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AUTO 97 :
CONTINUATION AND BIFURCATION SOFTWARE
FOR ORDINARY DIFFERENTIAL EQUATIONS
(with HomCont)

Eusebius J. Doedel
Concordia University
Montreal, Canada

Alan R. Champneys
University of Bristol
United Kingdom

Thomas F. Fairgrieve
Ryerson Polytechnic University
Toronto, Canada

Yuri A. Kuznetsov
CWI, Amsterdam
The Netherlands

Björn Sandstede
Weierstraß-Institut
Berlin, Germany

Xianjun Wang
Concordia University
Montreal, Canada

March 29 1998

• Contents
• Installing AUTO.
  • Installation.
  • Restrictions on Problem Size.
  • Compatibility with Older Versions.
• Overview of Capabilities.
  • Summary.
  • Algebraic Systems.
  • Ordinary Differential Equations.
  • Parabolic PDEs.
  • Discretization.
• How to Run AUTO.
  • User-Supplied Files.
    • The equations-file xxx.f
    • The constants-file r.xxx
  • User-Supplied Subroutines.
  • Arguments of STPNT.
  • User-Supplied Derivatives.
  • Running AUTO using Command Mode.
    • Basic commands.
    • Plotting commands.
    • File-manipulation.
    • Diagnostics.
    • File-editing.
    • File-maintenance.
    • HomCont commands.
    • Copying a demo.
    • Pendula animation.
    • Viewing the manual.
  • Output Files.
• The Graphics Program PLAUT.
  • Basic PLAUT-Commands.
  • Default Options.
  • Other PLAUT-Commands.
  • Printing PLAUT Files.
• Graphical User Interface.
  • General Overview.
    • The Menu bar.
    • The Define-Constants-buttons.
    • The Load-Constants-buttons.
    • The Stop- and Exit-buttons.
  • The Menu Bar.
    • Equations-button.
    • Edit-button.
    • Write-button.
    • Define-button.
    • Run-button.
    • Save-button.
    • Append-button.
    • Plot-button.
    • Files-button.
    • Demos-button.
    • Misc.-button.
    • Help-button.
  • Using the GUI.
  • Customizing the GUI.
    • Print-button.
    • GUI colors.
    • On-line help.
• Description of AUTO-Constants.
  • The AUTO-Constants File.
  • Problem Constants.
    • NDIM
    • NBC
    • NINT
    • JAC
  • Discretization Constants.
    • NTST
    • NCOL
    • IAD
  • Tolerances.
    • EPSL
    • EPSU
    • EPSS
    • ITMX
    • NWTN
    • ITNW
  • Continuation Step Size.
    • DS
    • DSMIN
    • DSMAX
    • IADS
    • NTHL
    • NTHU
  • Diagram Limits.
    • NMX
    • RL0
    • RL1
    • A0
    • A1
  • Free Parameters.
    • NICP, ICP
    • Fixed points.
    • Periodic solutions and rotations.
    • Folds and Hopf bifurcations.
    • Folds and period-doublings.
    • Boundary value problems.
    • Boundary value folds.
    • Optimization problems.
    • Internal free parameters.
    • Parameter overspecification.
  • Computation Constants.
    • ILP
    • ISP
    • ISW
    • MXBF
    • IRS
    • IPS
  • Output Control.
    • NPR
    • IID
    • IPLT
    • NUZR
• Notes on Using AUTO.
  • Restrictions on the Use of PAR.
  • Efficiency.
  • Correctness of Results.
  • Bifurcation Points and Folds.
  • Floquet Multipliers.
  • Memory Requirements.
• AUTO Demos : Tutorial.
  • Introduction.
  • ab : A Tutorial Demo.
  • Copying the Demo Files.
  • Executing all Runs Automatically.
  • Executing Selected Runs Automatically.
  • Using AUTO-Commands.
  • Plotting the Results with PLAUT.
  • Following Folds and Hopf Bifurcations.
