Re: ODEs and phase portraits

*To*: mathgroup at smc.vnet.net*Subject*: [mg14728] Re: ODEs and phase portraits*From*: Selwyn Hollis <shollis at peachnet.campus.mci.net>*Date*: Wed, 11 Nov 1998 17:53:34 -0500*Organization*: fair*References*: <728kp1$ehc@smc.vnet.net>*Sender*: owner-wri-mathgroup at wolfram.com

Dear 'lord, This should help: f[epsilon_][x_,y_]:= -epsilon(x^2-1)y-x system[epsilon_,{x0_,y0_}]= {x'[t]==y[t], y'[t]==f[epsilon][x[t],y[t]], x[0]==x0, y[0]==y0} <<Graphics`PlotField` field=PlotVectorField[{y,f[1][x,y]},{x,-3,3},{y,-3,3}, ScaleFunction->(1&),ColorFunction->(Hue[0]&)]; soln={x[t],y[t]}/.NDSolve[system[1,{0,.1}],{x,y},{t,0,10}] curve=ParametricPlot[Evaluate[soln],{t,0,10}, PlotStyle->Thickness[.008]]; Show[field,curve]; --sh phantomlord at my-dejanews.com wrote: > I am trying to write a function such that I have the following ODE: > x''(t)+epsilon*(x(t)^2-1)*x'(t)+x(t)==0 [1] where epsilon is to be one > of the parameters in the function. I want to beable to draw out the > phase portraits for the equation for different values of epsilon. > > To deduce the phase portraits in a mathematical procedure I multiply > equation[1] by dx(t)/dt and integrate w.r.t. t. To do this in > Mathematica is trivial, so I'll skip past this - it is the next step > that I would like assistance with: > > Q:Is there a way that I can decompose the result of the above (I'll call > it [2]) into the corresponding pair (below) of ODEs to deduce the > trajectory of the phase portrait? > > x'(t) = y(t) > y'(t) = f(x,y) > some f(x,y) function of x and y. > > Perhaps there is a built in function that will allow me to do this? If > not do, can anybody suggest another way to do this? > > Also after plotting the trajectory is there any way to determine the > direction of it in Mathematica? > > thanks for you time. > Paul > > -----------== Posted via Deja News, The Discussion Network ==---------- > http://www.dejanews.com/ Search, Read, Discuss, or Start Your Own -- ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Dr. Selwyn Hollis Associate Professor of Mathematics Armstrong Atlantic State University Savannah, GA 31419 USA <http://www.math.armstrong.edu/faculty/hollis/> ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~