Van der Pol - Maple Help

Van der Pol ODEs

Description

 • The general form of the Van der Pol ODE is given by the following:
 > Van_der_Pol_ode := diff(y(x),x,x)-mu*(1-y(x)^2)*diff(y(x),x)+y(x)=0;
 ${\mathrm{Van_der_Pol_ode}}{≔}\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right){-}{\mathrm{\mu }}{}\left({1}{-}{{y}{}\left({x}\right)}^{{2}}\right){}\left(\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right){+}{y}{}\left({x}\right){=}{0}$ (1)
 See Birkhoff and Rota, "Ordinary Differential Equations", p. 134.
 The second order Van der Pol ODE can be reduced to a first order ODE of Abel type as soon as the system succeeds in finding one polynomial symmetry for it (see symgen):
 $\left[\left[{\mathrm{_2nd_order}}{,}{\mathrm{_missing_x}}\right]{,}{\mathrm{_Van_der_Pol}}\right]$ (2)
 > symgen(Van_der_Pol_ode, way=3);
 $\left[{\mathrm{_ξ}}{=}{1}{,}{\mathrm{_η}}{=}{0}\right]$ (3)
 From which, giving the same indication directly to dsolve you obtain the reduction of order
 > ans := dsolve(Van_der_Pol_ode,way=3);
 ${\mathrm{ans}}{≔}{y}{}\left({x}\right){=}{\mathrm{_a}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{&where}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\left[\left\{\left(\frac{{ⅆ}}{{ⅆ}{\mathrm{_a}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{_b}{}\left({\mathrm{_a}}\right)\right){}{\mathrm{_b}}{}\left({\mathrm{_a}}\right){+}{\mathrm{_b}}{}\left({\mathrm{_a}}\right){}{{\mathrm{_a}}}^{{2}}{}{\mathrm{\mu }}{-}{\mathrm{\mu }}{}{\mathrm{_b}}{}\left({\mathrm{_a}}\right){+}{\mathrm{_a}}{=}{0}\right\}{,}\left\{{\mathrm{_a}}{=}{y}{}\left({x}\right){,}{\mathrm{_b}}{}\left({\mathrm{_a}}\right){=}\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right\}{,}\left\{{x}{=}{\int }\frac{{1}}{{\mathrm{_b}}{}\left({\mathrm{_a}}\right)}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{\mathrm{_a}}{+}{\mathrm{_C1}}{,}{y}{}\left({x}\right){=}{\mathrm{_a}}\right\}\right]$ (4)
 For the structure of the solution above see ODESolStruc. Reductions of order can also be tested with odetest
 > odetest(ans,Van_der_Pol_ode);
 ${0}$ (5)
 The reduced ODE is of type Abel, and can be selected using either the mouse, or the following:
 > reduced_ode := op([2,2,1,1],ans);
 ${\mathrm{reduced_ode}}{≔}\left(\frac{{ⅆ}}{{ⅆ}{\mathrm{_a}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{_b}{}\left({\mathrm{_a}}\right)\right){}{\mathrm{_b}}{}\left({\mathrm{_a}}\right){+}{\mathrm{_b}}{}\left({\mathrm{_a}}\right){}{{\mathrm{_a}}}^{{2}}{}{\mathrm{\mu }}{-}{\mathrm{\mu }}{}{\mathrm{_b}}{}\left({\mathrm{_a}}\right){+}{\mathrm{_a}}{=}{0}$ (6)
 $\left[{\mathrm{_rational}}{,}\left[{\mathrm{_Abel}}{,}{\mathrm{2nd type}}{,}{\mathrm{class A}}\right]\right]$ (7)