MATH383: Solving hLSDEs, direction fields, repeated eigenvalues

Overview

• Direction fields

• Phase portraits

• Repeated eigenvalues

Direction fields

• In ${ℝ}^n$ the hLSDE ${𝒚}^{\text{'}}=𝑨𝒚$ defines a direction field ${𝒚}^{\text{'}}=\left(\begin{array}{lll}{y}_1^{\text{'}}& \dots & y_n^{\text{'}}\end{array}\right)$, useful for qualitative characterization of the solution from knowledge of eigenvalues of $𝑨$

• Mass-dampener-spring system $m{u}^{\text{'}\text{'}}+c{u}^{\text{'}}+ku=0$, set $v=u^{\text{'}}$, $m>0$, $2\gamma =c/m$, $\kappa =k/m$

  ${\displaystyle \{\begin{array}{l}{u}^{\text{'}}=v\\ v^{\text{'}}=-\kappa u-2\gamma v\end{array}.,𝒚=\left(\begin{array}{l}u\\ v\end{array}\right),𝒚\left(0\right)=\left(\begin{array}{l}1\\ 0\end{array}\right),{𝒚}^{\text{'}}=𝑨𝒚,𝑨=\left(\begin{array}{ll}0& 1\\ -\kappa & -2\gamma \end{array}\right)}$

• Characteristic polynomial

  ${\displaystyle p_{𝑨}\left(\lambda \right)=\det (\lambda 𝑰-𝑨)=\left|\begin{array}{ll}\lambda & -1\\ \kappa & \lambda +2\gamma \end{array}\right|={\lambda }^2+2\gamma \lambda +\kappa }$

• Roots of characteristic polynomial ${\lambda }_{1,2}=-\gamma \pm \sqrt{{\gamma }^2-{\kappa }^2}$, $\gamma ⩾0$, $\kappa >0$, $\{e^{{\lambda }_{1}t},e^{{\lambda }_{2}t}\}$

  ${\displaystyle \mathrm{No}\mathrm{dampening}:\begin{array}{lll}\gamma =0,& \mathrm{Under}-\mathrm{damped}:\kappa >\gamma >0& \mathrm{Over}-\mathrm{damped}:\gamma >\kappa \end{array}}$

(%i9) 

plotdf([v,-(kappa*u+2*gamma*v)],[u,v],[trajectory_at,1,0],[u,-1,1],[v,-1.5,1.5],
[nsteps,2500],[tstep,.001],[direction,forward],[parameters,"kappa=1,gamma=0"],[sliders,"kappa=.1:2,gamma=0:4"],[versus_t,1])$

(%i10) 

Repeated eigenvalues

• If eigenvalue ${\lambda }_k$ is an $m-$multiple root of the characteristic polynomial, the independent solutions become

  ${\displaystyle y_k=e^{{\lambda }_{k}t},y_{k+1}=t{e}^{{\lambda }_{k}t},\dots ,y_{k+m-1}=t^{m-1}{e}^{{\lambda }_{k}t}}$

(%i185) 

A: matrix([11,-25],[4,-9])$ Y: matrix([x],[y])$ Yp: A.Y$ p: factor(charpoly(A,lambda));

(%i189) 

plotdf([Yp[1][1],Yp[2][1]],[trajectory_at,2,0],
[x,-2,2],[y,-2,2],[direction,forward],
[versus_t,1])$

(%i190)