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\begin{document}

\begin{slide}
\heading{Bifurcations of the Forced van der Pol Equation}

\begin{center}
John Guckenheimer \\
Cornell University\\
\end{center}
\vspace{0.4in}
\begin{center}
Joint work with \\
Kathleen Hoffmann and Warren Weckesser\\
\end{center}
\end{slide}

\begin{slide}
\heading{Forced van der Pol Equation}
\begin{eqnarray*}
   \eps \dot{x} & = & y + x - \frac{x^3}{3}  \\
   \dot{y} & = & - x + a \sin(2\pi \theta) \\
   \dot{\theta} & = & \omega
\end{eqnarray*}
Classical model in dynamical systems theory: first example of ``chaos''
\begin{itemize}
\item
Analysis by Cartwright and Littlewood, Grasman, Takens, ...
\item
Emphasis upon stable subharmonic orbits of different periods
\item
Levinson's piecewise linear modification the precursor to Smale's horseshoe
\end{itemize}
{\bf No} published calculations for $\eps < 0.01 ?$
\end{slide}

\begin{slide}
\begin{center}
\includegraphics[height=3in]{fvdp3.ps}
\end{center}
\end{slide}
 
\begin{slide}
\heading{Bifurcation with Multiple Time Scales}
Dynamical systems theory examines generic (persistent) phenomena
\begin{itemize}
\item
Special structure (multiple time scales) changes meaning of generic
\item
Forced van der Pol system provides case study for bifurcations
of relaxation oscillations
\item
Limited understanding of bifurcation in slow-fast systems
\item
Numerical methods for multiple time scales are problematic
\end{itemize}
\end{slide}

\begin{slide}
\heading{Slow-fast Systems}
\begin{eqnarray*}
\eps \dot{x} & = & f(x,y)  \qquad x \in R^m\\
\dot{y} & = & g(x,y) \qquad y \in R^n
\end{eqnarray*}
Two-time scales
\begin{itemize}
\item
Limit $\epsilon = 0$ is {\bf differential algebraic equation} 
\end{itemize}
Time rescaling produces {\bf slowly varying} system
\begin{eqnarray*}
x' & = & f(x,y)  \qquad x \in R^m\\
y' & = & \eps g(x,y) \qquad y \in R^n
\end{eqnarray*}
\begin{itemize}
\item
For fixed $y$, flow in $x$ is {\bf fast subsystem}
%\begin{eqnarray*}
%\eps \dot{x} & = & y + x - x^3/3 \\
%\dot{y} & = & - x \\
%\end{eqnarray*}
\end{itemize}
\end{slide}

\begin{slide}
\begin{center}
{\large \bf Van der Pol Cycle}
\end{center}
\begin{center}
\includegraphics[height=2in]{vdp_cycle.eps}
\end{center}
Segments of the cubic characteristic and horizontal segments form limit of
periodic orbits as $\epsilon \rightarrow 0$
\end{slide}

\begin{slide}
\heading{Terminology}
Standing assumption: limit sets of fast subsystems are equilibria
\begin{itemize}
\item
{\bf Critical manifold}: set of equilibria from fast subsystems 
\item
{\bf Slow manifold}: invariant manifold on which flow has speed $O(\eps)$
\item
{\bf Slow flow}: flow on critical manifold derived by rescaling time 
and eliminating fast variables
\item
{\bf Fold}: singularities of the projection of the critical manifold 
onto the slow variables
\item
{\bf Junctions}: where slow and fast segments of a trajectory meet
\item
{\bf Relaxation oscillation}: periodic orbit with slow and fast segments
\end{itemize}
\end{slide}

\begin{slide}
\heading{Slow-fast Flow}
\begin{center}
\includegraphics[height=3in,angle=270]{ttfig4c.ps}
\end{center}
\end{slide}
 
\begin{slide}
\heading{Forced van der Pol Slow Flow}
On the critical manifold
\begin{eqnarray*}
   \dot{y} - (x^2 -1) \dot{x} & = &0
\end{eqnarray*}
\begin{itemize}
\item
Rescale by $h(x,y,\theta) = x^2 - 1$
\item
Eliminate $y$ from rescaled slow equations
\end{itemize}
\begin{eqnarray*}
   \theta' & = & \omega (x^2 - 1) \\
   x' & = & - x + a \sin(2\pi \theta) 
\end{eqnarray*}
\begin{itemize}
\item
Jumps from fold curves $x = \pm 1$ to $x = \mp 2$
\item
Symmetry: $x \rightarrow -x$, $\theta \rightarrow \theta + 0.5$
\end{itemize}
\end{slide}

\begin{slide}
\begin{center}
\includegraphics[height=2in]{ficnf.ps} \\
Flow past generic fold
\end{center}
\end{slide}