  • Relabeling Solutions in the Data-Files.
  • Plotting the 2-Parameter Diagram.
  • Converting Saved PLAUT Files to PostScript.
  • Using the GUI.
  • Abbreviated AUTO-Commands.
• AUTO Demos : Fixed points.
  • enz : Stationary Solutions of an Enzyme Model.
  • dd2 : Fixed Points of a Discrete Dynamical System.
• AUTO Demos : Periodic solutions.
  • lrz : The Lorenz Equations.
  • abc : The A B C Reaction.
  • pp2 : A 2D Predator-Prey Model.
  • lor : Starting an Orbit from Numerical Data.
  • frc : A Periodically Forced System.
  • ppp : Continuation of Hopf Bifurcations.
  • plp : Fold Continuation for Periodic Solutions.
  • pp3 : Period-Doubling Continuation.
  • tor : Detection of Torus Bifurcations.
  • pen : Rotations of Coupled Pendula.
  • chu : A Non-Smooth System (Chua's Circuit).
  • phs : Effect of the Phase Condition.
  • ivp : Time Integration with Euler's Method.
• AUTO Demos : BVP.
  • exp : Bratu's Equation.
  • int : Boundary and Integral Constraints.
  • bvp : A Nonlinear ODE Eigenvalue Problem.
  • lin : A Linear ODE Eigenvalue Problem.
  • non : A Non-Autonomous BVP.
  • kar : The Von Karman Swirling Flows.
  • spb : A Singularly-Perturbed BVP.
  • ezp : Complex Bifurcation in a BVP.
• AUTO Demos : Parabolic PDEs.
  • pd1 : Stationary States (1D Problem).
  • pd2 : Stationary States (2D Problem).
  • wav : Periodic Waves.
  • brc : Chebyshev Collocation in Space.
  • brf : Finite Differences in Space.
  • bru : Euler Time Integration (the Brusselator).
• AUTO Demos : Optimization.
  • opt : A Model Algebraic Optimization Problem.
  • ops : Optimization of Periodic Solutions.
  • obv : Optimization for a BVP.
• AUTO Demos : Connecting orbits.
  • fsh : A Saddle-Node Connection.
  • nag : A Saddle-Saddle Connection.
  • stw : Continuation of Sharp Traveling Waves.
• AUTO Demos : Miscellaneous.
  • pvl : Use of the Subroutine PVLS.
  • ext : Spurious Solutions to BVP.
  • tim : A Test Problem for Timing AUTO.
• HomCont.
  • Introduction.
  • HomCont Files and Subroutines.
  • HomCont-Constants.
    • NUNSTAB
    • NSTAB
    • IEQUIB
    • ITWIST
    • ISTART
    • NREV, IREV
    • NFIXED, IFIXED
    • NPSI, IPSI
  • Restrictions on HomCont Constants.
  • Restrictions on the Use of PAR.
  • Test Functions.
  • Starting Strategies.
  • Notes on Running HomCont Demos.
• HomCont Demo : san.
  • Sandstede's Model.
  • Inclination Flip.
  • Non-orientable Resonant Eigenvalues.
  • Orbit Flip.
  • Detailed AUTO-Commands.
• HomCont Demo : mtn.
  • A Predator-Prey Model with Immigration.
  • Continuation of Central Saddle-Node Homoclinics.
  • Switching between Saddle-Node and Saddle Homoclinic Orbits.
  • Three-Parameter Continuation.
  • Detailed AUTO-Commands.
• HomCont Demo : kpr.
  • Koper's Extended Van der Pol Model.
  • The Primary Branch of Homoclinics.
  • More Accuracy and Saddle-Node Homoclinic Orbits.
  • Three-Parameter Continuation.
  • Detailed AUTO-Commands.
• HomCont Demo : cir.
  • Electronic Circuit of Freire et al.
  • Detailed AUTO-Commands.
• HomCont Demo : she.
  • A Heteroclinic Example.
  • Detailed AUTO-Commands.
• HomCont Demo : rev.
  • A Reversible System.
  • An -Reversible Homoclinic Solution.
  • An -Reversible Homoclinic Solution.
  • Detailed AUTO-Commands.
• References
• About this document ...