%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % chap3.tex %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \chapter{NUMERICAL CONSTRUCTION OF THE RETURN MAP}\label{ch:numerics} \section{Overview of Numerical Methods Used} A nice small table is shown in Table~\ref{table:SSparams} \begin{table}[h] \begin{center} \begin{tabular}{p{0.5in}p{.25in}p{.25in}p{.25in}p{.3in}p{.25in}p{.25in}p{.25in}} \hline &\vspace{1pt} $\beta_1$ &\vspace{1pt} $\alpha_1$ &\vspace{1pt} $\beta_2 $ &\vspace{1pt}\hspace{3pt} $\alpha_2$ &\vspace{1pt} $\gamma$ &\vspace{1pt} $\tau_1$ &\vspace{1pt} $\tau_2$ \\ \hline $(+,+)$ & 3 & $-1$ & $0.5$&$-3$& $0.7$& $0.9$& 1 \\ $(+,-)$ & 4 & $-1$ & $-1$ & $-0.7$& 1& 1 & $0.3$ \\ \hline \end{tabular}\end{center} \caption{(SS) Parameter sets} \label{table:SSparams} \end{table} \subsection{More Figures} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Our, by now familiar, caricature of the (SS) is shown in Figure~\ref{fig:auto_description}, which is an example of a {\bf picture} environment 'drawn' figure. \begin{figure}[t] \begin{center} \begin{picture}(400,200) %{\linethickness{4pt} \put(0,30){\vector(1,0){399}} %x-axis \put(100,0){\vector(0,1){200}} %y-axis \put(400,27){$x$} \put(104,196){$y$} \put(80,200){\line(5,-3){310}} \put(0,130){\line(5,-3){200}} \put(202,-2){$\Gamma_{-}$} \put(393,0){$\Gamma_{\HC}$} \put(300,140){$x_f = \widetilde{\phi} (x_0)$} \qbezier(82,80)(110,90)(168,147) \put(79,77){$\bullet $} \put(165,143){$\bullet $} \put(173,147){$(x_0,y_0) $} \put(34,75){$(x_f,y_f) $} \put(130,95){$T_s(x_0) $} \end{picture} \caption{A caricature of $\widetilde{\phi}$ (SS), projected onto the $(x,y)$-plane.} \label{fig:auto_description} \end{center} \end{figure} Figure~\ref{fig:domain_issues_stable} was created using the program {\bf xfig}. Xfig allows you to export the graphics in a number of different formats (including .eps) and further allows you to choose the color of the background, including trasparent. The transparent choice is great for preparing talks in .pdf (using prosper or pdfslide). \begin{figure}[hb] \begin{center} \pspicture(0,0)(12,8) % \psgrid[gridcolor=blue,subgriddiv=1,griddots=10,gridlabels=7pt](0,0)(12,8) \rput[bl](0.5,0){\includegraphics[angle=0,width=11.5cm]{stable_SS.eps}} \rput[l](11.75,0.95) {$x$} \rput[l](1.0,8.0) {$y$} \rput[l](0.6,3.4){\red $\Gamma_-$} \rput[l](10.7,1.8){\red $\Gamma_{\HC}$} \psdot(5.3,3.5) \rput[l](5.1,3.8){${\bf x}_e$} \psdot[dotscale=0.8](2.63,7.75) \rput[l](2.75,7.85){\footnotesize $x_0$} \psdot[dotscale=0.8](2.47,2.36) \rput[r](2.4,2.15){\footnotesize $\widetilde \phi(x_0)$} \psdot[dotscale=0.8](3.77,6.85) \rput[l](3.74,7.13){\footnotesize $x_a$} \psline[linestyle=dotted,arrowsize=0.15]{->}(4.4,-0.1)(4.4,0.6) \rput[r](4.2,0.2){\footnotesize $\widetilde \phi(x_c)$} \psdot[dotscale=0.8](9.1,2.75) \rput[l](9.0,2.5){\footnotesize $x_c$} \endpspicture \caption[A generic example of a single stable equilibrium on ${\mathcal S}_L$.] {A generic example of a single stable equilibrium on ${\mathcal S}_L$ in the band between $\Gamma_-$ and $\Gamma_{\HC}$. We utilize this figure to assist in the general discussion of factors effecting the domain and range of $\widetilde \phi$} \label{fig:domain_issues_stable} \end{center} \end{figure} Figure~\ref{fig:PhiTilde} was generated using AUTO then read into Matlab, plotted and exported as an .eps figure. \begin{figure}[h] \begin{center} \pspicture(0,0)(14,6) %\psgrid[subgriddiv=1,griddots=10,gridlabels=7pt](0,0)(14,7) \rput[bl](-0.6,0){\includegraphics[angle=0,width=7.5cm]{PhiTilde_p_p.eps}} \rput[bl](7.1,0){\includegraphics[angle=0,width=7.5cm]{PhiTilde_p_n.eps}} \rput[l](3.2,-0.2){$x$} \rput[l](11.2,-0.2){$x$} \rput[l](6.95,3.65){$\widetilde{x}$} %\rput[l](7.2,0.35){\LARGE $x$} %\rput[l](0.4,7){\LARGE $\widetilde{x}$} \endpspicture \caption[Map $\widetilde \phi(x)$ generated numerically using AUTO.]