\begin{slide}
\heading{The Singular Limit: Classical Theory}
Relate slow flow and fast trajectories to trajectories of full flow?
\begin{itemize}
\item
Theorem: On regular sheets of critical manifold, slow flow trajectories are
singular limits of trajectories of full system
\item
Theorem: Flow along jumps of regular folds bounding stable slow manifolds
are limits of trajectories of full system
\item
{\bf Folded singularities}: singular points of slow flow on fold curves
of critical manifold
\item
{\bf Canards}: trajectories that flow along unstable sheets of critical 
manifold form folded singularities
\end{itemize}
\end{slide}

\begin{slide}
\begin{center}
\includegraphics[height=2in]{vdps_5.ps} \\
Forced van der Pol slow flow: $a = 1.5 \; \omega = 1$
\end{center}
\end{slide}

\begin{slide}
\heading{Slow Flow Geometry}
Folded singularities if $a > 1$: $x = \pm 1$, $\sin(2\pi \theta) = \pm 1/a$
\begin{itemize}
\item
Folded saddles and nodes 
($1< a < \sqrt{1+ 1/(256 \pi^2 \omega^2)}$) or foci 
($a > \sqrt{1+ 1/(256 \pi^2 \omega^2)})$
\item
Stable and unstable manifolds of folded saddles yield discontinuities of 
return map 
\item
Asymptotics of return map differ on two sides of discontinuities: zero and infinite slopes
\end{itemize}
{\bf Tin} (tangency inflow) points if $a > 2$: $x = \pm 2$, $\sin(2\pi \theta) = \pm 2/a$
\begin{itemize}
\item
Tin points become quadratic turning points in return maps
\end{itemize}
\end{slide}

\begin{slide}
\begin{center}
\includegraphics[height=2in]{vdps_7.ps} \\
Forced van der Pol slow flow: $a = 8 \; \omega = 1$
\end{center}
\end{slide}
 
\begin{slide}
\begin{center}
\includegraphics[height=2in]{fvdp3d_a20_o5.00.ps} \\
Forced van der Pol 3d flow: $a = 20 \; \omega = 5$
\end{center}
\end{slide}

\begin{slide}
\heading{The Half Return Map $H$}
$H$: flow from $x = 2$ to $x = 1$, jump and apply symmetry
\begin{itemize}
\item
Fixed points give symmetric periodic orbits with only two jumps
\item 
Three parameter regimes
\begin{itemize}
\item
$a<1$: circle diffeomorphisms - invariant tori
\item
$1<a<2$: discontinuous monotone maps of circle into arc
\item
$2<a$: discontinuous maps of circle into arc with turning points
\end{itemize}
\item
Conjecture: at most $3$ fixed points
\end{itemize}
\end{slide}

\begin{slide}
\begin{center}
\includegraphics[height=2in]{vbif2.eps} \\
Forced van der Pol slow flow: critical trajectories
\end{center}
\end{slide}
 
\begin{slide}
\begin{center}
\includegraphics[height=2in]{ha_0.5_1.eps} \\
Forced van der Pol half-return map: $a = 0.5 \; \omega = 1$
\end{center}
\end{slide}

\begin{slide}
\begin{center}
\includegraphics[height=2in]{ha_1.5_1.eps} \\
Forced van der Pol half-return map: $a = 1.5 \; \omega = 1$
\end{center}
\end{slide}

\begin{slide}
\begin{center}
\includegraphics[height=2in]{ha20o5.eps} \\
Forced van der Pol half-return map: $a = 20 \; \omega = 5$
\end{center}
\end{slide}

\begin{slide}
\heading{Return Map Bifurcations}
Repetitive families indexed by {\bf circuit number}
\begin{itemize}
\item
Homoclinic bifurcations at points of discontinuity for return map
\begin{itemize}
\item
Left: saddle jumps to first intersection of stable manifold with $x=2$ 
\item
Right: $p_{1u}$ jumps to intersection of stable manifold with $x=2$
\end{itemize}
\item
Saddle-node bifurcations of return map
\begin{itemize}
\item
Types max and min: local maximum/minimum of $H(x) - x$
\end{itemize}
\item
Nodal homoclinic bifurcations where  node jumps to intersection of its strong 
stable manifold with $x=2$
\item
Heteroclinic bifurcation where  node jumps to stable manifold of saddle
\end{itemize}
\end{slide}
 
\begin{slide}
\begin{center}
\includegraphics[height=2in]{bd1.eps} \\
Bifurcation diagram of forced van der Pol slow flow
\end{center}
\end{slide}

