IPS

This constant defines the problem type :

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IPS=0 : An algebraic bifurcation problem. Hopf bifurcations will not be detected and stability properties will not be indicated in the fort.7 output-file.

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IPS=1 : Stationary solutions of ODEs with detection of Hopf bifurcations. The sign of PT, the point number, in fort.7 is used to indicate stability : $-$ is stable , + is unstable.
(Demo ab.)

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IPS=$-$1 : Fixed points of the discrete dynamical system $u^{(k+1)}=f(u^{(k)},p ),$ with detection of Hopf bifurcations. The sign of PT in fort.7 indicates stability : $-$ is stable , + is unstable. (Demo dd2.)

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IPS=$-$2 : Time integration using implicit Euler. The AUTO-constants DS, DSMIN, DSMAX, and ITNW, NWTN control the stepsize. In fact, pseudo-arclength is used for ``continuation in time''. Note that the time discretization is only first order accurate, so that results should be carefully interpreted. Indeed, this option has been included primarily for the detection of stationary solutions, which can then be entered in the user-supplied routine STPNT.
(Demo ivp.)

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IPS=2 : Computation of periodic solutions. Starting data can be a Hopf bifurcation point (Run 2 of demo ab), a periodic orbit from a previous run (Run 4 of demo pp2), an analytically known periodic orbit (Run 1 of demo frc), or a numerically known periodic orbit (Demo lor). The sign of PT in fort.7 is used to indicate stability : $-$ is stable , + is unstable or unknown.

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IPS=4 : A boundary value problem. Boundary conditions must be specified in the user-supplied routine BCND and integral constraints in ICND. The AUTO-constants NBC and NINT must be given correct values. (Demos exp, int, kar.)

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IPS=5 : Algebraic optimization problems. The objective function must be specified in the user-supplied routine FOPT. (Demo opt.)

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IPS=7 : A boundary value problem with computation of Floquet multipliers. This is a very special option; for most boundary value problems one should use IPS=4. Boundary conditions must be specified in the user-supplied routine BCND and integral constraints in ICND. The AUTO-constants NBC and NINT must be given correct values.

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IPS=9 : This option is used in connection with the HomCont algorithms described in Chapters 20-26 for the detection and continuation of homoclinic bifurcations.
(Demos san, mtn, kpr, cir, she, rev.)

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IPS=11 : Spatially uniform solutions of a system of parabolic PDEs, with detection of traveling wave bifurcations. The user need only define the nonlinearity (in routine FUNC), initialize the wave speed in PAR(10), initialize the diffusion constants in PAR(15,16,$\cdots$), and set a free equation parameter in ICP(1). (Run 2 of demo wav.)

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IPS=12 : Continuation of traveling wave solutions to a system of parabolic PDEs. Starting data can be a Hopf bifurcation point from a previous run with IPS=11, or a traveling wave from a previous run with IPS=12. (Run 3 and Run 4 of demo wav.)

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IPS=14 : Time evolution for a system of parabolic PDEs subject to periodic boundary conditions. Starting data may be solutions from a previous run with IPS=12 or 14. Starting data can also be specified in STPNT, in which case the wave length must be specified in PAR(11), and the diffusion constants in PAR(15,16,$\cdots$). AUTO uses PAR(14) for the time variable. DS, DSMIN, and DSMAX govern the pseudo-arclength continuation in the space-time variables. Note that the time discretization is only first order accurate, so that results should be carefully interpreted. Indeed, this option is mainly intended for the detection of stationary waves. (Run 5 of demo wav.)

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IPS=15 : Optimization of periodic solutions. The integrand of the objective functional must be specified in the user supplied routine FOPT. Only PAR(1-9) should be used for problem parameters. PAR(10) is the value of the objective functional, PAR(11) the period, PAR(12) the norm of the adjoint variables, PAR(14) and PAR(15) are internal optimality variables. PAR(21-29) and PAR(31) are used to monitor the optimality functionals associated with the problem parameters and the period. Computations can be started at a solution computed with IPS=2 or IPS=15. For a detailed example see demo ops.

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IPS=16 : This option is similar to IPS=14, except that the user supplies the boundary conditions. Thus this option can be used for time-integration of parabolic systems subject to user-defined boundary conditions. For examples see the first runs of demos pd1, pd2, and bru. Note that the space-derivatives of the initial conditions must also be supplied in the user supplied routine STPNT. The initial conditions must satisfy the boundary conditions. This option is mainly intended for the detecting stationary solutions.

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IPS=17 : This option can be used to continue stationary solutions of parabolic systems obtained from an evolution run with IPS=16. For examples see the second runs of demos pd1 and pd2.