2: phase curve...