{Map $\widetilde \phi(x)$ generated numerically using AUTO: the solid line shows $\widetilde{x}=\widetilde \phi(x)$ for a range of initial $x$ values; $(+,+)$ case on left, $(+,-)$ case on right. The dotted line is $y=x$, added just for reference. Parameter values for each computation are tabulated in Table~\ref{table:SSparams}} \label{fig:PhiTilde} \end{center} \end{figure} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{figure}[h] \begin{center} \pspicture(0,0)(14,11.5) % \psgrid[subgriddiv=1,griddots=10,gridlabels=7pt](0,0)(14,11) \rput[bl](-1.3,6.0){\includegraphics[angle=-90,width=8.3cm]{Run2.ps}} \rput[bl](-1.3,0){\includegraphics[angle=-90,width=8.3cm]{Run2_zoom.ps}} \rput[bl](6.3,6.0){\includegraphics[angle=-90,width=8.3cm]{Run3.ps}} \rput[bl](6.3,0){\includegraphics[angle=-90,width=8.3cm]{Run3_zoom.ps}} \rput[l](0.2,6.9){\large \bf A} \rput[l](8.1,6.9){\large \bf B} \rput[l](2.8,5.9){$x$} \rput[l](10.4,5.9){$x$} \rput[l](6.85,9.65){$y$} \rput[l](5.0,9.95){\footnotesize $p=1$} \rput[l](4.55,6.75){$\Gamma_-$} \rput[l](5.9,7.4){$\Gamma_{\HC}$} \rput[l](11.95,6.7){$\Gamma_-$} \rput[l](13.5,7.5){$\Gamma_{\HC}$} \rput[l](0.2,1.0){\large \bf C} \rput[l](7.9,1.0){\large \bf D} \rput[l](3.2,-0.1){$x$} \rput[l](10.85,-0.1){$x$} \rput[l](6.85,3.5){$y$} %\psline[linestyle=dashed,dash=2pt 2pt](13.3,3.8)(13.3,2.7) \psline[linecolor=yellow,linewidth=0.03](-0.2,9.05)(6.65,9.95) \psline[linestyle=dashed,dash=2pt 4pt]{->}(1.97,9.1)(1.97,6.5) \psline[linestyle=dashed,dash=2pt 4pt]{->}(9.03,9.03)(9.03,6.5) \psline[linestyle=dashed,dash=2pt 3pt]{<-}(1.65,0.5)(1.65,2.8) \rput[l](1.8,0.9){\footnotesize $\bar x \approx 0.266$} \psline[linestyle=dashed,dash=2pt 3pt]{<-}(9.42,0.6)(9.42,4.25) \rput[l](9.6,0.9){\footnotesize $\bar x \approx 0.553$} \endpspicture \caption[Verification of the fixed point $\bar x=\phi(\bar x)$ obtained in the map composition for both the $(+,+)$ and $(+,-)$ cases.]{Verification of the fixed point $\bar x=\phi(\bar x)$ obtained in the map composition, illustrated in Figure~\ref{fig:PhiTilde}. The two figures on the left are the $(+,+)$ case and the two on the right are the $(+,-)$ case. In {\bf A} we see the projection of two trajectories of (FULL) onto the $(x,y)$-plane appearing to converge toward the superimposed line $p=1$. In {\bf C} we have enlarged the region of interest, about the map fixed point $\bar x \approx 0.266$. Also shown in both of these figures are the curves $\Gamma_-$ and $\Gamma_{\HC}$. In {\bf B} we also see the projection of two trajectories of (FULL) onto the $(x,y)$-plane but in the $(+,-)$ case. The two projected trajectories wind onto a cycle in the $(x,y)$-plane. In {\bf D} we are able to discern the map fixed point of $\bar x \approx 0.553$} \label{fig:fixed_pts} \end{center} \end{figure} \begin{figure}[h] \begin{center} \pspicture(0,0)(16,10.3) % \psgrid[gridcolor=blue,subgriddiv=1,griddots=10,gridlabels=7pt](0,0)(15,11) \rput[bl](-1.0,0){\includegraphics[angle=-90,width=16cm]{burst_tangent.ode.ps}} \rput[l](13.2,0.2) {$x$} \rput[r](0.7,7.0) {$y$} \rput[l](10.1,1.2){\red $\Gamma_-$} \rput[l](12.95,1.3){\red $\Gamma_{\HC}$} \rput[l](12.8,4.3){\blue $\widetilde {\bf C}_{\SS}$} \psdot[dotscale=0.9](8.84,4.97) \rput[r](8.9,4.7){\footnotesize $x_{\HC}$} \rput[r](2.6,7.36){\footnotesize $\widetilde \phi(x_{\HC})$} \psdot[dotscale=0.8](2.49,7.64) \rput[l](8.9,7.3){\blue \scriptsize ${\bf F} \cdot {\bf N}_- \big\vert_{\Gamma_{u}}>0$} \rput[l](9.0,3.1){\blue \scriptsize ${\bf F} \cdot {\bf N}_- \big\vert_{\Gamma_{u}}<0$} % \psline[arrowsize=0.16]{->}(8.9,4.3)(9.07,4.8) \psline[linestyle=dotted,arrowsize=0.17]{->}(2.7,9.0)(2.6,8.97) \psline[linestyle=dotted,arrowsize=0.19]{->}(12.55,2.8)(12.35,3.0) \psline[linecolor=red,linewidth=.5pt](1.23,8.62)(2.3,7.77) \endpspicture \caption[Several (SS) trajectories along with a plot of the curve $\widetilde C_{\SS}$ in the $(+,-)$ case.]{Several (SS) trajectories, in the $(+,-)$ case, along with a plot of the curve $\widetilde C_{\SS}$ parametrized by $\widetilde R_{\SS}(u) =(\widetilde X_{\SS}(u), \widetilde Y_{\SS}(u))$ for $u