\begin{slide}
\begin{center}
\includegraphics[height=2in]{bd_zoom1.eps} \\
Bifurcation diagram of forced van der Pol slow flow: a cusp of saddle-nodes
\end{center}
\end{slide}

\begin{slide}
\begin{center}
\includegraphics[height=2in]{bd_zoom2.eps} \\
Bifurcation diagram of forced van der Pol slow flow: near a is one
\end{center}
\end{slide}

\begin{slide}
\begin{center}
\includegraphics[height=2in]{h1_hmap.eps} \\
Forced van der Pol half-return map at left homoclinic bifurcation: 
\end{center}
\end{slide}
 
\begin{slide}
\begin{center}
\includegraphics[height=2in]{h2_1sf.eps} \\
Forced van der Pol half-return map at right homoclinic type bifurcation
of type 1
\end{center}
\end{slide}
 
\begin{slide}
\begin{center}
\includegraphics[height=2in]{h2_3sf.eps} \\
Forced van der Pol half-return map at right homoclinic bifurcation of 
type 3
\end{center}
\end{slide}
 
\begin{slide}
\begin{center}
\includegraphics[height=2in]{h2_2sf.eps} \\
Forced van der Pol half-return map at right homoclinic bifurcation of
type 2
\end{center}
\end{slide}
 
\begin{slide}
\begin{center}
\includegraphics[height=2in]{sn_max_hmap.eps} \\
Forced van der Pol slow flow near max saddle-node
\end{center}
\end{slide}

\begin{slide}
\begin{center}
\includegraphics[height=2in]{sn_min_hmap.eps} \\
Forced van der Pol slow flow near min saddle-node
\end{center}
\end{slide}

\begin{slide}
\heading{Return Map Codimension Two Bifurcations: Fixed Points}
From the bifurcation diagram 
\begin{itemize}
\item
Saddle-node equilibrium with homoclinic trajectory
\begin{itemize}
\item
$ a =1$
\item
Weak stable orbits jump before reaching saddle-node point
\end{itemize}
\item
Cusp: weakly unstable 
\item
Transversal crossings of codimension one bifurcations
\begin{itemize}
\item
saddle-nodes with left homoclinics 
\item
left and right homoclinics  
\end{itemize}
\end{itemize}
\end{slide}

\begin{slide}
\begin{center}
\includegraphics[height=2in]{vdps_4.ps} \\
Forced van der Pol flow near saddle-node point
\end{center}
\end{slide}
 
\begin{slide}
\begin{center}
\includegraphics[height=2in]{hslope1_6.eps} \\
Forced van der Pol half-return map at saddle-node \\
$a =  1 \; \omega = 1.6$
\end{center}
\end{slide}
 
\begin{slide}
\begin{center}
\includegraphics[height=2in]{hslope1.00005_f.eps} \\
Forced van der Pol half-return map near folded saddle: 
$a =  1.00005 \; \omega = 1.6$
\end{center}
\end{slide}
 
\begin{slide}
\heading{Bifurcations of Period 2 Orbits}
Add period 2 bifurcations to bifurcation diagram
\begin{itemize}
\item
Many similarities with fixed point bifurcations
\item
New types of codimension 2 homoclinic bifurcations
\item
Interaction with codimension 2 bifurcation of fixed points: transversal 
crossing of right and left homoclinic curves
\item
Regions with chaotic invariant set
\end{itemize}
\end{slide}
 
\begin{slide}
\begin{center}
\includegraphics[height=2in]{bd2.eps} \\
Bifurcation diagram with period 2 oribts
\end{center}
\end{slide}

\begin{slide}
\begin{center}
\includegraphics[height=2in]{bddet2.eps} \\
Bifurcation diagram: period 2 orbit details
\end{center}
\end{slide}

\begin{slide}
\begin{center}
\includegraphics[height=2in]{p2_214.eps} \\
Half return map and second iterate along homoclinic curve
\end{center}
\end{slide}

\begin{slide}
\begin{center}
\includegraphics[height=2in]{p2_263.eps} \\
Half return map and second iterate along homoclinic curve
\end{center}
\end{slide}

\begin{slide}
\begin{center}
\includegraphics[height=2in]{p2_266.eps} \\
Half return map and second iterate along homoclinic curve
\end{center}
\end{slide}

\begin{slide}
\begin{center}
\includegraphics[height=2in]{p2_329.eps} \\
Half return map and second iterate at intersection of homoclinic curves
\end{center}
\end{slide}

\begin{slide}
\begin{center}
\includegraphics[height=2in]{p2_320.eps} \\
Half return map and second iterate with chaos
\end{center}
\end{slide}

\begin{slide}
\begin{center}
\includegraphics[height=2in]{vchaos1.eps} \\
Chaotic return map in small interval
\end{center}
\end{slide}

