Summary.

AUTO can do a limited bifurcation analysis of algebraic systems

$$f( u , p ) = 0 , \qquad f(\cdot,\cdot) , u \in {\rm R}^n.$$ (2.1)

The main algorithms in AUTO, however, are aimed at the continuation of solutions of systems of ordinary differential equation (ODEs) of the form

$$u'(t) = f\bigl( u(t) , p \bigr) , \qquad f(\cdot,\cdot) , u(\cdot) \in {\rm R}^n,$$ (2.2)

subject to boundary (including initial) conditions and integral constraints. Above, $p$ denotes one or more free parameters,

These boundary value algorithms also allow AUTO to do certain stationary solution and wave calculations for the partial differential equation (PDE)

$$u_t = D u_{xx} + f( u , p ), \qquad f(\cdot,\cdot) , u(\cdot) \in {\rm R}^n,$$ (2.3)

where $D$ denotes a diagonal matrix of diffusion constants.

The basic algorithms used in AUTO, as well as related algorithms, can be found in HBK:77 HBK:77, HBK:86 HBK:86, DoKeKe:91a DoKeKe:91a, DoKeKe:91b DoKeKe:91b.

Below, the basic capabilities of AUTO are specified in more detail. Some representative demos are also indicated.