\begin{slide}
\heading{Bifurcations of Forced van der Pol}
From the singular limit back to the slow-fast system
\begin{itemize}
\item
Uniformization and asymptotic analysis of folded singularities
\item
Canards are trajectories that flow onto unstable sheet of critical manifold
\item
Benoit analysis of folded saddles
\item
Canards at folded nodes have not been characterized
\begin{eqnarray*}
\eps \dot{x} & = & y - x^2 \\
\dot{y} & = & a z + b x \\
\dot{z} & = & 1
\end{eqnarray*}
\end{itemize}
\end{slide}
 
\begin{slide}
\begin{center}
\includegraphics[height=2in]{fnode_1.ps} \\
Folded node trajectories: x vs. z
\end{center}
\end{slide}
 
\begin{slide}
\begin{center}
\includegraphics[height=2in]{fnode_2.ps}  \\
Folded node trajectories: x vs. y
\end{center}
\end{slide}
 
\begin{slide}
\heading{Canards in the Forced van der Pol System}
Canards beginning at folded saddles
\begin{itemize}
\item
Follow (un)stable manifold of folded saddle to a jump point
\item
Two families of jumps parallel to $x$ axis : up and down
\item
Multiple canards: jump lands on stable manifold of a folded saddle
\item 
Construct canard return map with ``horseshoe'':
two stable periodic orbits and hyperbolic solenoid
\item
Circuit numbers of rectangles in construction differ
\end{itemize}
\end{slide}

\begin{slide}
\begin{center}
\includegraphics[height=2in]{vcanard2.eps} \\
Canard location in forced van der Pol: $a = 4 \; \omega = 1.55$ \\
\textcolor{red}{Red:} stable manifolds \\  
\textcolor{magenta}{Magenta:} unstable manifold (extended past jump)   \\
\textcolor{green}{Green:} trajectories of tin points \\  
\textcolor{blue}{Blue:} canard and projections
\end{center}
\end{slide}
 
\begin{slide}
\begin{center}
\includegraphics[height=2in]{vcanard3.eps} \\
Return to $x = 1$ from canards in forced van der Pol: $a = 4 \; \omega = 1.55$ \\
\textcolor{blue}{Blue:} first return from jumps up  \\ 
\textcolor{red}{Red:} second return from jumps up \\
\textcolor{green}{Green:} first return from jumps down
\end{center}
\end{slide}
 
\begin{slide}
\heading{Generic Relaxation Oscillations}
Starting point: slow-fast decompositions
\begin{itemize}
\item
Classification of generic folds: uniformization
\item
Focus upon stable slow manifolds and ones with a single unstable direction
\item
Determine junctions of generic trajectories in generic systems
\end{itemize}
\end{slide}

\begin{slide}
\heading{Degenerate Decompositions: Codimension 1}
Degeneracies encountered in generic one parameter families of relaxation 
oscillations beginning at stable orbit
\begin{itemize}
\item
in-fold $\rightarrow$ out-fold
\item
folded saddle
\item
saddle-initiated canards
\item
Hopf bifurcation at folds
\item
initiation at cusp
\item
in-fold $\rightarrow$ in-fold
\item
jump maps an in-fold non-transversally to slow flow on sheet of slow manifold
at the end of the jump
\end{itemize}
\end{slide}

\begin{slide}
\heading{Geometric Models \& Asymptotic Analysis}
Case of in-fold $\rightarrow$ out-fold
\begin{itemize}
\item
Family of relaxation oscillations in which junction from 
fast segment to slow segment reaches an out-fold
\item
Return map is composition of transitions along slow and fast segments and
across jumps
\item 
Model system by normal forms in each region
\item
Special attention to transition past degeneracy on fold 1
\item 
Canard formation at degeneracy 
\item
Several cases: chaos is possible
\item
Bifurcations occur as canard formation begins 
\end{itemize}
\end{slide}

\begin{slide}
\begin{center}
\includegraphics[height=2in]{canard_io.ps} \\
Degenerate decomposition: in-fold $\rightarrow$ out-fold
\end{center}
\end{slide}

\begin{slide}
\heading{Bifurcations of Relaxation Oscillations}
Analysis of bifurcations subsidiary to degenerate decompositions 
\begin{itemize}
\item
Examine each type of degenerate bifurcation
\item
Study uniformizations and asymptotic expansions at degeneracies
\item
Build models for relaxation oscillations encountering degeneracies
\item
Maximal and multiple canards lead to hierarchy of exponentials
\item
Develop algorithms based upon asymptotic analysis
\end{itemize}
Initial focus upon classical example: forced van der Pol equation
\end{slide}

\end{document}